A Combined Monte Carlo and Group Theoretical Treatment

Research Collection

Doctoral Thesis

Continuous dipolar moments on regular lattices: a combined Monte Carlo and group theoretical treatment

Author(s): Schildknecht, Dominik

Publication Date: 2019

Permanent Link: https://doi.org/10.3929/ethz-b-000360043

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ETH Library diss. eth no. 26231

CONTINUOUSDIPOLARMOMENTSON REGULARLATTICES:ACOMBINEDMONTE CARLOANDGROUPTHEORETICAL TREATMENT

A dissertation submitted to attain the degree of doctor of sciences of eth zurich (Dr. sc. ETH Zurich)

presented by dominik schildknecht MSc ETH, ETH Zurich born on 23 September 1992 citizen of Waldkirch SG, Switzerland

accepted on the recommendation of Prof. Dr. Laura Jane Heyderman Prof. Dr. Nicola Ann Spaldin Prof. Dr. Frédéric Mila Dr. Peter Michael Derlet

I hereby declare that the following submitted thesis is original work which I alone have authored and which is written in my own words.

Title: Continuous dipolar moments on regular lattices: a combined Monte Carlo and group theoretical treatment Author: Dominik Schildknecht Supervisor: Prof. Dr. Laura J. Heyderman

With my signature I declare that I have been informed regarding normal academic citation rules and that I have read and understood the information on “Citation etiquette”. The citation conventions usual to the discipline in question here have been respected. Furthermore, I declare that I have truthfully documented all methods, data, and operational procedures and not manipulated any data. All per- sons who have substantially supported me in my work are identiﬁed in the acknowledgements. The above work may be tested electronically for plagiarism.

Zürich, July 22, 2019

iii

ABSTRACT

The leading order long-range interaction in magnetic systems is the dipolar interaction. Because of this, it strongly affects the properties of magnetic materials and the interaction between them. In the last decade, the understanding of the phenomena resulting from dipolar interactions could be advanced experimentally due to the ability to pattern arrays of nanomagnets into so-called artificial spin systems. The models describing such systems exhibit interesting physics such as continuous ground-state degeneracies, order-by-disorder transitions, or peculiar universality classes. Gaining a better theoretical understanding of these systems is the main objective of this thesis. The first question tackled by this thesis concerns the universality class of the finite-temperature phase transition of the dipolar-coupled XY spin model on the square lattice. The universality class is still under debate as it has been suggested in some papers that this model belongs to the two-dimensional Ising universality whilst in others it was claimed that the transition belongs to the universality class of the XY-model with a four-fold anisotropy. In previous literature, it was shown that the finite-size scaling method is insufficient to distinguish the two cases. Therefore, in this thesis, an alternative approach to determine critical exponents, using the Monte Carlo renormalization group method, is presented. Both methods were applied successfully to the two-dimensional Ising model, but only the finite- size scaling approach gave converged results for the dipolar-coupled XY model, whereas the Monte Carlo renormalization group simulations did not converge to any reasonable critical exponents. This latter lack of convergence suggests that the transition belongs to the marginal universality class of the XY-model with a four-fold anisotropy. However, there could be other reasons why the simulations did not converge so that this thesis cannot give a definitive determination of the universality class. Simultaneous to the theoretical efforts, experimental interest in artificial spin systems realizing the dipolar-coupled XY system has emerged. Such systems have recently been measured by the muon-spin rotation technique, which is a well-suited method for such experiments due to the sensitivity to local magnetic fields and their fluctuations. However, the interpretation of muon-spin rotation experiments is complex as it often requires a detailed model of the muon-spin precession. Here, such a model was ob- v tained by using the separation of the time scales between fluctuations of the muon spin and the individual nanomagnets as well as a mean-field calculation of the order parameter of the dipolar-coupled XY model. In this simple model, it was possible to correlate the depolarization of the muon spins with the emergence of an order parameter and therefore to long-range order. While many chapters of this thesis are concerned with the description of the effects of temperature, dipolar-coupled systems are also known to be sensitive to disorder. However, even for the well-studied dipolar-coupled XY model on the square lattice, the phase diagram for temperature and disorder has only been known qualitatively and not quantitatively. In this thesis, the quantitative phase diagram is derived by introducing proper order parameters and performing Monte Carlo simulations for two different types of disorder. The apparent similarity between the phase diagrams for the two types of disorder is argued to be a general feature of dipolar- coupled spin systems, originating from the competition of magnetic flux closure at local versus global length scales. While much of this thesis is concerned with analyzing the dipolar- coupled XY model on the square lattice, similar behavior has previously been observed for other geometries. Specifically, several dipolar-coupled spin systems are known where continuous ground-state degeneracies and order-by-disorder transitions arise. However, a fundamental reasoning for this behavior has not been developed so that, without explicit verification, it has been impossible to predict such features for new geometries. In this thesis, it was shown that, for a certain class of dipolar-coupled spin systems, the origin of these phenomena lies in the unusual symmetry of the Hamiltonian, specifically the point symmetry group of the underlying lattice. Therefore, in this thesis, a unifying theory of the ground-state degeneracy and the emerging order-by-disorder transition could successfully be provided for a large class of dipolar-coupled spin systems. In summary, the discoveries of this thesis provide a more profound insight into dipolar-coupled spin systems in general and specifically into the dipolar-coupled XY model on the square lattice. These results suggest new and exciting research directions, both in terms of theory and experiment, including the investigation of systems with interesting symmetries such as the triangular lattice or the kagome lattice, and the extension of the dipolar- coupled XY model phase diagram to other types of disorder. These ideas are applicable to artificial spin systems and more generally in condensed matter systems with predominantly dipolar interactions. vi ZUSAMMENFASSUNG

Die dominante langreichweitige Wechselwirkungen in magnetischen Sys- temen ist die Dipol-Wechselwirkung. Diese beeinflusst daher stark die Eigenschaften von magnetischen Materialien und die Wechselwirkung zwischen ihnen. Im letzten Jahrzehnt konnte das Verständnis der Phäno- mene, die sich aus dipolaren Wechselwirkungen ergeben, experimentell besser erforscht werden, da es möglich wurde Nanomagnete in sogenann- ten künstlichen Spinsystemen anzuordnen. Die theoretischen Modelle, die solche Systeme beschreiben, weisen ungewöhnliche Phänomene auf, wie zum Beispiel eine kontinuierliche Entartungen des Grundzustands, Ordnung-durch-Unordnungsübergänge oder ungewöhliche Universali- tätsklassen die das Skalenverhalten am Phasenübergang beschreiben. Ein besseres theoretisches Verständnis dieser Systeme ist daher das Hauptziel dieser Dissertation. Die erste in dieser Arbeit behandelte Frage betrifft die Universalitäts- klasse des Phasenübergangs im dipolar gekoppelten XY-Modell auf dem Quadratgitter bei endlicher Temperatur. Die Universalitätsklasse ist noch immer umstritten, da in einigen Arbeiten vermutet wurde, dass dieses Mo- dell dieselbe Universalitätsklasse wie das zweidimensionale Ising-Modell besitzt, während in anderen vermutet wurde, dass der Übergang besser beschrieben ist mit der Universalitätsklasse des XY-Modells mit einer vierfachen Anisotropie. In der bisherigen Literatur wurde gezeigt, dass die sogenannte “finite-size scaling” Methode ungenügend ist, um die beiden Fälle zu unterscheiden. In dieser Arbeit wird daher ein alternativer An- satz zur Bestimmung kritischer Exponenten unter Verwendung der Monte Carlo Renormalisierungsgruppe vorgestellt. Beide Methoden wurden er- folgreich auf das zweidimensionale Ising-Modell angewendet, aber nur die “finite-size scaling” Methode konvergierte für das dipolar gekoppelte XY-Modell, während die Monte Carlo Renormalisierungsgruppe nur unphysikalische kritische Exponenten lieferte. Dieser Mangel an Konver- genz deutet darauf hin, dass das Verhalten am Phaenübergang durch die marginalen Universalitätsklasse des XY-Modells mit einer vierfachen An- isotropie beschrieben wird. Es könnte jedoch noch andere Gründe geben, warum die Simulationen nicht konvergierten, so dass diese Arbeit keine endgültige Bestimmung der Universalitätklasse geben kann.

vii Gleichzeitig zu den theoretischen Bemühungen ist ein experimentelles Interesse an künstlichen Spinsystemen entstanden, die das dipolar gekoppelte XY-Modell realisieren. Solche Systeme wurden kürzlich mittels Myon- spinspektroskopieexperimenten gemessen, die aufgrund der Empﬁndlich- keit gegenüber lokalen Magnetfeldern und deren Fluktuationen für solche Experimente gut geeignet sind. Die Interpretation von solchen Experimen- ten ist jedoch komplex, da sie oft ein detailliertes Modell der Präzession des Myonspins erfordert. Hier wurde ein solches Modell entwickelt, mittels der Separation der Zeitskalen zwischen Fluktuationen des Myonspins und den einzelnen Nanomagneten, sowie einer Molekuarfeldberechnung des Orderparameters des dipolar gekoppelten XY-Modells. In diesem ein- fachen Modell war es möglich die Depolarisation der Myonspins mit dem Aufkommen eines Ordnungsparameters und damit mit der Langstrecken- ordnung in Verbindung zu bringen. Während sich viele Kapitel dieser Arbeit mit der Beschreibung der Aus- wirkungen von Temperatur befassen, ist ebenso bekannt, dass dipolar gekoppelte Systeme empﬁndlich auf Unordnung reagieren. Jedoch selbst für das gut untersuchte dipolar gekoppelte XY-Modell auf dem Quadratgit- ter ist das Phasendiagramm für Temperatur und Unordnung nur qualita- tiv und nicht quantitativ bekannt. Durch die Einführung geeigneter Ord- nungsparameter und durch Monte Carlo Simulationen wird in dieser Ar- beit das quantitative Phasendiamm für zwei verschiedene Arten von Un- ordnung hergeleitet. Es wird arumentiert, dass die Ähnlichkeit zwischen den Phasendiagrammen für die beiden Arten Unordnung ein allgemeines Merkmal dipolar gekoppelter Spinsysteme ist, die aus der Konkurrenz des Schliessens des magnetischen Flusses auf lokalen gegenüber globalen Län- genskalan resultiert. Während sich ein Grossteil dieser Arbeit mit der Analyse des dipolar gekoppelten XY-Modells auf dem Quadratgitter befasst, wurde zuvor ein ähnliches Verhalten für andere Geometrien beobachtet. Insbesonde- re sind mehrere dipolar gekoppelte Spinsysteme bekannt, bei denen eine kontinuierliche Entartungen des Grundzustands und Ordnung-durch- Unordnungsübergänge auftreten. Eine grundlegende Begründung für dieses Verhalten wurde jedoch nicht entwickelt, so dass es ohne explizite Überprüfung unmöglich war solche Merkmale für neue Geometrien vor- herzusagen. In dieser Arbeit wurde gezeigt, dass für eine bestimmte Klasse dipolar gekoppelter Spinsysteme der Ursprung dieser Phänomene in der ungewöhnlichen Symmetrie des Hamiltonians liegt, insbesondere in der Punktsymmetriegruppe des zugrundeliegenden Gitters. Daher konnte in viii dieser Dissertation eine vereinheitlichte Theorie der Entartung des Grund- zustands und des Ordnung-durch-Unordnungsübergangs für eine grosse Klasse dipolar gekoppelter Spinsysteme gegeben werden. Zusammenfassend lässt sich sagen, dass die in dieser Arbeit beschrieb- nenen Forschungsergebnisse einen tieferen Einblick in dipolar gekoppelte Spinsysteme im Allgemeinen und speziell in das dipolar gekoppelte XY-Modell auf dem Quadratgitter geben. Diese Ergebnisse lassen sowohl theoretisch als auch experimentell neue und aufregende Forschungs- richtungen erkennen, darunter die Untersuchung von Systemen mit interessanten Symmetrien wie dem Dreiecks- oder Kagomegitter, und die Betrachtung des dipolar gekoppelten XY-Modell Phasendiagramms auf andere Typen Unordnung. Diese Ideen sind sowohl auf künstliche Spinsysteme wie auch auf generelle Festkörpersysteme mit überwiegend dipolaren Wechselwirkungen anwendbar.

CONTENTS

1 introduction & motivation1 1.1 Motivation ...... 1 1.2 History ...... 2 1.2.1 Classical spin systems ...... 2 1.2.2 Classical dipolar-coupled spin systems ...... 3 1.3 Content of this thesis ...... 4 1.3.1 Scope ...... 4 1.3.2 Outline ...... 5 1.3.3 Contributions ...... 6

2 theory of classical spin systems9 2.1 Degrees of Freedom ...... 9 2.2 Hamiltonians ...... 10 2.2.1 Zeeman Hamiltonian ...... 11 2.2.2 Anisotropy Hamiltonians ...... 11 2.2.3 Heisenberg Hamiltonian ...... 12 2.2.4 Dipolar Hamiltonian ...... 13 2.3 Long-range and short-range interactions ...... 14 2.4 Frustration ...... 16 2.5 Thermodynamics ...... 17 2.5.1 Thermodynamic potential ...... 19 2.5.2 Partition Function ...... 20 2.5.3 Thermal average of an observable ...... 20 2.6 Phase transitions and critical exponents ...... 22

3 monte carlo methods 27 3.1 Mathematical approach ...... 27 3.2 Importance Sampling: Metropolis algorithm ...... 28 3.3 Parallel Tempering: an approach for highly frustrated systems 31 3.4 Monte Carlo simulations at criticality ...... 32 3.4.1 Finite-size scaling ...... 33 3.4.2 Monte Carlo renormalization group ...... 34 3.4.3 Determining Tc ...... 40 3.5 Temporal evolution of discrete degrees of freedom ...... 41

xi xii contents

4 applications of monte carlo simulations 43 4.1 Model nanomagnetic logic gates with kinetic Monte Carlo . . 43 4.1.1 Experiment on the nanomagnetic logic gates ...... 44 4.1.2 Theoretical model and its solution by kinetic Monte Carlo ...... 46 4.1.3 Conclusion & Outlook for the simulation of nanomagnetic logic gates ...... 49 4.2 Determination of Universality classes ...... 50 4.2.1 Ising-model in two dimensions ...... 50 4.2.2 Dipolar XY spins on the square lattice ...... 55 4.2.3 Conclusion of the Universality ...... 65

5 understanding the µ-spin rotation experiments 67 5.1 The µ-spin rotation as a local magnetic field probe ...... 67 5.1.1 General working principle of µ-spin rotation ...... 68 5.1.2 The µ-spin rotation experiments on Artificial Spin Ice 70 5.2 Experiment on dipolar XY spin systems ...... 70 5.3 Depolarization model of dipolar-coupled XY spin systems . . 75 5.3.1 Applicability of time averaging the magnetic field . . . 76 5.3.2 Mean-field description of dipolar XY spins on the square lattice ...... 77 5.3.3 Depolarization of the µ-spin due to the dXY Artificial Spin Ice ...... 79 5.4 Comparison of the model with the experimental results . . . 80 5.5 Conclusion & Outlook ...... 83

6 phase diagram of dipolar-coupled xy spins on the square lattice 85 6.1 Dipolar-coupled XY spins on disordered square lattices . . . . 86 6.2 Introducing proper order parameters ...... 88 6.3 Applicability of order parameters in disordered systems . . . 91 6.4 Non-disordered system & thermal Order-by-Disorder . . . . . 94 6.5 Temperature versus vacancy-density phase diagram ...... 96 6.5.1 Binder cumulant analysis & Phase diagram ...... 101 6.6 Temperature versus random-displacement phase diagram . . 107 6.7 Conclusion ...... 113

7 continuous ground-state degeneracy of classical dipoles on regular lattices 117 contents xiii

7.1 Simple example: dipolar-coupled XY spins on the square lattice ...... 118 7.2 Construction of the degenerate ground-states ...... 120 7.2.1 Using the translational invariance: the method of Luttinger and Tisza ...... 121 7.2.2 Continuous ground-state degeneracy as a result of the point symmetry group ...... 123 7.3 Application of our method ...... 125 7.3.1 Dipolar-coupled XY spins on the square lattice . . . . 125 7.3.2 Dipolar-coupled Heisenberg spins on the (distorted) cubic lattice ...... 129 7.3.3 Dipolar-coupled XY spins on the triangular lattice . . 134 7.4 Conclusion ...... 135

8 spin waves and order-by-disorder for classical dipoles on regular lattices 137 8.1 Spin waves in dipolar-coupled spin systems for simple lattices137 8.1.1 Spin waves for dXY spins on the chain lattice . . . . . 138 8.1.2 Spin waves for dXY spins on the square lattice . . . . . 144 8.2 Generalization of spin waves to other lattices ...... 151 8.2.1 Dispersion relation ...... 153 8.2.2 Free energy ...... 157 8.3 Application of the dipolar spin-wave theory ...... 161 8.3.1 Dipolar-coupled XY spins on the square lattice . . . . 161 8.3.2 Dipolar-coupled Heisenberg spins on the simple cubic lattice ...... 162 8.4 Conclusion ...... 167

9 conclusion & outlook 171 9.1 Universality class of dipolar-coupled XY spins on the square lattice ...... 172 9.2 µ-spin rotation experiments of dipolar-coupled XY spins . . . 173 9.3 Phase diagram of dipolar-coupled XY spins ...... 174 9.4 Symmetry analysis of dipolar-coupled spin systems ...... 175 9.5 Concluding Remarks ...... 177 a algorithms 179 a.1 Metropolis-Rosenbluth-Rosenbluth-Teller-Teller algorithm . . 179 a.2 Rejection-free kinetic Monte Carlo algorithm ...... 180 xiv contents

b mcrg interaction set 181

acknowledgments 183

bibliography 185

publications 203

notation 205 Glossary ...... 205 List of Abbreviations ...... 205 Frequently used Symbols ...... 206

list of figures 207

list of tables 216

curriculum vitæ 219 1 INTRODUCTION&MOTIVATION

1.1 motivation

Magnetism has excited humanity for millennia, providing us with one of the most fascinating natural forces. Early applications of magnetism included the compass, which enabled the traversal of the seas, and the discovery of unknown places. In modern times, the applications of magnets range from giant magnets in power generators and electric motors, down to tiny magnets used to store data in hard discs. The theoretical foundation for all these applications has an equally long history. Already Aristotle tried to understand magnetism, associating the magnetic force with some- thing mystical. Magnetism continued to remain a mystery and, it was only when Maxwell derived the field equations of electromagnetism [1], that a proper understanding of classical magnetic fields was achieved. Maxwell’s theory was incredibly successful in describing electromagnetic fields and served as an essential step in the development of special relativity by Einstein. However, the theory could not explain why some solids were magnetic. It was only later realized that magnetism in solids occurs as a quantum-mechanical effect [2, 3]. The first successful microscopic models of magnetism included intrinsic magnetic moments, i. e. “magnetic atoms”, coupled by a simple nearest-neighbor interaction [4, 5] — the exchange interaction predicted by a quantum mechanical treatment of interacting electrons [6–8]. Even today, most microscopic descriptions of magnetism rely on nearest-neighbor models, with some small adjustments such as next-nearest neighbor terms or Dzyaloshinskii-Moriya interactions. Nevertheless, Maxwell’s theory predicted that magnetic atoms are dipoles, and thus, there were early attempts to attribute the magnetism in solids to long-range order of microscopic dipole moments, which interacted magnetostatically. However, compared to the achievements of nearest-neighbor models in the description of magnetism in solids, the description of magnetism based on dipoles had only limited success as in most magnetic materials the nearest-neighbor exchange interaction is dominant over the dipolar interaction. Therefore, much of the research effort focused on nearest-neighbor models and dipolar systems soon

1 2 introduction & motivation

became neglected. Hence, our knowledge about dipolar systems is much less than our understanding of nearest-neighbor models. Nevertheless, the dipolar interaction is relevant, for example, to discuss the flux closure of the magnetic stray field of a finite-sized ferromagnet. In modern days, the study of dipolar interactions in spin systems became popular once again due to the importance of the dipolar interaction in the spin-ice pyrochlores [9]. These systems show many exciting features like an extensive ground-state degeneracy and hence a strong suppression of magnetic ordering. Moreover, the dipolar interaction is the dominant interaction in so-called Artificial Spin Ice (ASI). These systems were introduced initially with the desire for a deeper understanding of spin-ice physics. However, in contrast to the pyrochlores, ASI systems are not bound to a particular lattice geometry or limited in how a single moment anisotropy relates to this lattice. Hence, they provide an excellent playground to ex- plore exciting new physics. Inspired by recent advances in ASIs [10–12], the purpose of this thesis is to provide a better understanding of classical dipolar-coupled systems with continuous degrees of freedom. This objective will be achieved by various numerical and analytical methods, always with the purpose to gain a deeper insight into the fascinating physics that is displayed by classical dipolar spin systems.

1.2 history

1.2.1 Classical spin systems The ﬁrst successful microscopic theories of magnetism were not based on dipoles, but rather a simple nearest-neighbor model. In particular, Wilhelm Lenz formulated a model in order to explain ferromagnetism [4], which he gave to his student, Ernst Ising, who was able to compute the thermodynamic behavior exactly for the one-dimensional case [5]. Ising wrongly claimed that the model does not undergo a phase transition in any dimension, which he attributed to the oversimpliﬁed nearest-neighbor interaction. Peierls, however, proved him wrong by demonstrating the existence of a phase transition in higher dimensions [13]. Only shortly after their work, Heisenberg introduced more general models with continuous spins rather than discrete spins [14]. Eventually, more general O(n) models1 saw enhanced research interest [15, 16]. The nearest-

1 Here, O(n) model, also called the n-vector model, denotes a generalization of the nearest- neighbor models to general (classical) spins of dimension n. The O(1) model is the Ising 1.2 history3 neighbor interaction inherent to these models was shown to originate from the quantum treatment of interacting electrons [6–8, 17]. Hence, it was natural to extend the models to quantum systems, where operators rather than just vectors form the spin variables. This modiﬁcation facilitated an excellent description of experiments on spin-chains [18] (see Ref. [19] for a review on the theory) and fascinating new physics such as the exactly solvable Kitaev model [20].

1.2.2 Classical dipolar-coupled spin systems Even though the nearest-neighbor models have had significant success, the dipolar interaction cannot always be neglected. Indeed, early attempts to explain solid magnetism focused on dipolar interactions, as Maxwell already realized that there are no magnetic monopoles. Hence, the smallest building blocks of magnetism, i. e. “magnetic atoms”, had to consist of dipoles. Around the same time as Maxwell derived his field equations, experimental investigations of magnetic systems resulted in the discovery of the so-called adiabatic demagnetization [21], where certain systems cool down when a large magnetic field is removed. Hence, adiabatic demagnetization was later used in refrigeration down to temperatures below 100 mK [22– 24]. In the endeavor to find materials for even lower temperatures, many different paramagnetic salts were studied. This research led to the observation of a sharp increase of the specific heat in gadolinium sulfate [25], which was one of the first indications of the magnetic ordering of classical dipoles. In order to understand the phase transitions in the array of dipoles in the paramagnetic salts, several theoretical attempts were made, such as high- temperature expansions [26] or Lorentz-field methods [27]. However, it was not until the work of Luttinger and Tisza (LT), that the ground state of dipolar-coupled spin systems on regular lattices could be determined [28]. Belobrov et al. [29] realized that Luttinger and Tisza had obtained a continuous ground-state degeneracy, which is not protected by any global symmetry of the Hamiltonian. Since the degeneracy is not protected, infinitesimal perturbations are expected to lift the degeneracy. Indeed, Prakash and Henley found that disorder or thermal fluctuations induce a degeneracy breaking order-by-disorder transition [30]. This transition ensured that the

model, O(2) model is the XY model, and O(3) model is the Heisenberg model. The higher dimensional model are then enumerated by n. 4 introduction & motivation

system chooses configurations which do not have more symmetries than the Hamiltonian. Since then, the study of dipolar systems underwent a revival, as it was discovered that dipolar interactions could be used to model the spin-ice physics in pyrochlores [9]. Since the desire to understand the spin-ices per- sisted, both theoreticians and experimentalists searched for a way to go beyond the well-established spin-ice pyrochlores. This led to the experimental study of many other pyrochlores (see Ref. [31] for a review), to numerical investigations of the pyrochlores [32] and related geometries [33– 35], and to the development of Artificial Spin Ice (ASI). While the early ASI systems consisted of Ising-like moments on periodic lattices such as the square and the kagome lattice, the field has rapidly expanded since then. Currently, ASI systems include many different lattice geometries [36–42] and single particle anisotropies [10–12, 43–45] and, for most studies the focus has been on two-dimensional systems, although recently some three-dimensional ASI systems have been introduced [46–50]. What all ASI systems have in common is that the individual nanomagnets interact predominantly via the dipolar-interaction and that the magnetic moments are sufficiently large to behave classically.

1.3 content of this thesis

1.3.1 Scope The main intention of this thesis is to provide a better understanding of magnetic phases that arise in classical dipolar-coupled spin systems. This objective will be achieved mainly by using numerical simulations, analytical calculations, and by applying symmetry discussions based on group theory. Research in the field of ASI, in particular, work related to continuous in-plane spins on square lattices such as Ref. [11], initially motivated the topic of this thesis. Hence, many concepts are developed for the so-called dipolar-coupled XY (dXY) model. For this system, a qualitative phase diagram concerning temperature and disorder has been known [30], but the community lacks a quantitative phase diagram. Therefore, one goal of this thesis is to determine such a quantitative phase diagram for the dXY model on the square lattice. Many findings already exist for the dXY system on specific lattices, but the field lacks a general theory. Therefore, another primary goal of this thesis is to generalize the findings for the dXY systems to more general 1.3 content of this thesis5 geometries. Specifically, the aim is to extend the theoretical description to dipolar-coupled spin systems with continuous degrees of freedom in any dimension and lattice geometry. Hence, another primary objective of this thesis is the generalization of particular findings for the dXY model and to provide a deeper understanding of the underlying phenomena.

1.3.2 Outline The thesis itself is divided into several chapters, at the beginning of which a more detailed overview of the contents of the chapter is given. Here a brief summary of each chapter is provided to guide the reader through the thesis. In Chapter 2, a recapitulation of well-known physical concepts in statistical mechanics and thermodynamics is given. With a focus on classical spin systems and the possible interactions in these systems, the path from the microscopic description to the thermodynamic relations is described. Hence, the language and notation of this thesis are introduced in this chapter. In Chapter 3, the Monte Carlo algorithms used in the thesis are reviewed. After providing the mathematical foundation for Monte Carlo methods, the importance-sampling Monte Carlo algorithms are introduced. Further- more, specialized algorithms such as kinetic Monte Carlo (kMC) or the Monte Carlo renormalization group (MCRG) are described, which can be used to determine particular quantities. In Chapter 4, two applications of Monte Carlo algorithms are presented. Specifically, in Section 4.1, the application of kinetic Monte Carlo simulations for the interpretation of the experimental results of Ref. [51] is discussed. Furthermore, in Section 4.2, the attempts to obtain the universality class of the dXY spin system on the square lattice by finite-size scaling and Monte Carlo renormalization group simulations are presented. In Chapter 5, the experimental results of Ref. [11] on the dXY system obtained by µ-spin rotation measurements are presented, and it is discussed how theoretical modeling of the measurements provided a reasonable interpretation of the results. Explicitly, a criterion for probing only time-averaged quantities is developed. Here, the magnetic fields are determined by a mean-field calculation, which gives a theoretical prediction of the depolarization of the muon spin. It is shown that this approach yields a reasonable agreement with the experiment. In Chapter 6, the influence of temperature and disorder on the dXY system is addressed. Here, order parameters indicating the magnetic phases 6 introduction & motivation

that the dXY system on the square lattice can attain are derived by group theory. From extensive Monte Carlo simulations, the magnetic phase diagram is identified for two cases of disorder, leading to the first determination of the full phase diagram for this model. In Chapter 7, it is shown why many dipolar systems display a continuous ground-state degeneracy, even though dipolar systems do not admit a continuous symmetry in the Hamiltonian. By using the group and representation theory for a detailed symmetry analysis, it is shown that the continuous degeneracy arises as a consequence of the peculiar symmetries of the dipolar interaction. This analysis also demonstrates that the degeneracy is fairly fragile, so it is not expected to survive excitations. The effect of such excitations on this continuous ground-state degeneracy is analyzed in Chapter 8. First, some analytical results are derived for two simple systems, namely for dXY spins on a linear chain and a square lattice. These specific calculations introduce the main concepts of linear spin-wave theory, which are then used for the generalization to more general lattices using group theory. With this, it can be seen that some results for the simple lattices can be generalized. Finally, the thesis is concluded in Chapter 9 by a summary of the main achievements. Additionally, an outlook for the future of dipolar-coupled spin systems with continuous degrees of freedom is given.

1.3.3 Contributions Scientiﬁc advances often happen as a result of the contributions from several researchers, and the work presented in this thesis was obtained as part of research collaborations. For clarity, the contributions of the author of this thesis, Dominik Schildknecht, are explicitly listed for each chapter below. If results are presented, which were not obtained by the author of this thesis, it is clearly stated in the text.

• Chapter 4, Section 4.1: – Formulated the theoretical model with the help of Dr. Naëmi Leo. – Performed the kinetic Monte Carlo simulations by writing a wrapper around the code by Dr. Peter M. Derlet. – Helped with the preparation of the manuscript [51].

• Chapter 4, Section 4.2: – Implemented the Monte Carlo simulations for nearest-neighbor Ising systems and dXY systems on the square lattice. 1.3 content of this thesis7

– Analyzed the Monte Carlo results using ﬁnite-size scaling. – Implemented and tested the Monte Carlo renormalization group method for both systems.

• Chapter 5: – Derived and formulated the fast-fluctuation criterion. – Analyzed the results of the TRIM.SP simulation with the fast- fluctuation criterion to determine the fraction of muons that only the static contribution of the magnetic fields. – Performed the mean-field calculation for dXY spin systems on the square lattice. – Derived the depolarization of the muon spin using the magnetic stray field calculated from the mean-field approach and compared this to the experimental results. – Helped with the preparation of the manuscript [11].

• Chapter 6:

– Derived the order parameters Ms and Mmv. – Derived an applicability criterion for the order parameters in the presence of positional disorder. – Wrote the simulation code and performed large-scale simulations of the dXY system on the square lattice exposed to two different types of disorder. – Implemented the resampling procedure based on the Binder cumulants to determine the phase diagram for both types of disorder. – Prepared the manuscript [52] with contributions from all authors.

• Chapter 7 : – Reproduced the explicit ground-state calculations for the dXY model on the square lattice from previous literature. – Formulated the proof for the continuous ground-state degeneracy. – Prepared the manuscript [53] with contributions from all authors.

• Chapter 8: – Reproduced previously known results for spin-wave excitations on simple lattices. 8 introduction & motivation

– Formulated the generalization of these results to more general systems described by the theory of Chapter 7 in collaboration with Dr. Michael Schütt and Dr. Peter M. Derlet. 2 THEORYOFCLASSICALSPINSYSTEMS

In this chapter, thermodynamics and statistical physics are reviewed with a focus on classical spin systems. The concepts discussed in this chapter are well-known, and there exists an extensive repository of great books on these topics such as Refs. [54, 55] to which we want to refer the interested reader. In this thesis, the units are chosen so that the Boltzmann constant is kB = 1 (dimensionless). Therefore, temperature and energy are expressed in the same units.

2.1 degrees of freedom

Historically, the first microscopic models of magnetism were classical treatments, which have only later been extended to quantum mechanical operators (see Section 1.2.1). Here, however, it has to be clarified under which circumstances the classical limit is sufficient to describe a system. The system can be treated classically if the so-called large-S limit is fulfilled. One can consider the commutator of spin variables Sˆα, Sˆβ = ih¯ εαβγSˆγ,(2.1) where α, β, γ ∈ {x, y, z} and ε is the Levi-Civita symbol. Upon rescaling ˆ ˆ ~S = (Sˆx, Sˆy, Sˆz) to λ~S in Eq. (2.1), the right-hand side is linear in λ, and the left-hand side is quadratic in λ. Hence, for large λ (i. e., large spin), the commutator is close to 0, and quantum-mechanical fluctuations become negligible. The physical interpretation is that if ~Sˆ 2 h¯ 2, then quantum- mechanical fluctuations are negligible compared to the size of the spin, and hence a classical treatment of the magnetic moment as a unit vector is sufficient. In what follows, the large-S limit is always assumed so that the systems are treated entirely classically. This limit is sufficient for ASI, where the total spin of each island is typically O(105) or larger. Even though the focus of this thesis is entirely on classical entities, the normalized magnetic moment is still referred to as the spin, following the literature [15]. The classical analog of a spin is a vector on the unit sphere S2, i. e., a three-dimensional vector with unit length, which is referred to as a Heisen- berg spin. The dynamics of Heisenberg spins can be strongly restricted due 9 10 theory of classical spin systems

to the presence of anisotropies. Such anisotropies can either be described by additional terms in the Hamiltonian (see Section 2.2.2) or, assuming strong anisotropies, the degree of freedom can be restricted. For example, in systems with a strong in-plane anisotropy such as in Refs. [10–12], the adequate description is provided by spins on the unit-circle, which are referred to as XY spins. Such an XY spin is still a continuous degree of freedom since it is parametrized with a single angle, corresponding to the parametrization of the unit circle S1. In contrast, for an out-of-plane anisotropy or for elongated nanomagnets in ASI, the proper description relies on a spin which can only point along two directions. Such a spin is referred to as an Ising spin and is described by a point on the sphere embedded in one dimension S0 (the two points ±1 are equivalent to Z2). Since there is no continuous parameter changing the spin direction, such a spin is called discrete. In this thesis, a distinction is made between Ising spins and Ising-like spins whenever the Hamiltonian incorporates a mixing between real-space and spin-space as for the dipolar Hamiltonian (as seen in Section 2.2.4). If the easy-axis anisotropy deﬁning the spin direction is the same axis for all spins, then such spins are referred to as Ising spins. If, however, the anisotropy axis can change from one spin to another, such spins are referred to as Ising-like spins, which typically arise in ASI systems with discrete degrees of freedom, such as in the artiﬁcial square ice [37, 56, 57]. In addition to the Sd-spins (Heisenberg spins, XY spins, and Ising spins), other types of spins can be introduced, such as the p-state clock-model spin, which is an XY spin with strong p-fold anisotropy, such that the angle is restricted to p evenly spaced values. Additionally, p mutually orthogonal spins form the basis for the Potts-p-model spins, so that the lattice directions eˆx, eˆy, ··· can be chosen as a basis.

2.2 hamiltonians

The Hamiltonian deﬁnes the physics of a system, such as the time evolution and the thermodynamic behavior. It should be noted that one needs to differentiate between the Hamiltonian and the energy functional since they are in general inequivalent. However, this subtlety typically does not arise for classical spin systems, without an explicit time dependence. For the present work, it will be assumed that the energy functional is equivalent to the Hamiltonian. Here, we summarize the frequently used Hamiltonians of this thesis. 2.2 hamiltonians 11

2.2.1 Zeeman Hamiltonian Ferromagnets try to align with an external magnetic ﬁeld, which is why compass needles point towards the (magnetic) north pole. The alignment is explained by the Zeeman-energy, which describes the coupling of a mag- ~ ~ netic moment µiSi to the external magnetic ﬁeld Hext as ~ ~ HZeeman = −µ0 Hext · ∑ µiSi,(2.2) i

where the summation is over all spins in the sample. Here, µi is chosen so ~ that |Si| = 1 is dimensionless. Often the magnetic moment µ is separated into µ = MV where M is the magnetization density and V is the volume which exhibits this magnetization.1 The Zeeman-energy is minimized if all magnetic moments align with the external magnetic ﬁeld. It should be noted that this Hamiltonian is not time-reversal symmetric, i. e., it does not ~ ~ stay invariant under the operation Si 7→ −Si.

2.2.2 Anisotropy Hamiltonians In Section 2.1, it has been argued that anisotropies typically lead to a reduction of the degrees of freedom. This hard implementation of anisotropies is only valid when the anisotropies are inﬁnitely strong, or at least the energy scales associated with the anisotropies are much larger than any other energy scale (such as the temperature). Rather than taking this limit, anisotropies can be included in the Hamil- tonian by introducing terms favoring speciﬁc directions. For example, in- plane and out-of-plane spins can be modeled by Heisenberg spins with the anisotropy term

~ 2 HAnisotropy = K ∑(Si · eˆi) ,(2.3) i

where eî is the unit vector along the anisotropy direction. If eî = eˆz for all i, the anisotropy direction is always the same so that the so-called out-of- plane anisotropy is given if K < 0 and in-plane anisotropy if K > 0. To model Ising-like spins, however, eî depends on i, as eî aligns with the easy axis of the spin, which changes from spin to spin.

1 Here, the convention will be that M denotes a magnetization density in units of Am−1, and 2 µi denotes a magnetic moment with units of Am . The units of the external magnetic ﬁeld are therefore Am. 12 theory of classical spin systems

Additionally, other anisotropies can arise, such as the “weak” clock- model anisotropy formulated for XY spins by the in-plane angle θ as

Hp = hp ∑ cos (pθi) ,(2.4) i where p is the number of minimal energy states of this Hamiltonian. This Hamiltonian can arise, for example, from the crystal-ﬁeld anisotropy in a bulk crystal, and hence p is often the coordination number of the lattice [58, 59].

2.2.3 Heisenberg Hamiltonian Both, the Zeeman term (Section 2.2.1) and the anisotropy (Section 2.2.2) are interactions involving only the summation over a single spin. The Heisen- berg Hamiltonian, in contrast, couples different spins by having a summation over nearest-neighbor bonds so that ~ ~ HHeisenberg = J ∑ Si · Sj,(2.5) hiji where hiji denotes summation over nearest-neighbor pairs. Equation (2.5) is the so-called Heisenberg Hamiltonian, which is very similar to the Hamiltonian studied by Ising [4, 5]. If J < 0 then the Hamiltonian prefers the parallel alignment of spins and a ferromagnetic conﬁguration. If J > 0, then an antiferromagnetic alignment of spins is preferred. The sign of J can sometimes be derived from a microscopic theory when the low-energy sector can be described by a Heisenberg interaction. For example, when two electrons hybridize, then the Coulomb repulsion lifts the degeneracy between the singlet and triplet state leading to an effective Heisenberg Hamiltonian. If the triplet state is lower in energy, the spins lower the energy by aligning parallel resulting in a negative J. If, in contrast, the singlet state has lower energy, J becomes positive. Similarly one can derive an effective (antiferromagnetic) Heisenberg interaction from the Hubbard-model, a model used to study the on-site interactions between electrons in band theory [17, 60]. The Heisenberg interaction is often the dominant short-range interaction with coupling constants J often being of the order of several 100 K. Various extensions to the Heisenberg Hamiltonian in Eq. (2.5) exist: Inter- action with more distant neighbors such as the next-nearest neighbor can ~ ~ be included. Also, the product Si · Sj can be generalized to the quadratic ~ ~ form Si · A · Sj, where A is a matrix. This matrix is symmetric for space- inversion symmetric problems but can have an antisymmetric component 2.2 hamiltonians 13 by the inclusion of Dzyaloshinskii-Moriya interactions. Finally, A could also depend on the pair (i, j), such that even models of spin glasses can be written quite elegantly in a similar form to the Heisenberg Hamilto- nian [61–63].

2.2.4 Dipolar Hamiltonian ~ The Hamiltonian of dipolar interacting spins Si is given by D 1 H = ~S · ~S − 3 ~S · rˆ ~S · rˆ (2.6a) Dipol ∑ |~ |3 i j i ij j ij 2 i6=j rij  α β  D 1 rijrij β = δαβ − 3  SαS ,(2.6b) 2 ∑ ∑ |~r |3 |~r |2 i j i6=j α,β∈{x,y,z} ij ij where ~rij is the difference vector between the positions of the sites i and j, and rˆij is the normalization of ~rij to unit length. The Hamiltonian is α expressed in a symmetric quadratic form in Eq. (2.6b), where Si denotes ~ the α component of the spin Si. The dipolar interaction strength is given by D, which for ASI nanomagnets is given by

µ (MV)2 D = 0 ,(2.7) 4π where µ0 is the vacuum permeability, and MV is the magnetic moment of a single ferromagnetic element with magnetization M and volume V. ~ As Eq. (2.6) incorporates terms such as Si · rîj some care has to be given when defining the scalar product between vectors in spin-space and in real-space. In general, the dimension d of the scalar product is the dimension of the space, where the spin and the lattice can be embedded. One can consider that rîj is placed on the lattice, and hence inherits the lattice dimension dlattice. Furthermore, spins are described by vectors on a (dspin − 1)-sphere, i. e., the dimension of a spin is one larger than the number of degrees of freedom describing the spin. As an example, Heisenberg spins are described by two angles serving as the degrees of freedom so Heisenberg that the spin dimension of a Heisenberg spin is dspin = 3. The second term in Eq. (2.6) is also responsible for the peculiar symmetry group of the dipolar Hamiltonian. It supports more symmetries than the Ising model, but fewer symmetries than the Heisenberg model. The symmetry group of the Hamiltonian is given by Z2 × T × P, namely time-reversal symmetry, translational invariance, and the point symmetry 14 theory of classical spin systems

~ ~ group. The Hamiltonian is unchanged under the operation Si 7→ −Si for all i, which constitutes the time-reversal symmetry Z2. Furthermore, ~ T ~ ~ ~ (~ri, Si) 7→ (~ri0 , Si0 ) = (~ri − t, Si0 ) (2.8) is a symmetry of Eq. (2.6) when~t is a lattice vector. This symmetry is the so- called translational invariance T , which leaves Eq. (2.6) invariant because only relative coordinates appear in the Hamiltonian. Both Z2 and T are often found for other spin Hamiltonians as well so that their implications are well understood. The unusual symmetry of the dipolar Hamiltonian is P, the point-symmetry group of the underlying lattice. To explain this symmetry, the vector representation V has to be introduced. This is the representation of the point symmetry group that transforms a vector according to the point symmetry group. For simplicity, we assume that all directions in spin-space and real-space are shared (implying that dspin = dlattice), then both ~ri and ~ Si transform under the point symmetry group element g ∈ P as ~ P ~ (~ri, Si) 7→ V(g)~ri, V(g)Si ,(2.9) where the representation has to act on the position and the spin simulta- ~ neously to ensure that the second term in Eq. (2.6a) (the term −3(Si · rîj) ~ (Sj · rîj)) stays invariant. If dspin 6= dlattice, then the relevant space is the em- ~ bedding space that already appears in the scalar product Si · rîj in Eq. (2.6a). It should be noted, however, that Eq. (2.9) also has to respect the spin di- ~ rections so that V(g)Si has to be described by another vector of the spin space. In other words, the point symmetry group is not allowed to change the spin-space to an orthogonal space but, rather, is only allowed to change one spin to another spin in the same manifold. Therefore, it is often sufficient to restrict the argument to the lower of the two dimensions and only work with projections. As an example, for in-plane XY spins on the cubic lattice, dspin = 2 and dlattice = 3. However, the Hamiltonian describing this system only supports the symmetry of the square lattice layers, since a rotation out-of-plane changes the plane of the XY spins, which is hence not a symmetry of the complete Hamiltonian. Therefore, the system can be described in d = 2 dimensions.

2.3 long-range and short-range interactions

One classification of Hamiltonians is based on the range of their interactions. Many common Hamiltonians, such as the Heisenberg Hamiltonian 2.3 long-range and short-range interactions 15 of Eq. (2.5) are short-ranged. However, the most prominent Hamiltonian of this thesis, the dipolar Hamiltonian as defined in Eq. (2.6), has interaction tails, i. e., the interaction between two spins is non-vanishing even for large distances. Therefore, the influence of long-range interactions on the behavior of the system has to be discussed. A more extensive discussion of the thermodynamics of long-range interacting systems can be found in Ref. [64]. First, the following toy model Hamiltonian is introduced to define the difference between short-range and long-range interactions, ~ ~ H = ∑ Ji−jSi · Sj, where Ji−j ≡ J(~ri −~rj),(2.10) i6=j which is a generalization of the Heisenberg Hamiltonian given in Eq. (2.5) to include more than nearest-neighbors. Here, ~ri is the position of the ith site so that δ~r = ~ri −~rj denotes the difference vector between the two sites. The Hamiltonian in Eq. (2.10) can have different ranges depending on the functional dependence of J(δ~r). If there are only a finite number of J(δ~r), which are non-zero, such as in the Heisenberg Hamiltonian, the interaction is truly short-range. If the J decay exponentially fast, i. e., J(δ~r) ∼ exp(−δ~r/λ), the system is still considered to be effectively short- range, even though the interactions never truly vanish. In contrast, the Hamiltonian in Eq. (2.10) has long-range interactions, if J(δ~r) decays slower than exponentially. A typical case is an algebraic decay of the coupling constants: J(δ~r) ∼ |δ~r|−σ. Here, one should differentiate between strong and weak long-range interactions by considering if the interaction leads to a super-extensive energy, i. e., whether the energy grows faster than the system size when the system is enlarged. The energy of a single spin in a large system of size L in dimension d is given by  1  − , d < σ Z L Z L  σ d 1 − − −  E ∝ rd 1dr = rd 1 σdr ≈ = .(2.11) rσ log L, d σ 1 1   Ld−σ d−σ , d > σ

Therefore, strong long-range interaction occurs in cases when d > σ. This situation leads to many interesting phenomena, such as ensemble inequiv- alence or the sample-shape dependence of the ground state (see Ref. [64] for a review). For the other case, namely d < σ, the long-range interaction is only weak so that the Hamiltonian can be truncated in many situations. 16 theory of classical spin systems

However, some artifacts can remain [65–67], so that the truncation has to be chosen carefully. For the speciﬁc case of dipolar interactions given by the Hamiltonian in Eq. (2.6), σ = 3. Hence, in one- and two-dimensional systems, the interaction can be truncated. However, in three-dimensional systems, the interaction becomes strongly long-ranged as seen in Eq. (2.11), where the single- site energy diverges logarithmically with respect to the system size L. Therefore, simulations of such systems need to take the long-range nature into account. This strong long-range interaction further leads to a sample- shape dependent Hamiltonian, H 7→ H − DM˜ 2/N, where D˜ is the demagnetization factor, and M is the magnetization. While the Hamiltonian is sample-shape dependent, the free energy has to be sample-shape independent in the thermodynamic limit according to Grifﬁth’s theorem [68, 69], which implies that the ground-state of a three-dimensional macroscopic system cannot be a single ferromagnetic domain.

2.4 frustration

Another classification of Hamiltonians is based on their frustration, which denotes the inability of a system to minimize all pairwise interaction energies simultaneously, i. e., there are competing interactions in the Hamil- tonian. One should differentiate between classical frustration and quantum frustration. The typical example of quantum frustration is the Ki- taev model, leading to a quantum spin liquid and Majorana fermions [20]. For classical frustration, one can further differentiate between geometric frustration and disorder-induced frustration [70, 71]. Geometric frustration gives a plethora of interesting phenomena, such as incommensurate phases [72–74] or highly degenerate ground-states [75], which often implies the inhibition of ordering down to very low temperatures [76]. It can also lead to constraints on local fluctuations such as these seen in the fa- mous pyrochlores [77–79]. If strong disorder is present, often spin glass behavior is observed [61–63]. The first model that results in strong (geometrical) frustration is the Ising model with antiferromagnetic interactions on a non-bipartite lattice such as the triangular lattice. A sketch for this is shown in Fig. 2.1. Here, one can imagine drawing a loop on the lattice. One starts at an arbitrary site on the lattice and fixes the spin. Then, the next spin in the loop is fixed so that the mutual energy between this spin and the first spin is minimized. This procedure is then repeated for every spin in the loop. Since loops on non- 2.5 thermodynamics 17 ? J > 0 J > 0

J > 0

Figure 2.1: Sketch of a highly frustrated triangle of Ising-spins with an antiferromagnetic interaction. The lower left spin is set to point up without loss of generality. Next, the lower right spin has to point down in order to minimize its energy. Finally, the last spin can either minimize its interaction with the left or the right spin, but not with both simultaneously, so that this spin is frustrated.

bipartite lattices, such as the triangular lattice, can have an odd length, upon returning to the first site, the associated spin would, for antiferromagnetic interactions, need to point in a direction opposite to its initial direction so that the system is frustrated. This frustration leads to an extensive ground-state entropy,2 i. e., a degeneracy growing with system size, which inhibits long-range order at any finite temperature [76]. Also the dipolar interaction itself defined by the Hamiltonian in Eq. (2.6) ~ ~ is geometrically frustrated. The first term of the Hamiltonian (Si · Sj) prefers an antiferromagnetic alignment between all the spins, which can also lead to odd loops as in triangular antiferromagnetic Ising model. A model only incorporating only this term is for example given by out-of- plane moments. Such moments placed on the triangular lattice exhibits an ordered phase and a spin-liquid phase typical for frustrated systems [80]. ~ ~ The second term in the Hamiltonian (−3(Si · rîj)(Sj · rîj)), however, prefers a ferromagnetic alignment of spins but is only effective for the spin components along the bond. Hence, a rule of thumb for favorable configurations is the so-called “head-to-tail” rule: Dipoles align ferromagnetically if they can align along their bond and antiferromagnetically if they are forced to be orthogonal. This rule is illustrated with dogs in Fig. 2.2.

2.5 thermodynamics

Even though the entire system is fully described by a Hamiltonian as in Section 2.2, it is impossible to treat a macroscopically large sample with

2 Or equivalently to an exponentially growing number of ground-state conﬁgurations. 18 theory of classical spin systems

(a) Ferromagnetic and along (b) Antiferromagnetic and the bond orthogonal to the bond

Figure 2.2: Illustration of the “head-to-tail” rule. Dipoles behave like these dogs, i. e., if the spins can align with their bond, then they prefer a parallel alignment. If the spins are forced to be orthogonal to their connecting vector, then the spins prefer to align antiferromagnetically. (Courtesy of Dr. Naëmi Leo)

O(1023) spins. To gain quantitative insight into such systems, it is vital to consider the thermodynamic variables of the macroscopic system that emerge from the statistical mechanics of the underlying microscopic variables. The relationship between the microscopic dynamics and the thermodynamics of the system is provided by the ergodicity hypothesis, which states that time-averaged quantities and ensemble-averaged quantities are identical. This hypothesis led to the advent of thermodynamics in the 19th century [81]. The thermodynamic description relies on the use of macroscopic quantities rather than a description using the microscopic degrees of freedom. The thermodynamic variables always come in pairs. Out of this pair, one value can be held constant, the so-called natural variable, whereas the other value responds to changes in the thermodynamic environment. The pair typically consists of an extensive quantity (growing linearly with system size) and an intensive quantity (independent of system size). The common pairs of thermodynamic variables in a thermodynamic system are the volume V and the pressure p, entropy and temperature (S, T), particle number and chemical potential (Ni, µi) for every particle species i, and magnetization and external magnetic ﬁeld (M~ , H~ ). In these examples, the ﬁrst variable is extensive and the second variable is intensive. 2.5 thermodynamics 19

2.5.1 Thermodynamic potential The relationship between the thermodynamic variables is defined by the thermodynamic potential. One typically separates the thermodynamic potentials by the natural variables that are set by the experiment and the variables that are free to evolve. Nevertheless, average quantities typically agree in the thermodynamic limit among the different potentials due to the so-called ensemble equivalence [82]. However, it is often advantageous to describe the system by the (fixed) natural variables and the associated thermodynamic potential.3 In this section, these thermodynamic potentials and the corresponding experimental conditions are described. If the system is entirely isolated from its environment, then all extensive variables (energy, particle number, volume, magnetization, . . . ) are fixed either due to boundary conditions or due to conservation laws. Hence, the internal energy U is a conserved quantity and is, therefore, best suited to describe such a system. Even though such systems are rarely encountered in condensed matter physics, it serves as a pedagogic example of thermodynamic potentials. If an experiment is conducted at a constant temperature, i. e., the system is attached to a heat bath, then energy is no longer a conserved quantity and can be exchanged between the heat bath and the system. The thermodynamic potential best suited to describe such a system is the so-called Helmholtz free energy. Additionally, in magnetic systems, experiments are often conducted at a fixed external magnetic field. The thermodynamic potential corresponding to this situation is also often called the free energy of the system. The convention in this thesis will be that the thermodynamic potential with fixed particle number N, volume V, external magnetic field H~ , and temperature T is called the free energy F. For the sake of completeness, also the grand potential, or Landau potential, Ω is introduced. This potential arises when particle number N is no longer conserved so that the natural variable is the chemical potential µ rather than the particle number. This description corresponds formally to an experiment where the system can exchange particles with a reservoir, and arises typically in the treatment of quasi-particles and excitations.

3 It should be noted, however, that due to ensemble equivalence, any thermodynamic potential is sufﬁcient to describe a particular system, independent of the experimental conditions. However, in order to compare the calculation with an experiment with ﬁxed external parameters, a Legendre-transformation has to be applied in order to obtain the thermodynamic potential that describes the experiment. 20 theory of classical spin systems

2.5.2 Partition Function The potentials describe the thermodynamics of the system, but they do not represent the microscopic details. Statistical mechanics, in contrast, con- nects the Hamiltonian, and hence the microscopic description, to the thermodynamics of a system. Central for this connection is the concept of the so-called partition function: For every potential, there exists an associated partition function. First, the internal energy U and the associated microcanonical ensemble are considered. In such systems, every extensive quantity is ﬁxed, for example by conservation laws. Therefore, every conﬁguration ω has the same probability p(ω) given that it has the correct energy, volume, and particle number. The partition function itself takes the role of a normalization constant, with the microcanonical partition function given by Z z(E, V, N) = dω δ[E − E(ω)] (2.12)

( ) = 1 so that the probability of a state is p ω z δE,Eω δV,Vω δN,Nω . If a heat bath is attached, then the energy is no longer a conserved quantity, but states occur with a probability proportional to exp[−βE(ω)], −1 where β = (kBT) is the inverse temperature once natural units are used, in which kB = 1. The partition function also needs to incorporate the changed probability, which leads to the canonical partition function Z Z(β, V, N) = dω e−βE(ω).(2.13)

It is generally true that the relevant partition function can be obtained from the microcanonical partition function in Eq. (2.12) via Laplace- transformations. In Table 2.1, the connection between the different partition functions and their respective potentials is summarized. For most purposes in this thesis, the free energy is the relevant thermodynamic potential.

2.5.3 Thermal average of an observable The thermal average of an observable O in the canonical ensemble can be obtained from the partition function by considering that a linear term can be added to the Hamiltonian, i. e., H 7→ H + hO. This Hamiltonian is identical to the original Hamiltonian in the limit of h → 0. The variable h 2.5 thermodynamics 21

Table 2.1: Summary of the different partition functions and their corresponding thermodynamic potentials. Here ω is a phase-space element and β is the inverse temperature. Ensemble fixed Formula thermodynamic natural potential variables R micro- E, V, N z = dω δ[E − E(ω)] S = kB log z canonical R −βE(ω) canonical T, V, N Z = dω e F = −kBT log Z R −β[E(ω)+µN(ω)] grand T, V, µ Z = dω e Ω = −kBT log Z canonical is called the conjugate field to O since, if O is the magnetization, then h is the external magnetic field. Therefore,

1 Z hOi = lim dω O(ω) exp − βH(ω) (2.14a) h→0 Z Ω 1 Z 1 ∂ = lim dω − exp − βH(ω) (2.14b) h→0 Z Ω β ∂h 1 ∂ log Z = − lim (2.14c) β h→0 ∂h deﬁnes the thermal average of the observable O. It should be noted that the thermal average hOi is the value of O that is measured in an experiment where the system is coupled to a heat bath. In an equivalent manner to Eq. (2.14), the integration can be carried out in one dimension only, if the density of states ν(E) is known, since

Z ∞ hOi = dEO(E)ν(E) exp (−βE) ,(2.15) E0 where E0 is the ground-state energy and O(E) is the microcanonical average of O. The weight ν(E) exp (−βE) takes both the growing phase space and the exponential suppression of higher energies into account. If a sad- dle point analysis is applied to

ν(E) exp (−βE) = exp (−β(E − T log ν(E))) ,(2.16) 22 theory of classical spin systems

it can be observed that the main contribution to the integral in Eq. (2.15) comes from the part of the integration domain where E − T log ν(E) is minimal. Hence, for most systems, conﬁgurations where E ∼ T contribute most to the thermal averages. It should be noted that going from Eq. (2.14) to Eq. (2.15) is merely a change of variables so that, without prior knowledge about ν(E), it is equally hard to determine the thermal average from Eq. (2.14) and Eq. (2.15).4

2.6 phase transitions and critical exponents

Finally, many properties discussed in this thesis are related to phase transitions. A phase transition is most generally deﬁned as a discontinuous change in the properties of a system when an external parameter p is varied continuously. Often, the external parameter p is the temperature, such as in the example of the water to ice transition at 0 ◦C or for the ferromagnet to paramagnet transition that occurs at the Curie temperature. However, as other parameters such as pressure or magnetic ﬁeld can also induce a phase transition, the tuning parameter is denoted with p rather than T in this section. Even though all phase transitions come with some discontinuities, there are different types of discontinuities arising in phase transitions as realized by Ehrenfest [83]: A system is said to have a transition of nth order at p = pc, if

− ∂n 1F ∂nF < ∞, but = ∞,(2.17) ∂pn−1 ∂pn p=pc p=pc

where F is the free energy. In practice, one often only needs to differentiate between a discontinuous first order phase transition, such as the freezing of water to ice, and a continuous higher order transitions such as most magnetic ordering transitions. Moreover, there exist so-called crossover phenomena, which do not fit into Ehrenfest’s classification scheme, as they do not constitute proper phase transitions. However, properties of a system change dramatically during a crossover without any discontinuity in the derivatives of the free energy. It should be noted that phase transitions can only happen in the thermodynamic limit so that, in finite systems, the discontinuities are smeared out, and instead of a phase transition only a crossover occurs.

4 Hence, there is no free lunch. 2.6 phase transitions and critical exponents 23

Often a phase transition is associated with a spontaneous symmetry breaking, i. e., the Hamiltonian possesses a certain symmetry, which is no longer present in the system after it undergoes a phase transition. For example, for a ferromagnet, the time-reversal symmetry is broken. Hence, the high-temperature phase is typically called the symmetric phase and the low-temperature phase is called the broken symmetry phase. The concept of spontaneous symmetry breaking can be quantified by the so-called order parameter φ. The order parameter vanishes in the high-temperature phase and has a finite value whenever the symmetry of the Hamiltonian is broken. For a ferromagnet, the order parameter is the magnetization, which is 0 above the Curie-temperature and has a finite value below. It should be noted that the order parameter φ is extensive and the conju- gated field hφ is intensive. However, often the order parameter density is considered, i. e., the order parameter normalized by the system volume, to make simulations or samples with different system sizes comparable. These concepts form the basis for the modern approach to phase transitions provided by the Renormalization Group (RG) theory. The RG treatment fundamentally relies on the scale-free nature of systems at criticality, so that a rescaling of the length scales in a system only has a limited impact on the thermodynamics. The rescaling can either be achieved in momentum-space by integrating out high-momentum modes [84, 85], or in real-space by a block spin transformation [86], which will be introduced in more detail in Section 3.4.2. A major result of RG theory is that certain thermodynamic observables around the phase transition obey power laws, which are described by critical exponents that are defined in Table 2.2. The critical exponents are defined in such a way that they are typically positive, i. e., if a minus sign appears in the scaling law, the quantity typically diverges at the phase transition. Furthermore, RG theory shows that the critical exponents are not independent and that two are sufficient to determine the other exponents by the so-called scaling relations [87]:

δ + 1 νd = 2 − α = 2β + γ = β(δ + 1) = γ , and (2.18a) δ − 1 γ δ − 1 2 − η = = d .(2.18b) ν δ + 1

Possibly the most fundamental result of RG theory when applied to phase transitions is that many phase transitions behave the same, even though the microscopic physics might be entirely different. Indeed, the physics of phase transitions is universal: critical exponents do not depend 24 theory of classical spin systems

Table 2.2: Definition of some thermodynamic observables and their critical behavior around a phase transition upon changing the external parameter p, where p is often the temperature. Here, F is the free energy and hφ is the field conjugate to the order parameter. All quantities are considered in the limit hφ → 0. Furthermore, the order parameter is extended to φ 7→ φ(~x) so that the order parameter is hφ(~x)i. Quantity Definition Scaling Validity β ∝ (pc − p) p < pc Order parameter hφi = −∂hφ F 1 ∝ |hφ| δ p = pc

T 2 α Heat Capacity Cv = − V ∂T F ∝ |p − pc|

−γ Susceptibility χφ = ∂hφ hφi ∝ |p − pc|

Correlation ∝ |~x|−(d−2+η) p = p G(|~x|) = hφ(0)φ(~x)i c Function ∝ exp(−|~x|/ξ) p > pc

−ν Correlation ξ ∝ |p − pc| Length 2.6 phase transitions and critical exponents 25 on microscopic details but only on the dimension of the system, the symmetry that is broken by the transition and whether the interaction is short- or long-range [84, 85, 88, 89]. Finally, the RG theory can also be applied to finite systems, which do not undergo a phase transition but rather a crossover due to their finite size. However, if the system is large enough, it can behave as if it would be infinite. The theory behind this observation is the so-called finite-size scaling theory [90], which is comprehensively reviewed in Ref. [91]. For this thesis, it is sufficient to summarize finite-size scaling in the following way: Finite-size scaling essentially distinguishes two regimes based on the correlation length ξ a system would have if it were infinite. If L ξ, the correlations do not experience the finiteness of the system so that the system behaves as if it would be infinite. In contrast, if L ξ, the correlation length is capped by the size of the system, and strong deviations from the thermodynamic limit are expected. Finite-size scaling will be discussed in Section 3.4.1 in more detail, once Monte Carlo simulations on finite lattices have been introduced.

3 MONTECARLOMETHODS

In this chapter, we introduce the Monte Carlo (MC) algorithms that are used in this thesis. A more exhaustive discussion can be found in textbooks such as Ref. [92]. The code used for this thesis is based on the ALPS project [93–96] and is hosted on GitHub.1 Simply speaking, Monte Carlo algorithms are methods using (pseudo-) random numbers to approximate a solution to a problem, which is often in the form of an integral. Monte Carlo methods predominantly come with the advantage that they do not require any assumptions or uncontrolled approximations. Instead, everything limiting the accuracy, such as the system size or the statistical sample size, can be increased, so that with sufﬁcient time on a computer, the problem can be solved to any required accuracy.

3.1 mathematical approach

From a mathematical context, Monte Carlo algorithms are based on the simple equation Z dr f (r) = vol(Ω)h f iΩ,(3.1) Ω

where h· · · iΩ represents the average over the volume Ω. The average of a function can be approximated by sampling the function N times since

1 N 1 h f iΩ = ∑ f (xi) + O √ ,(3.2) N i=1 N

where the xi are uniformly distributed random numbers in Ω. This integration method is known as the “crude method” [92].The error decreases statistically with the square root of the sample size. Hence, to obtain a 10-fold increase in precision of a result, the algorithm has to be run 100 times longer. In low-dimensional integration, quadrature rules such as Simpson’s rule easily beat this convergence. However, the convergence order decreases with dimension for quadrature rules, whereas Monte Carlo

1 https://github.com/domischi/mcpp 27 28 montecarlomethods

methods are independent of the dimension, always converging with a relative error proportional to √1 . N √ Even though the convergence goes as O( N), the prefactor for the convergence can vary widely. To obtain a better prefactor for Monte Carlo simulations, many different methods have been devised. Examples include the splitting of the integration interval into sub-intervals focusing on the more signiﬁcant intervals or the so-called control-variate Monte Carlo method if an approximation f˜ ≈ f is known that is easy to integrate [97, 98]. For typical problems of physics, where the integrand is typically only signiﬁ- cant in a small region of Ω, the method of choice is importance-sampling Monte Carlo.

3.2 importance sampling: metropolis algorithm

In situations, where the thermal average of an observable is to be determined, the crude method of Section 3.1 is inefficient as ν(E)e−βE is strongly peaked so that most configurations have close to 0 weight. The crude approach would amount to choosing a completely random point in phase space. As no correlation occurs between the elements of the simulation, such a situation corresponds to the infinite temperature case. In principle, such a Monte Carlo integration would still work; however, most of the time is spent in regions where the Boltzmann factor exp(−βE) is essentially 0. To circumvent this problem, Metropolis et al. [99] proposed to sample physical configurations, i. e., configurations which would resemble an actual physical configuration at a given temperature. In their paper, Metropolis et al. were interested in the determination of the canonical ensemble of two-dimensional particles interacting with a hard-core pair-potential  ∞, |~xi − ~xj| < 2rhc V(~xi,~xj) = ,(3.3) 0, otherwise

where rhc is the radius of the particles and ~xi is the position of the ith particle. As described in Section 2.5.2, they computed

Z N " N 2 !# 2 2 pi Z = d rid pi exp −β + V(ri, rj) ,(3.4) 4N ∏ ∑ ∑ R i=1 i=1 2m i6=j where their integral is 4N-dimensional since each of the N particles comes with 2 positional and 2 momentum degrees of freedom. Due to the high 3.2 importance sampling: metropolis algorithm 29

dimensional integration, Monte Carlo methods are essential for evaluating such integrals. The crude method, amounting to randomly placing the particles inside the domain, is highly inefficient due to a significant probability that two particles overlap. In such a case, the energy of a configuration is infinite and hence the Boltzmann weight of this configuration vanishes. Instead of randomly picking positions of the sites, the simulations of Metropolis et al. start with a regular lattice of the particles. Then, an up- 0 date to the position ~ri of the ith particle to a new position ~ri is proposed and, if the particle is not overlapping with any other particle, then the energy remains finite, and the proposed update is accepted. If there is an overlap with other particles, however, then the move is discarded, and the particle remains at position ~ri. Hence, the simulation follows a Markov chain,2 generating physically meaningful configurations over time. This algorithm contained the main idea of modern importance-sampling Monte Carlo algorithms, namely obtaining new physically meaningful configurations by changing an original configuration and accepting the change if the new configuration is physical. This procedure is the so-called “Metropolis- Rosenbluth-Rosenbluth-Teller-Teller-algorithm” [100], often just called the Metropolis algorithm. The simulation generates a Markov chain, which provides physically meaningful configurations after a thermalization period (also called burn- in period) if the algorithm obeys detailed balance and ergodicity. Typically, the transition probability W is separated into a product of the proposal probability T and the acceptance probability A. The proposal probability T ensures that the update is in some sense local in phase space and has the following (Markov) properties:

• Ergodicity: Every conﬁguration can be obtained after a ﬁnite number of steps.

• Normalization: The sum of all possible paths from a conﬁguration X is 1, formally given by:

∀X : ∑ TX→Y = 1. (3.5) Y

2 A chain of events

X1 → X2 → · · · ,

describes a Markov chain if the probability of going from Xi to Xi+1 only depends on the state Xi and not on any of the previous events. This means that the Markov chain is a stochastic process without memory. 30 montecarlomethods

• Reversibility: The proposed transitions are reversible:

TX→Y = TY→X.(3.6)

In order to construct the full transition probability W = T · A that fulﬁlls the required detailed balance condition, the acceptance probability A is used to ensure the condition

peq(X)WY→X = peq(Y)WX→Y, ⇒ peq(X)AY→X = peq(Y)AX→Y,(3.7)

which is required to ensure that the correct distribution is simulated. For thermal conﬁgurations,

peq(X) = exp(−βE(X))/Z (3.8)

is the equilibrium distribution. For spin systems, a similar approach to Metropolis et al. [99] can be applied. Analogously, an initial configuration with non-vanishing Boltzmann- factor is chosen. One possible choice can be the ground state. However, since the energy for spin systems typically does not diverge, as in the case of hard-core particles, any configuration can serve as the starting configu- ~ ~ 0 ration. In the next step, a single-spin update Si 7→ Si is proposed, and the change in energy that results from this update is computed to be

∆E = E0 − E.(3.9)

Finally, a random number r in the interval [0, 1] is drawn and compared to Boltzmann-factor, which leads to the Metropolis dynamics:  Accept: ~S 7→ ~S 0, ∆E < 0 or r < exp(−β∆E) i i (3.10) ~ ~ Refuse: Si 7→ Si, otherwise.

Hence, the acceptance probability of the Metropolis algorithm is given by

AX→Y = min [1, exp(−β∆EX→Y)] ,(3.11)

which fulﬁlls the detailed balance condition given in Eq. (3.7). A pseudo- code implementation of this algorithm is provided in AppendixA. 1. Importance sampling, as discussed here, is the simplest version of entire classes of Monte Carlo algorithms. Modiﬁcations can be made to all steps. For example, rather than the Metropolis probability from Eq. (3.11), 3.3 parallel tempering 31

Glauber dynamics [101] can be used.3 As an alternative to single spin-ﬂip updates, cluster updates can be introduced, which accelerate the decorre- lation of the system dramatically [103–105]. Also, instead of the canonical ensemble, other physical ensembles can be considered [106], and even unphysical ensembles have proven to be useful for frustrated systems [107– 111]. A more extensive survey of different methods can be found in textbooks such as Ref. [92].

3.3 parallel tempering: an approach for highly frustrated systems

For highly frustrated systems such as spin glasses, the approach outlined in Section 3.2 is typically insufficient because finding non-trivial updates to the system is difficult. Therefore, in early work, attempts were made to find low-temperature properties of highly frustrated systems by lowering the temperature from the high-temperature phase slowly [112, 113]. These attempts relied on the careful analysis of the thermalization of the systems to assure that the system would only fall out of equilibrium at sufficiently low temperatures so that meaningful conclusions could be obtained at the relevant temperatures. A better approach to slowly lowering the temperature is provided by parallel-tempering Monte Carlo [114, 115] (also called exchange Monte Carlo). Here, Ns replicas of the same system are simulated simultaneously at different temperatures T. The replicas evolve independently according to a regular Metropolis algorithm. However, every once in a while, two replicas with inverse temperatures, β and β0, and energies, E and E0, exchange their temperatures with a probability 0 0 pexchange = min 1, (β − β )(E − E ) .(3.12) A sketch of this procedure is given in Fig. 3.1. In modern spin glass work, the primary simulation technique is parallel-tempering Monte Carlo [116, 117]. In current studies, the parallel- tempering Monte Carlo method was extended with an adaptive search for

3 In Glauber dynamics Eq. (3.11) is replaced by 1 A → = [1 − tanh(β∆E → )], X Y 2 X Y which has the advantage that it ensures ergodicity for high temperatures, but thermalization is typically worse than with the Metropolis algorithm. This method is also equivalent to the heat-bath method [102], which does not require branching in the algorithm, and is thus faster on modern computers. 32 montecarlomethods

∆β∆E pexchange = min 1, e

T = T T = T ··· i i+1 ··· E = Ei E = Ei+1

Figure 3.1: Sketch illustrating the working principle of parallel-tempering Monte Carlo. The blue boxes illustrate independent simulations at a ﬁxed temperature T = Ti, where the conﬁgurations have energy Ei. Every once in a while the simulations are allowed to exchange their temperatures according to Eq. (3.12). Here, ∆E = Ei+1 − Ei and ∆β = βi+1 − βi

the inverse temperature grid β1, β2, ··· , βNs so that, during the simulation, the temperatures are exchanged regularly. Typically, such simulations reveal that a higher temperature resolution is required in the vicinity of phase transitions and towards lower temperatures [96].

3.4 monte carlo simulations at criticality

Having established that classical spin systems can be simulated using Monte Carlo algorithms, it remains to be discussed what observables should be measured. Many systems that we will analyze in this thesis undergo a phase transition upon changing a given tuning parameter. For the sake of simplicity of this section, we exclusively take temperature as the tuning parameter (cf. Section 2.6, where now p = T). For phase transitions, the critical exponents are universal quantities associated with the symmetries of the system (see also Section 2.6). Hence, it would be desirable to extract the critical exponents from simulations, which is, in general quite tricky since phase transitions only happen in inﬁnite systems, i. e., in the thermodynamic limit inaccessible to Monte Carlo simulations. Nevertheless, some methods exist to extract estimates for the critical exponents even from ﬁnite-sized Monte Carlo simulations. Two such approaches are discussed in the next sections. 3.4 monte carlo simulations at criticality 33

3.4.1 Finite-size scaling The standard approach to extract the critical exponents of second-order phase transitions from Monte Carlo simulations is called finite-size scaling (FSS). The FSS method is based on the observation that the dependence of an observable x, such as the order parameter, on the temperature T is changed from a true scaling-law dependence in an infinite system to finite- size scaling-law in a finite system. Namely,

θ − θ h 1 i x ∝ (T − Tc) 7→ x ∝ L ν Fx (T − Tc)L ν ,(3.13) where L is the length of the finite system, Tc is the critical temperature, and Fx is the finite-size scaling function, which has the asymptotic behavior  yθ, y 1 Fx[y] = (3.14) 1, y 1 and continuously interpolates between these two extremes. Equa- tions (3.13) and (3.14) can be derived from the observation that around −ν T ∼ Tc the correlation length behaves as ξ(T) ∝ |T − Tc| . However, in a finite system, the correlation length cannot grow larger than the length of the system L [91]. Therefore, the correlation length is limited and, hence, finite-size scaling of observables as described by Eqs. (3.13) and (3.14) is expected. In practice, Monte Carlo simulations are performed for several lattice sizes L and values of the temperature T. The averages for x in such a simulation, denoted as xL(T), are then plotted in a rescaled coordinate 1 system, with (T − Tc)L ν (the argument of Fx) plotted on one axis and θ xL L ν [the value of Fx according to Eq. (3.13)] on the other axis. Given the correct values of Tc, θ and ν, all curves for different L collapse into a single curve. The method is a straight-forward approach and can provide useful insight into the universality class of the phase transition. However, this technique has several shortcomings. The most essential restrictions are as follows: the collapse is only in the vicinity of T = Tc. This temperature region, where FSS can be applied, can become small, due to model-specific non-universal corrections. Furthermore, the collapse of the curves is not rigorously doable via a fitting procedure but is typically accomplished using a “best-fit-by-eye” approach, and the errors are strongly correlated. Finally, for this approach to work optimally, good statistics around T = Tc is required. However, close to Tc, the statistics 34 montecarlomethods

is typically poor due to the so-called critical slowing down of simulations around second-order phase transitions. This phenomenon of critical slowing down denotes the divergence of the time the system needs for a significant change to the configuration to occur. Hence, single spin-flip updates become increasingly inefficient at updating the state of the system.

3.4.2 Monte Carlo renormalization group A more sophisticated approach to determining the critical exponents is given by combining Monte Carlo simulations with the powerful tools of RG flows. Ma developed the original idea for this method [118], and Swendsen later refined it [119]. For brevity, only Swendsen’s MCRG algorithm will be introduced. In contrast to FSS presented in Section 3.4.1, the MCRG method relies on one simulation only, which is performed as close to criticality as possible and in a system as large as possible. Then, a real-space rescaling procedure is applied, which reduces the system size in every direction by a rescaling factor b. This rescaling procedure is described in detail in Section 3.4.2.1. After the rescaling, changes in coupling constants in the renormalized system are measured as detailed in Section 3.4.2.2. Finally, from the change in coupling constants, the critical exponents can be extracted as detailed in Section 3.4.2.3. This procedure is typically iterated several times, as results are expected to improve with more iterations. However, as the Monte Carlo simulation is carried out on a finite lattice, and the system size is reduced by a factor of b with every iteration, only a very limited number of iterations can be carried out before the system is too small to obtain meaningful averages. The MCRG method outlined here has proven to be a highly precise tool for estimating the critical exponents accurately, but it also has been shown that the convergence of the method crucially depends on specific details of the simulation [120–123]. This makes the MCRG method less robust than FSS. However, with a good intuition about the input parameters, the MCRG method can be successfully applied to various systems, as can be seen in the overview of previous MCRG studies given in Table 3.1.

3.4.2.1 Rescaling Procedure As the name implies, the MCRG method is a renormalization method. Hence, with this method, the changes of the Hamiltonian are analyzed, when the intrinsic length scales are rescaled. However, in contrast to the 3.4 monte carlo simulations at criticality 35

Table 3.1: An overview of previous MCRG studies of classical spin systems in various dimensions. Dim. Degree of Freedom Reference 1.89 Ising [124] 2 Ising [118, 120, 122, 125] 2 Heisenberg [126] 2 general q-state Potts model [127] 2 4-state Potts model [121] 2 4, 5, 6-state clock model [128] 3 Ising [123, 129, 130] 3 XY [131] 3 stacked triangular XY & Heisenberg [132] 4 Ising [133] probably more often applied momentum-space renormalization [134], the MCRG method relies on real-space renormalization, as introduced by Kadanoff via his block-spin procedure [86]. In contrast to the momentum- space renormalization, however, the real-space renormalization can be carried out in various ways. The block-spin procedure of Kadanoff can be generalized to a simple recipe that does some sort of (weighted) averaging of spins, and is typically carried out locally in a Kadanoff-block. Hence, the number of degrees of freedom is reduced, so that this step of the MCRG algorithm is referred to as the reduction step. In most of the cases, the reduced spin should describe the same degree of freedom as before, e. g., a collection of Ising spins maps to an Ising spin. Here, we introduce some particular reduction schemes, which deﬁne maps from a collection of spins to a reduced number of spins via block-spin transformations. For continuous spins, such as XY spins or Heisenberg spins, a block of N spins, represented by the symbol , can be considered. Then, a large class of reduction schemes can be formulated as the transformation ~ ~ ~ ~ 0 ∑i∈ wiSi S1, ··· , SN 7→ S = ,(3.15) ~ ∑i∈ wiSi 36 montecarlomethods

~ 0 where S denotes the reduced spin and w~ = (w1, ··· , wN) denotes the weights that parametrize the reduction scheme. For continuous spins, the block is commonly chosen to be a d-dimensional cube of side length 2 as this can be easily implemented, and it provides more renormalization iterations than reduction schemes with larger blocks. Several particular choices for w~ are known under specific names. For example, the reduction scheme where w~ = (1, 0, 0, 0) is known as the “decimation rule”, as the reduced spin is always the lower left corner of a plaquette [135]. In contrast, if w~ = (1, 1, 1, 1) the so-called “block spin rule” is obtained, which takes the (vectorial) average of the spins [86]. For antiferromagnetic systems, it can be advantageous to use w~ = (+1, −1, +1, −1) to avoid cancellation [135, p. 30]. Likewise, Eq. (3.15) can also be applied to Ising spins. However, in this situation, the denominator in Eq. (3.15) has a significant probability of vanishing, in which case the reduced spin becomes ill-defined. Hence, and w~ need to be chosen so that the denominator does not vanish. One reduction scheme that accomplishes this requirement uses the block spin rule on a 3 × 3 block so that the denominator cannot become 0 as it involves an odd number of spins. This scheme is the so-called “majority rule”, which is illustrated in Fig. 3.2a. However, since the scaling factor is b = 3, i. e., rather large for RG methods, this procedure rescales very fast resulting in less MCRG iterations for the considered finite system size. Hence, more economical rescaling procedures have been invented, which reduce the system size by a smaller factor b. As before, the simplest reduction scheme for a 2 × 2 block is decimation but, as this scheme results in the discarding of plenty of information, the decimation scheme shows a relatively slow convergence [120]. An improved 2 × 2 reduction scheme is given by the so- called ”tie-breaker rule“, which applies the majority rule to a 2 × 2 block. If the sum of the four spins is 0, then the lower left corner is chosen as the tie-breaker. This rule can again be brought into the form of Eq. (3.15) with a weight vector w~ = (1 + ε, 1, 1, 1) where 0 < ε < 2. This procedure is shown in Fig. 3.2b. In general, all procedures should eventually converge to the same critical exponents. However, it was noted early on that the speed of convergence strongly depends on the rescaling procedure [120, 122, 123]. 3.4 monte carlo simulations at criticality 37

⇒

(a) Majority Rule

⇒

(b) Tie-Breaker Rule

Figure 3.2: Illustration of two rescaling procedures for Ising spins. In Fig. 3.2b, the sum of the spins in the lower left and the upper right block is 0. Here, the lower left spin is applied as the tie-breaker. 38 montecarlomethods

3.4.2.2 Truncation of the interaction space Formally speaking, a general translational-invariant spin Hamiltonian is given by

βH = ∑ KαSα,(3.16) α where the inverse temperature is absorbed into the deﬁnition of the coupling constants Kα. Here, Sα are general translation-invariant spin interactions such as

SHeisenberg = ∑ SiSj, or SPlaquette = ∑ SiSi+xˆSi+xˆ+yˆSi+yˆ.(3.17) hiji i

0 Using the MCRG algorithm, one then tries to find the set of Kα which would produce the same configuration that was obtained by applying the rescaling procedure of Section 3.4.2.1. The original MCRG algorithm by Ma [118] attempted to fit a Hamiltonian of the form Eq. (3.16) to the configuration and to extract the coupling constants in this way. Swendsen [119], however, realized that it is sufficient to measure specific correlation func- 0 tions as demonstrated in Section 3.4.2.3. Since the space of Kα, i. e., the interaction space spanned after just one rescaling, is considerably large, the effective Hamiltonian has to be truncated significantly. The truncated Hamiltonian typically consists of only a few interactions (O(10)) such x x as nearest-neighbor interactions (Sα = ∑hiji Si Sj ), next-nearest-neighbor x x terms (Sα = ∑hhijii Si Sj ) or a plaquette term as in Eq. (3.17). Generally, it is considered an art to guess the relevant interactions, although keeping very few terms is often sufficient for well-converged results [122, 123].

3.4.2.3 Extracting critical exponents from rescaled configurations With a rescaling procedure, as introduced in Section 3.4.2.1, a Hamiltonian as given by Eq. (3.16) maps to a new Hamiltonian H0. Equivalently the 0 coupling constants are mapped to new values, i. e., Kα 7→ Kα = β(Kα). If the system is simulated sufficiently close to the phase transition, then the linearized flow equations are sufficient to study the RG flow and hence the critical behavior of the system. These linearized flow equations are described by

0 0 2 ∂Kα Kα = ∑ TαβKβ + O(K ), where Tαβ = ,(3.18) β ∂Kβ 3.4 monte carlo simulations at criticality 39 which can be obtained from Monte Carlo simulations by using the chain rule and the ﬂuctuation-dissipation like theorems as follows:

Tαβ D E z }| { D E (n) ( ) (n) ∂ Sγ ∂K n ∂ Sγ = α ,(3.19a) (n−1) ∑ (n−1) (n) α ∂Kβ ∂Kβ ∂Kα D ( )E ∂ S n γ D (n) (n− )E D (n)ED (n− )E where = S S 1 − S S 1 ,(3.19b) (n−1) γ β γ β ∂Kβ D ( )E ∂ S n γ D (n) (n)E D (n)ED (n)E and = S S − S S ,(3.19c) (n) γ α γ α ∂Kα in which h· · · i denotes the thermodynamic average, so that right-hand side terms of Eqs. (3.19b) and (3.19c) are connected correlators [119]. Further- more, the superscript (n) indicates that the renormalization was applied n times to the system. Hence, the superscript (0) denotes the original Monte Carlo data, and (1) after rescaling the system once. Finally, for a time-reversal symmetric Hamiltonian, the odd and even sector of Tαβ do not mix, i. e., if Sα contains an even number of spins while Sβ includes an odd number of spins, then hSαSβi = hSβi = 0. Hence, Tαβ decays into a block-diagonal form. Both sectors provide one relevant eigenvalue, λe,o, for the even and odd sectors, respectively. These eigenvalues are then related to the critical exponents via

log b log λo ν = , and η = d + 2 − 2 ,(3.20) log λe log b where b is the scaling factor of the rescaling procedure.

3.4.2.4 Some remarks on the MCRG method The MCRG method, as outlined above, has proven to be very successful in the past, cf. the many references given in Table 3.1. However, only very few groups published using this method. This is mainly due to the complexity of this method that makes a successful implementation of this method non-trivial. Furthermore, when the MCRG method is applied to a new problem, many parameters have to be carefully chosen. This includes the Monte Carlo parameters such as system size and temperature, as well as the 40 montecarlomethods

MCRG parameters such as the reduction scheme and the interaction set. In previous literature it can be seen that the convergence crucially depends on the combination of all these parameters [92, 120, 122–124], so that finding a good combination of parameters is essential in the success or failure of the simulations. Finally, for systems with a marginal operator, i. e., operators which do not grow or shrink upon renormalization, the MCRG method can only provide limited insight [127, 136], as the MCRG method is based on the linearized flow equations shown in Eq. (3.18) that fail to incorporate marginal operators. Special reduction schemes have been proposed for such cases in order to mitigate problems [121]: When the marginal operator is identified, an appropriate reduction scheme can be devised that renormal- izes all interactions except those in the marginal direction. However, these reduction schemes lack generalizability, as they require substantial previous knowledge about the marginal operator. As such, the MCRG method has only limited applicability in marginal systems.

3.4.3 Determining Tc For both FSS and MCRG method, a self-consistent determination of the critical temperature can be implemented. However, this approach is un- desirable since such simulations are typically long and computationally expensive. Hence, Tc is typically determined in an independent simulation beforehand. A fast but not very accurate determination of Tc relies on inspection of the order parameter φ as a function of T. The phase transition is then located where the order parameter goes towards zero. A slightly better method is to consider the temperature dependence of the susceptibility χ associated with φ since it diverges at Tc in the thermodynamic limit. Hence, χ is peaked around Tc in a ﬁnite-sized simulation. However, ﬁnite-size effects not only smear the divergence to a peak, but also slightly shift the position of the peak away from the true Tc. A nice compromise between the computationally expensive analysis methods via FSS and MCRG and the imprecise method of inspecting of φ or χ as a function of temperature is given by a so-called Binder cumulant analysis [137]. Here, one can consider

hφ4i U = 1 − ,(3.21) 3hφ2i2

which is the so-called Binder cumulant. By using FSS as shown in Eq. (3.13), it becomes clear that U is the ratio between two different scaling functions. 3.5 temporal evolution of discrete degrees of freedom 41

However, its only dependence on the ﬁnite-size is in the argument of the scaling functions so that at T = Tc the value of U is universal since

− 4β h 1 i ν F ( − ) ν hφ4i L φ4 T Tc L F 4 [0] = = φ 4β h i .(3.22) hφ2i2 − 2 1 F 2 [ ] T=Tc L ν F (T − T )L ν 2 0 φ2 c φ T=Tc Therefore, the curves of the cumulants U for different L will (up to corrections to scaling) cross at a single point, indicating the critical temperature.

3.5 temporal evolution of discrete degrees of freedom

Even though most of the thesis is concerned with continuous spins, some projects included discrete degrees of freedom in the form of Ising-like spins. More specifically, one project involved the study of the dynamics of a small cluster of Ising-like spin (see Section 4.1), which was carried out using the so-called kinetic Monte Carlo (kMC) algorithm [138]. Here, we discuss the kMC algorithm, which is also known under different names such as the Bortz-algorithm, the rejection-free kinetic Monte Carlo or the N-fold way [92]. Historically, the algorithm was invented to accelerate regular Monte Carlo simulations since, in every update, a new configuration is obtained [138]. However, nowadays the algorithm is used to predict the long-time dynamics of systems when it is possible to determine the transition rates of single transitions. Examples where kMC simulations bridge time-scales, include protein folding [139] or structural changes in materials induced through various effects such as temperature or irradiation [140]. In order to apply the kMC strategy, the system should only possess a discrete number of possible states, and all transition rates between the states have to be known. Here, we denote the transition rate from state i to state j with Γij, so that the characteristic time for this transition to happen is t = 1 . If no transition between i and j is possible, then the rate ij Γij is 0, which implies that it would take an infinite time for this transition to occur. The algorithm relies on the assumption that the system can be well described by a Poisson process, i. e., that the system can sample each pathway independently. The rejection-free algorithm, as described here, obtains a new configuration of the system at every step, in contrast to the algorithm presented in Section 3.2. Every update is performed in two steps: First, the algorithm determines which transition is executed by randomly choosing a 42 montecarlomethods

transition, where the probability of a speciﬁc transition to happen is proportional to its rate. In a second step, the time is updated with the transition time. This transition time is generated as an exponentially distributed random-number with the rate of this distribution being the total transition rate to any accessible state so that the average transition time given −1 by δt = ∑j Γij. A pseudo-code implementation of the kMC algorithm is given in Algorithm 2. 4 APPLICATIONSOFMONTECARLOSIMULATIONS

This chapter is split into two parts: In Section 4.1, we discuss the theoretical contribution to the publication [51], where kinetic Monte Carlo (kMC) was used to model the time-dependence of small assemblies of Ising-like spins which form nanomagnetic logic gates. In Section 4.2, the ﬁnite-temperature phase transition of the dipolar-coupled XY (dXY) system on the square lattice is analyzed, as there have been controversial results in the literature concerning the universality class [30, 141–145].

4.1 model nanomagnetic logic gates with kinetic monte carlo

The results presented in this chapter are published in [51]: “Engineering Relaxation Pathways in Building Blocks of Artificial Spin Ice for Computation”, H. Arava, N. Leo, D. Schildknecht, J. Cui, J. Vijayakumar, P. M. Derlet, A. Kleibert, and L. J. Heyderman, Phys. Rev. Appl. (2019). Computation in modern computers is typically performed by semicon- 3 ducting transistors, which use an energy of 1.2 · 10 kBT per operation [146]. This value is much larger than the theoretical limit, the so-called Landauer limit of energy dissipation. This limit is log(2)kBT per operation [147], which evaluates at room temperature to 2.87 zJ = 18 meV. In contrast to the semiconducting transistor-based logic gates, nanomagnetic logic gates can operate with an energy dissipation close to the Landauer limit [148, 149], which makes them a possible alternative to semiconductors for low- power computation. However, in order to make nanomagnetic computation a viable substitute for transistor-based logic gates, the systems have to be tunable to their needs and they should have a reliable operation. To demonstrate the required tunability, the nanomagnetic logic gate design in Ref. [150] is modified to operate in a tunable stochastic manner. These logic gates are assemblies of small clusters of Ising-like spins inspired by ASI systems. To to use these gates for computation, they were initialize by applying a large magnetic field. Then, they are allowed to relax thermally until the computation is finished. In the experiment, the thermal relaxation is carried out by increasing the temperature of the system until

43 44 applications of monte carlo simulations

x ~ Hbias (a) (b)

Figure 4.1: Sample designed by Hanu Arava [51]. In Fig. 4.1a, the design schematic is shown, illustrating how the original gate (for d = 0) is modiﬁed by a displacement d. In Fig. 4.1b an SEM image of one of the structures with d = 0 is shown. (Courtesy of Hanu Arava).

the magnets are thermally active whereas, in the simulations, the dynamics of the system is analyzed with the kMC algorithm. Under the right circumstances, the system evolves to one of its ground states. A measurement then determines the state of the nanomagnets, which serves as the output of the gate. In this work, the output of such gates is changed by a modification to the geometry of the nanomagnetic assembly. The modified logic gate design by Hanu Arava is described in Section 4.1.1. Then, a model for the thermally driven dynamics is proposed in Section 4.1.2. Two solution strategies for the single spin-flip evolution equations are presented: either a matrix exponential has to be computed, or kMC simulations can be applied. With kMC simulations, the probability of reaching the low-energy states after thermal relaxation is computed and compared to the experimental results. From the comparison, microscopic parameters can be determined, and it can be concluded that the experiment is well described by the model. Specifically, it is shown that the modification of the nanomagnetic structure indeed al- ters the output behavior of the gate. Finally, a summary of this section and an outlook for the future of this field are provided in Section 4.1.3.

4.1.1 Experiment on the nanomagnetic logic gates Generally, the experiments in Ref. [51] apply the following protocol: A nanomagnetic logic gate is initialized in a well-defined state by applying 4.1 model nanomagnetic logic gates with kinetic monte carlo 45 a large field which magnetizes the nanomagnets along the field direction. Subsequently, the magnetic field is removed, and the system is kept at an elevated temperature. The higher temperature enables the system to exit metastable states due to thermal activation. Hence, the gate tries to find its most stable configuration, which is its ground state. After keeping the system at elevated temperatures for a specific time, the output of the gate is determined by measuring the magnetic configuration of the system. The system typically evolves preferentially into one of the two ground states, as the system only has sufficient time to get into the ground state that is more accessible from the initial configuration. Although the energy barrier between the two ground states is finite, as expected for a finite system, it is sufficiently large so that the system does not undergo a transition to the other ground state during the time of the experiment. Therefore, the system initialized in a particular state relaxes to the two ground states with an unequal probability. The gate considered in this section is based on a well-characterized nanomagnetic circuit with 11 nanomagnets (see Fig. 4.1a for d = 0) previously proposed by Arava et al. [150]. This gate shows a pseudo-NAND gate operation with high reliability after thermal relaxation from a field-saturated state. The gate has subsequently been modified by Hanu Arava to demonstrate the possibility of adjusting the output of the gate by a modification in the geometry. The design of the sample is shown in Fig. 4.1a, where the change of the initial design leads to a modification in the probability to reach a specific ground state of the system. This modification is achieved by the introduction of the displacement d to change the energy landscape associated with the gate. The design is then implemented by nanolitho- graphic patterning of the permalloy nanomagnets on the substrate. An SEM image of a sample with d = 0 is shown in Fig. 4.1b. The magnetic configuration of the samples as a function of the displacement d is measured with Photoemission Electron microscopy (PEEM) at the SIM beamline at the Swiss Light Source [151]. The system is initialized in a field-saturated state shown in Fig. 4.2a. Then, the field is removed, and the sample is held at a temperature of 460 K. After three hours, the probability of being in one of the ground states G1,2 is measured by counting the number of gates in a particular state out of a sample of 51 gates for each displacement d. The notation G1,2 corresponds to the configurations shown in Figs. 4.2b and 4.2c, respectively. The probability of being in G1,2 after this thermal protocol as a function of d is shown as points with error bars in Fig. 4.2. It can be observed that 46 applications of monte carlo simulations

(a) Initial state (b) Ground state G1 (c) Ground state G2

0.8

0.6 G1, theory G2, theory 0.4 G1, experiment G Probability 2, experiment 0.2

0 0 20 40 60 80 d [nm]

(d) Probability of being in conﬁguration G1,2 after the experimental protocol as a function of displacement d.

Figure 4.2: The nanomagnetic system depicted in Fig. 4.1a is initialized in the configuration shown in Fig. 4.2a. The magnetic field is then removed, and the system evolves for three hours at 460 K. After the three hours, the probability of ending up in G1,2 as defined in Figs. 4.2b and 4.2c is measured. This procedure results in the displacement- dependent probabilities shown in Fig. 4.2d.

the probabilities for G1 and G2 do not add up to 1, as there some gates that were not completely relaxed during the experiment, so that they remained in a higher energy state. The data has signiﬁcant statistical uncertainty since for each displacement d only 51 gates are measured. Therefore, it is not conclusive that there exists a dependence of the probabilities on the displacement d. However, as the displacement d changes the energy landscape, such a dependence of the output probabilities is expected.

4.1.2 Theoretical model and its solution by kinetic Monte Carlo Due to the statistical uncertainty present in the experimental data, the dependence of the output probabilities on the displacement d is not evident. Therefore, a model of the time-evolution of the nanomagnets is necessary 4.1 model nanomagnetic logic gates with kinetic monte carlo 47

to validate the modification of the output by the displacement parameter d. First, we need to describe the single moment reorientation. Here, it is assumed that the magnetization of the nanomagnet can only align along the long axis of the nanomagnet and that temperature induced random spin flips determine the dynamics of the system. Hence, the dynamics can be approximated by harmonic transition state theory (see Ref. [140] for a review), where the transition rates between different configurations are given by an Arrhenius form [152, 153] Ej − Ei ν → = ν exp −β E + ,(4.1) i j 0 b 2

where Eb is the energy required to overcome the single particle barrier and Ei and Ej are the energies of the configurations before and after a single spin-flip transition, respectively. The factor 1/2 arises under the assump- 1 tion of a symmetric triangular transition path [154]. The energies Ei have been computed in the point-dipole approximation, i. e., their physical ex- tent was neglected.2 A sketch of the transition and the involved energies is given in Fig. 4.3. Finally, ν0 is the so-called attempt rate, which is determined by how often a transition with a vanishing energy barrier would happen in unit time. For our simulation of this system, the transition rate 9 −1 ν0 is chosen to be 10 s according to Ref. [155]. To model the nanomagnetic logic gate, we further assume that no co- operative spin flips occur so that the individual spin flips are independent. Hence, the only possible transitions from each configuration are single spin-flips. Furthermore, since the nanomagnets are Ising-like spins, the possible configuration space is discrete and contains 2N states, where N is the number of nanomagnets in the gate. Therefore, the geometry of the phase space corresponds to an N-dimensional hypercube, where the only allowed transitions are along the bonds of the hypercube [153]. Therefore, the dynamics of the system is described by the rate matrix Γ = [νi→j](i,j)∈{1,··· ,2N }, so that the time-dependent distribution follows the Master equation ∂ ~P = Γ~P, ~P(t = 0) = (0, ··· , 0, 1, 0, ··· , 0).(4.2) ∂t 1 For chemical reactions, a similar relation between the activation energy and the energy difference is referred to as the Bell-Evans-Polanyi principle. 2 The energy scales could be more accurately estimated either by splitting the nanomagnet up into smaller parts, which is typically used in micromagnetic simulations, or by the inclusion of higher order multipoles. 48 applications of monte carlo simulations

Eb i ∆Ei→j ∆Ei→j 2 j

Reaction coordinate

Figure 4.3: A simple sketch, explaining the process from a state i to a state j. Here, ∆Ei→j < 0, so that the ﬁnal state is lower in energy. Furthermore, Eb is the single particle activation energy, i. e., the energy barrier an isolated nanomagnet has to overcome to reorient its magnetization.

The initial value ~P(0) is 1 for the ﬁeld-saturated initial conﬁguration shown in Fig. 4.2a and 0 otherwise. Equation (4.2) can be solved analytically as

~P(t) = exp(tΓ)~P(0),(4.3)

which is the unique solution to Eq. (4.2). It should be noted that this approach is computationally costly for large systems, as the required computation time grows exponentially with the number of nanomagnets N. However, for modeling the logic gate presented in Fig. 4.1a, N = 11, so that computing the matrix exponential in Eq. (4.3) is still feasible. Never- theless, extensions to the problem, such as coupling several gates together, become impossible to simulate due to the exponential scaling of the configuration space. This problem of an exponentially large configuration space can be alleviated using Monte Carlo (MC) methods. Specifically, the time evolution of discrete degrees of freedom can be obtained from kMC simulations as introduced in Section 3.5. In contrast to the matrix exponential approach in Eq. (4.3), kMC simulations have the disadvantage of only providing one possible relaxation path per simulation, rather than the full distribution at once. Therefore, the simulations have to be repeated several times to obtain 4.1 model nanomagnetic logic gates with kinetic monte carlo 49 statistics on the relaxation path. However, this is still faster than the matrix exponential, as kMC simulations focus on the physically relevant part of the system given by the low-energy transition paths. The kMC code is written by Dr. Peter Derlet and has been used previously [152, 153]. The system is then simulated with input parameters corresponding to the experiment (T = 460 K). Furthermore, Eb and Hbias were adjusted to obtain the best agreement with experiments. The optimal agreement was obtained with Eb = 626 meV and Hbias = 50 µT in the direction indicated in Fig. 4.1a. Since the output of the kMC code is stochastic, the code has to be invoked many times, and the average of the outputs has to be taken. Here, 1.2 · 105 simulations have been averaged to obtain observables with negligible error. The results of the simulation are compared with the measurements in Fig. 4.2 where the experimental data is shown as points with error bars, and the simulation results are shown as solid lines. With the optimal choice of Eb and Hbias, a good agreement between the experimental data and the prediction by the simulation is observed. Furthermore, the simulation shows that a modification of the gate can change the output probabilities. It should be noted that the outcome of the simulations is rather sensitive to the precise values, as they enter the transition rates in Eq. (4.1) exponentially. However, the qualitative feature of decreasing the probabilities of G1,2 with a larger d is robust to changes in Eb and Hbias.

4.1.3 Conclusion & Outlook for the simulation of nanomagnetic logic gates To summarize, small assemblies of Ising-like spins have been considered, which are designed to act as logic gates. The design of the gates can be modified to tune the output probabilities. However, due to low statistics in the experimental data, it is not evident if the displacement d is capable of changing the output. Hence, the dynamics of the system was modeled using single spin-flip transitions so that the time dependence could be obtained from kMC simulations. The model could be fit to the experimental results, and it could be determined that the displacement d can indeed modify the output. In conclusion, kMC simulations are indispensable to design and verify the behavior of thermally activated logic gates based on nanomagnets. For small systems, such as the ones considered here, the time evolution could still be carried out analytically by computing the matrix exponential defined in Eq. (4.3). However, for larger systems, such as an integrated circuit with several gates, the only viable way to simulate the behavior is using the 50 applications of monte carlo simulations

kMC algorithm. Here, it has been shown that kMC simulations of the system enable the veriﬁcation of the logic design and the fast prototyping of integrated circuits. Hence, we hope that this type of simulations facilitates a better understanding of nanomagnetic logic gates.

4.2 determination of universality classes

As a second application of Monte Carlo (MC) simulations, we attempt to determine the universality class of dipolar-coupled XY (dXY) spins on the square lattice. Previous studies showed contradicting results [30, 141–145, 156], indicating two different universality classes, which are, however, hard to differentiate using the standard numerical approach of finite-size scaling (FSS). Hence, to improve previous results, a more accurate method is required to extract critical exponents from Monte Carlo simulations. Such a method is given by the Monte Carlo renormalization group (MCRG) algorithm as introduced in Section 3.4.2. Since this method is known to provide highly accurate estimates of the critical exponents, it seems feasible to apply the MCRG method to the dXY model on the square lattice in order to differentiate the two universality classes. First, however, the implementation of the methods used in this chapter need to be validated on a model with a known solution. For this purpose, the example of the Ising model on the square lattice is considered, as this system admits an analytic solution [157, 158]. For the Ising model, the FSS collapse is performed in Section 4.2.1.1, and the MCRG method is applied in Section 4.2.1.2. After gaining some confidence in the application of the two methods, both are applied to the dXY system. In Section 4.2.2.1, a first estimate of the critical exponents is determined by FSS. Then, an attempt to improve these estimates with the MCRG method is given in Section 4.2.2.2. However, it is observed that the MCRG method fails to converge to a physical estimate of critical exponents. Possible reasons for the lack of convergence are discussed in Section 4.2.3.

4.2.1 Ising-model in two dimensions As with any non-trivial simulation, the methods and their implementation have to be validated on a simple model before their application on a more complicated system. The arguably best model for this purpose is the Ising model on the square lattice, as it admits a ﬁnite-temperature phase transition, which is understood analytically [157, 158]. Therefore, the Ising 4.2 determination of universality classes 51

Table 4.1: Summary of the critical exponents (deﬁned in Table 2.2) and the critical temperature for the Ising model with nearest-neighbor coupling J on the square lattice.

Tc/J α β γ ν η 2 √ ≈ 2.269 0 (log) 1 7 1 1 log(1+ 2) 8 4 4

model is used to validate the implementations of the FSS method and the MCRG algorithm.

4.2.1.1 Finite-size scaling In order to apply FSS, Monte Carlo data on several lattices with different system sizes has to be obtained. The Monte Carlo data used for the subsequent FSS collapse is displayed in Fig. 4.4: The magnetization density, the susceptibility, and the heat capacity are shown in Figs. 4.4a to 4.4c, respectively, where one can observe that all quantities show the expected behavior. Namely, the susceptibility and the heat capacity are strongly peaked around Tc, and the magnetization density exhibits a sharp increase when the temperature is decreased below the critical temperature Tc. Moreover, 3 the estimate for Tc = 2.263 ± 0.004 according to the Binder cumulants displayed in Figs. 4.4d and 4.4e agrees with the exact critical temperature of 2.269 given in Table 4.1. Furthermore, the Monte Carlo data admit an excellent FSS collapse using the exact critical exponents reported in Table 4.1 as shown in Fig. 4.5. It should be noted that the heat capacity data does not admit a complete scaling collapse since α = 0 so that the heat capacity only diverges logarithmically.

4.2.1.2 Monte Carlo renormalization group Finite-size scaling is a powerful method for determining critical exponents in many cases, but it fails to provide very high precision estimates with less than 1% uncertainty for critical exponents. Therefore, a second method

3 It should be noted, that for the estimate of Tc by the Binder cumulant the data for L = 12 and L = 16 is not considered, as the corrections to scaling are large for such small systems [92]. Furthermore, the error is obtained by considering the standard deviation of all crossings for different pairs of curves. However, the statistical error of each of the curves is not considered, so that the error reported here is underestimated. 52 applications of monte carlo simulations

1 60 12 L=12 L= L=16 L=16 40 L=24 L=24 χ M 0.5 L=32 L=32 20 L=48 L=48 L=64 L=64 0 0 0 1 2 3 4 0 1 2 3 4 T T (a) MC data for the magnetization density (b) MC data for the susceptibility

30 L=12 0.6 L=12 L=16 20 L=16 L=24 0.4

V L=24 U C L=32 L=32 10 0.2 L=48 L=48 L=64 0 L=64 0 0 1 2 3 4 0 1 2 3 4 T T (c) MC data for the heat capacity (d) MC data for the Binder Cumulant

0.65

0.60 L=12 0.55 L=16 U L=24 0.50 L=32 0.45 L=48 L=64 0.40 2.10 2.15 2.20 2.25 2.30 2.35 2.40 T (e) Same data as in Fig. 4.4d zoomed to the region of interest

Figure 4.4: Monte Carlo (MC) results for the Ising model on the square lattice with size L: (a) order parameter, (b) susceptibility, (c) heat capacity, (d) Binder cumulant, with a zoom-in of the relevant region in (e). The estimate of the critical temperature read off from the Binder cumulant agrees with the exact value, Tc ≈ 2.27. 4.2 determination of universality classes 53

0.05 1.75 L=12 L=12 1.50 0.04 L=16 L=16 ν ν 1.25 / / L=24 0.03 γ β L=24 1.00 L=32 − L 0.02 L=32 ML 0.75 L=48 χ L=48 0.50 0.01 0.25 L=64 L=64 0.00 0.00 −20−15−10−5 0 5 10 15 20 25 −40−30−20−10 0 10 20 30 40 TL1/ν TL1/ν (a) FSS collapse for the magnetization density (b) FSS collapse for the susceptibility

50 0.7 L=12 L=16 L=24 0.6 L=12 40 ν 0.5

/ L=32 L=48 L=64 L=16 α 30 0.4 − L=24 L U 0.3

V 20 0.2 L=32 C 10 0.1 L=48 0 0.0 L=64 −0.1 −40−30−20−10 0 10 20 30 40 −20−15−10 −5 0 5 10 15 20 TL1/ν TL1/ν (c) FSS collapse for the heat capacity (d) FSS collapse for the Binder cumulant

Figure 4.5: FSS collapse of the Monte Carlo data for the two-dimensional Ising model shown in Fig. 4.4 using the critical exponents from Ta- ble 4.1. As expected, the data admits for an excellent scaling collapse, especially for the larger lattices, where corrections to scaling are smaller. 54 applications of monte carlo simulations

to obtain estimates of critical exponents from Monte Carlo data is implemented, which provides higher precision in the determination of critical exponents. This high-precision estimates of critical exponents can be obtained using the MCRG method that, in contrast to FSS, has the advantage of providing numerical values for the critical exponents in a mathematically controlled way, since it does not require a “by-eye-fitting” as FSS does. However, as the MCRG method is very sensitive to the critical exponents, it is likewise susceptible to perturbations such as a wrong estimate for Tc, and therefore, also the MCRG method needs some fine-tuning. In this section, the MCRG method is applied to the example of the Ising model. As described in Section 3.4.2 the effective Hamiltonian obtained after renormalization has to be projected into a finite-dimensional space called the interaction set. Here, up to three different interaction sets are used: Swendsen’s original interaction set [119], which consists of three even interactions (nearest-neighbor, next-nearest-neighbor, four-spin plaquette) and one odd interaction (field-like term). Furthermore, Wilsons’s interaction set is used, which consists of 100 even and 100 odd interactions. Finally, a new interaction set for the even sector with 22 interactions is introduced. Out of these interactions, 14 are quadratic up to 6 lattice spac- ings away. Furthermore, 4 four-spin interactions and 4 six-spin interactions have been included. The interaction set is tabulated in TableB. 1. No new interaction set is provided for the odd sector, as we could not find a suitable interaction which would improve upon the previously known ones. In general, it is non-trivial to find an acceptable interaction set, capturing the main RG flow, while being sufficiently small not to cause numerical instabilities. The three interaction sets are then used for the MCRG simulations. For these simulations, periodic boundary conditions are applied to the square lattice which has N = 642 sites. The lattice is reduced up to three times using the Ising tie-breaker rule. The simulation is performed for several temperatures, T, around Tc, where the temperature difference T − Tc is called the detuning. This detuning is introduced to analyze the stability of the MCRG method regarding the uncertainty in Tc, which naturally arises in systems where Tc has to be determined from simulations and is not known analytically. After thermalizing for 50 · 103 lattice sweeps, 15 · 103 measurements are obtained. Between the measurements, 25 lattice sweeps are performed to decorrelate the measurements. The results of this simulation, in particular the MCRG estimates for the critical exponents ν and η versus the temperature, are shown in Fig. 4.6. 4.2 determination of universality classes 55

An excellent agreement between the MCRG estimates for the critical exponents and the analytically critical exponents can be seen. Specifically, at the third iteration, both, Swendsen’s and Wilson’s interaction set provide a less than 1% deviation from the exact critical exponents for no detuning. In other words, the estimates from Swendsen’s interaction set (orange points) and Wilson’s interaction set (blue points) are very close to the exact critical exponents marked with a black cross at Tc. However, at larger detuning, i. e., away from T = Tc, more MCRG iterations worsen the result, which is expected since the RG fixed-point associated with the phase transition is unstable. Therefore, any initial detuning is exponentially increased with every MCRG iteration. Our newly provided interaction set is reasonably competitive in the first two iterations but, since it contains longer ranged interactions, the lattice obtained in the last RG step is too small to host all interactions and therefore no estimate for the critical exponents could be obtained. In comparison, all three interaction sets behaved similarly well for the Ising model, even though there is a vast difference in complexity. However, going to even larger interaction sets makes the algorithm numerically ill-behaved, as the MCRG matrices defined in Eq. (3.19) become ill-conditioned. Therefore, smaller interaction sets are typically preferred over larger ones, as long as the relevant interactions are included. To summarize, precise estimates for the critical exponents of the two- dimensional Ising model can be obtained. However, the precision crucially depends on the input parameters. Namely, the estimate for Tc has to be very accurate, especially for later MCRG iterations. Furthermore, not all interaction sets provided convergence, such that, for example, no competitive interaction set could be found in the odd sector. In conclusion, under the right circumstances, the MCRG algorithm can provide highly precise estimates for critical exponents but fails to do so for a wrong set of parameters.

4.2.2 Dipolar XY spins on the square lattice Having established in Section 4.2.1 that high-precision estimates for critical exponents can be obtained for the Ising model, the same procedure is now attempted for the truncated dipolar-coupled XY (tdXY) model on the square lattice, as there is an ongoing debate concerning which universality class this model belongs to. In this section, we attempt to determine the universality class using FSS described in Section 4.2.2.1 and the MCRG algorithm described in Section 4.2.2.2. 56 applications of monte carlo simulations

1.3 0.32 1.2 0.3

1.1 0.28 0.26 ν 1 η Wilson (1975) 0.24 0.9 Swendsen (1979) 0.22 Wilson (1975) 0.8 This work 0.2 Swendsen (1979) exact exact 0.7 0.18 2.1 2.15 2.2 2.25 2.3 2.35 2.4 2.45 2.1 2.15 2.2 2.25 2.3 2.35 2.4 2.45 T T (a) 1st MCRG iteration

1.3 0.32 1.2 0.3

Figure 4.6: MCRG results for the Ising model on the square lattice, with a detuning from the critical temperature. Points where the MCRG procedure did not converge, are not shown. Here, the exact critical exponents are indicated at the exact critical temperature with a black cross. The blue data points correspond to MCRG estimates for the critical exponents using Wilson’s interaction set [89], and the orange data points using Swendsen’s interaction set [119]. Additionally, a new interaction set was used for the even sector which is plotted with green data points. This interaction set is tabulated in TableB. 1. The comparison of the MCRG estimates to the exact values should be made at T = Tc ≈ 2.269. 4.2 determination of universality classes 57

Hence, the truncated dipolar-coupled XY (tdXY) model on the square lattice is simulated for several temperatures and lattice sizes. Since each spin degree of freedom is described by a vector on the unit circle, the spin can be parametrized by the polar angle, i. e., the angle between the vector and the x-axis. This parametrization can be used in the Hamiltonian in Eq. (2.6), to obtain the Hamiltonian D 1 1 3 XY = − − + + − HDipol ∑ 3 cos θi θj cos θi θj 2ψij ,(4.4) 2 i6=j rij 2 2 rij6rcut

where ψij denotes the polar angle of the vector~rij and θi is the polar angle ~ of Si. Here, a is the nearest-neighbor distance so that the intrinsic energy scale of the system is given by D/a3 so that we use natural units in which D = a = 1. Furthermore, the Hamiltonian in Eq. (4.4) is truncated to accelerate the simulations. As discussed in Ref. [67], the truncation has to be chosen carefully in order to avoid artifacts. Indeed, if rcut = 1, the system behaves as if it would undergo a Berezinskii–Kosterlitz–Thouless (BKT) transition [65]. However, any truncation larger than a nearest-neighbor truncation seems to be sufficient, as the estimates of the critical exponents do not seem to change with a larger cutoff radius according to a preliminary FSS analysis for rcut ∈ {1.8, 2, 3}). For the remainder of this chapter, we focus on the tdXY model with rcut = 2, which includes 12 neighbors per site, assuming that this is sufficient to obtain the critical exponents of the dXY model without truncation. The (t)dXY model undergoes a thermally driven continuous phase transition if it is placed on the square lattice [30]. As the square lattice is two- dimensional, the interactions are effectively short-range. Since short-range magnetic systems in two-dimensions only exhibit one of the following be- haviors [59]: • Ising-universality class • (finite-size) Berezinskii–Kosterlitz–Thouless (BKT) universality class4

4 This universality class only appears for ﬁnite systems, either experimentally or numerically. Indeed, this universality class disappears in the thermodynamic limit and is replaced with a topological phase transition [159, 160], that does not break any symmetry in agreement with the Mermin-Wagner theorem [161]. Therefore, there is no order parameter or critical exponents associated with this transition. However, in ﬁnite systems or systems with periodic boundary conditions, such models exhibit similar behavior to a second order phase transition associated with a spontaneous symmetry breaking [162] so that critical exponents can be assigned. 58 applications of monte carlo simulations

Ising (ﬁnite-size) BKT [58] 1 β = 1 + c 8 0.23 8 h4 | |

Figure 4.7: Illustration of the behavior of the critical exponent β in the XYh4 model. According to [58, 59], β is in the interval [1/8, 0.23]. The critical exponents of the dXY model on the square lattice is speculated 1 to lie close to or at the point β = 8 .

• XY-model with a four-fold anisotropy (XYh4) universality class

The dXY model on the square lattice has to be in one of these three universality classes. Non-magnetic systems additionally exhibit Potts-model behavior. However, this is not relevant for the dXY model. The XYh4 universality class arises if a two-dimensional XY spin model, coupled via the Heisenberg Hamiltonian from Eq. (2.5), has in addition a local four-fold anisotropy [Eq. (2.4) with p = 4 and anisotropy strength h4]. In the limit h4 → 0, the BKT transition is recovered, and in the limit, h4 → ∞ an Ising-universality class transition is observed [128, 163–165]. For intermediate values, i. e., 0 < h4 < ∞, the anisotropy ﬁeld h4 acts as a marginal operator, continuously tuning the critical exponent β between the two extremal values. This behavior is summarized in a sketch in Fig. 4.7. In the (t)dXY model, each of the spins has an XY degeneracy. However, the relevant symmetry of the Hamiltonian is rather associated with a four- fold rotation axis, which makes the BKT universality class unlikely [58, 59]. Indeed, in previous publications, it was indicated that the phase transition of the dXY system is either an Ising [30, 141, 142, 156] or an XYh4 universality class [143–145]. Based on symmetry arguments, the XYh4 universality class seems to be more likely to describe the transition of the dXY model on the square lattice. Indeed, the dipolar Hamiltonian in Section 2.2.4 inherits the symmetry of the underlying lattice and in the square lattice system, the symmetry group is C4v. The fourfold rotation symmetry suggests that the transition is in the XYh4 universality class rather than in the Ising universality class. However, as the XYh4 universality class coincides with the Ising- universality class in the limit of h4 → ∞, it remains unclear whether the limit is saturated or not. If the limit is saturated, the dXY model on the square lattice undergoes a proper Ising transition, but if h4 is ﬁnite, the transition belongs to the XYh4 universality class. 4.2 determination of universality classes 59

Figure 4.8: The magnetic order of the (t)dXY spin system on the square lattice, at a ﬁnite but low temperature, is given by one of four striped conﬁgurations along the lattice directions. Two are depicted here, the other two follow by time reversal.

4.2.2.1 Finite-size scaling The first attempt to differentiate between the two universality classes relies on FSS. In order to apply FSS, the order parameter of the transition has to be identified first. Since the (t)dXY model on the square lattice orders in a striped phase as depicted in Fig. 4.8,

1 yi xi mstag = ∑ ((−1) cos(θi), (−1) sin(θi)) ,(4.5) N i can be used as the order parameter of the transition [143]. Here, θi is the ~ polar angle of the spin Si, and N is the number of spins in the simulated system. Hence, mstag is the staggered magnetization density, which is normalized so that mstag|T=0 = 1. This order parameter will be used as the basis to derive other order parameters in Chapter 6, where the phase diagram of the tdXY model on the square lattice for temperature and positional disorder will be determined. The Monte Carlo results for the tdXY model on the square lattice are shown in Fig. 4.9. The results for the order parameter are presented in Fig. 4.9a and for the associated susceptibility in Fig. 4.9b. Since the model is reasonably frustrated, parallel-tempering simulations are employed and a relative large number of lattice sweeps (20 · 103) are necessary to thermalize the system. In total, 50 · 103 lattice sweeps are carried out with a measurement happening every 15 lattice sweeps. By locating the crossing of the Binder cumulant curves in Figs. 4.9c and 4.9d, the best estimate for the critical temperature (at rcut = 2) is determined to be Tc = 0.968 ± 0.01. As Tc is cutoff dependent, this value cannot be compared directly to literature values, which used no cutoff or a different cutoff. However, this estimate for Tc can be compared to literature values for Tc where a different rcut was applied. It is expected that that Tc decreases when rcut is increased, as the 60 applications of monte carlo simulations

frustration in the system grows. Since the estimate of Tc(rcut = 2) obtained in this thesis is between previous results of Tc(rcut = 1) ≈ 1.48 [65] and Tc(rcut = ∞) ≈ 0.7 [142, 143], our estimate is consistent with the expectation of the growing frustration. Using the estimate of the critical temperature Tc, an FSS collapse of the data can be attempted, analogous to the collapse of the data for the Ising model presented in Section 4.2.1.1. The collapse of the data is relatively poor with Ising critical exponents (cf. Table 4.1), as shown in Fig. 4.10. Indeed, the collapse is not as good as the collapse seen in Fig. 4.5. This lack of a good collapse can have different reasons: the determination of Tc presented here is based on a Binder cumulant approach. How- ever, as Tc shifts due to corrections to scaling, the real Tc might be different. However, as no temperature could be found where the collapse is signiﬁ- cantly better, the possibility that a wrong estimate for the temperature was used seems unlikely. Another explanation why the Monte Carlo data for the tdXY model collapses poorly compared to the Ising model is given by the possibility that the temperature region where universal scaling behavior can be observed is smaller in the tdXY model than in the Ising model, which could be enhanced due to the tail of the dipolar interaction and the inherent frustration. The other non-universal behavior that could emerge is due corrections to scaling, whose effects could be reduced by implementing simulations of considerably larger systems. Finally, wrong estimates for critical exponents can prohibit the FSS collapse. Indeed, a large statistics FSS analysis of the tdXY model with several different rcut might reveal that a weak dependence of the critical exponents on the cutoff persists so that the tdXY model would exhibit a different universality class than the dXY model. If the critical exponents of the (t)dXY model are not Ising critical exponents, but XYh4 critical exponents with h4 < ∞, the behavior observed in Fig. 4.10 is expected. While the last reason seems the most physical, no set of estimates for the critical exponents, obeying the scaling relations in Eq. (2.18), provided a signiﬁcantly better collapse than the collapse with Ising critical exponents shown in Fig. 4.10. Hence, to conclude, if the critical exponents differ from Ising critical exponents, the critical exponents have to be determined with higher precision than achievable by FSS.

4.2.2.2 Monte Carlo renormalization group In order to determine whether there is a deviation from the Ising critical exponents, the MCRG algorithm can be employed to provide more accu- 4.2 determination of universality classes 61

1 40 L=12 0.8 L=12 L=16 0.6 L=16 L=24 χ stag 20 L=32 m 0.4 L=24 L=32 L=48 0.2 L=48 0 0.5 1 1.5 0.5 1 1.5 T T

(a) MC results for the order parameter mstag (b) MC results for the susceptibility corresponding to mstag

0.6600 0.6575 0.6550 0.6 L=12 0.6525 L=12 L=16 U 0.6500 U 0.5 16 L=24 0.6475 L= L=24 L=32 0.4 0.6450 L=32 L=48 0.6425 L=48 0.6400 0.5 1 1.5 0.94 0.95 0.96 0.97 0.98 0.99 T T (c) MC results for the Binder Cumulant (d) MC results for the Binder Cumulant (zoomed)

Figure 4.9: Monte Carlo (MC) results for the tdXY model on the square lattice with size L and a cutoff radius of rcut = 2: (a) order parameter, (b) associated susceptibility, (c) Binder cumulant, with a zoom-in of the relevant region in (d). The estimate for the critical temperature is Tc = 0.968 ± 0.01 determined by the Binder cumulant. 62 applications of monte carlo simulations

1.4 1.2

ν 1.0

/ L=12 β

L 0.8 L=16 0.6

stag L=24

m 0.4 L=32 0.2 L=48 0.0 −15 −10 −5 0 5 10 15 TL1/ν

(a) FSS collapse for the order parameter mstag

·10−2 0.05 L=12 0.04 L=16 ν /

γ 0.03 L=24 −

L 0.02 L=32 χ L=48 0.01 0.00 −8 −6 −4 −2 0 2 4 6 8 10 TL1/ν (b) FSS collapse for the susceptibility corresponding to mstag

0.65 0.60 L=12 0.55 L=16

U 0.50 0.45 L=24 0.40 L=32 0.35 L=48

−15 −10 −5 0 5 10 15 TL1/ν (c) FSS collapse for the Binder Cumulant

Figure 4.10: FSS collapse of the Monte Carlo data for the tdXY model on the square lattice. Here, Ising critical exponents are used as well as our 3 best estimate for the critical temperature Tc/(D/a ) = 0.968 ± 0.01. The non-scaled Monte Carlo data is shown in Fig. 4.9. 4.2 determination of universality classes 63

rate estimates of the critical exponents. This proves more difficult than anticipated, as will be shown in this section. Similar to Section 4.2.1.2, a Monte Carlo simulation of the tdXY model is performed at the critical temperature. However, since there is no analytic result, the critical temperature is determined from the Binder cumulant crossing in Fig. 4.9d to be Tc = 0.968 ± 0.01. To test against deviations of the estimate for the critical temperature from the true critical temperature, again a detuning in the temperature is applied analogously to Sec- tion 4.2.1.2. The same three interaction sets as those used in Section 4.2.1.2 are implemented but, instead of the tie-breaker rule, two different reduction schemes are applied, namely the 2 × 2 decimation rule and the 2 × 2 block spin rule introduced in Section 3.4.2.1. The results of applying the MCRG method to the tdXY model on the square lattice are presented in Fig. 4.11, where no convergence can be observed. From Section 4.2.2.1, it is known that the dXY model on the square lattice shows a critical behavior that is described by critical exponents close to those of the Ising model. However, the MCRG predictions for the critical exponents are far away from the Ising critical exponents for η and very noisy for ν as seen in Fig. 4.11, leading to the conclusion that the algorithm did not converge.5 The lack of convergence can be due to two possible reasons: either it is a technical issue, or there is a deeper physical reason for the inability of MCRG to find physical estimates of critical exponents. The simplest explanation is that there is a technical reason for this failure. As seen in Sec- tion 4.2.1.2, the MCRG method crucially depends on details of the setup, such as the interaction set, the reduction scheme, and the correct estimate of the critical temperature. As the temperature has been varied broadly around the best estimate for the true Tc, i. e., more than ten times the estimated error of Tc, it is unlikely that a wrong estimate for Tc produces this lack of convergence. The reduction schemes are more likely to provide an issue, since the tdXY model on the square lattice has a magnetic unit cell which is larger than a structural unit cell, such that real-space renormalization schemes need to be crafted more carefully. Nevertheless, two different reduction schemes have been tested, which should have worked, but neither of them showed any convergence. Finally, the possibly most significant technical issue is the interaction sets used. By using an interaction set, the RG flow is projected into a finite-dimensional space. However, it is

5 The exponents η and ν are derived from the odd and even sector of the matrix T in Eq. (3.18) respectively. These two sectors do not mix, as the model is time-reversal symmetric so that T is block-diagonal. The details have been discussed more extensively in Section 3.4.2.3. 64 applications of monte carlo simulations

1.3 0.5

1.2 Decimation, Wilson (1975) 0.4 Blockspin, Wilson (1975) 1.1 0.3 Decimation, Swendsen (1979) ν 1 η Blockspin, Swendsen (1979) 0.2 Decimation, This work 0.9 Blockspin, This work 0.8 0.1

0.7 0 0.8 0.85 0.9 0.95 1 1.05 1.1 1.15 0.8 0.85 0.9 0.95 1 1.05 1.1 1.15 T T (a) 1st MCRG iteration

1.3 0.5

1.2 Decimation, Wilson (1975) 0.4 Blockspin, Wilson (1975) 1.1 0.3 Decimation, Swendsen (1979) ν 1 η Blockspin, Swendsen (1979) 0.2 Decimation, This work 0.9 Blockspin, This work 0.8 0.1

0.7 0 0.8 0.85 0.9 0.95 1 1.05 1.1 1.15 0.8 0.85 0.9 0.95 1 1.05 1.1 1.15 T T (b) 2nd MCRG iteration

1.3 0.5

1.2 Decimation, Wilson (1975) 0.4 Blockspin, Wilson (1975) 1.1 0.3 Decimation, Swendsen (1979) ν 1 η Blockspin, Swendsen (1979) 0.2 Decimation, This work 0.9 Blockspin, This work 0.8 0.1

0.7 0 0.8 0.85 0.9 0.95 1 1.05 1.1 1.15 0.8 0.85 0.9 0.95 1 1.05 1.1 1.15 T T (c) 3rd MCRG iteration

Figure 4.11: MCRG results for the tdXY model on the square lattice, with a detuning in the critical temperature. Points where the MCRG procedure did not converge, are not shown. Especially for η in the first iteration (shown in Fig. 4.11a), the procedure did not seem to converge at all and only one point is given. In contrast, the data for ν is randomly scattered. It is expected that the values for ν are close to Ising critical 1 values, and the values of η should be exactly 4 for both possible universality classes of the dXY system [59]. The Ising critical exponents are indicated with a dashed line. From the figures it is therefore evident that no convergence can be observed. 4.2 determination of universality classes 65 a priori unclear what the relevant interaction space is, so that if there was a relevant interaction missing, no convergence could have been obtained. There is also a possible physical reason as to why no convergence could be observed. Namely, if the system is indeed in the universality class of an XYh4 model then the system crucially depends on a marginal operator. For marginal transitions, previous literature showed that the MCRG algorithm does not converge without special tricks in the reduction scheme. These tricks typically renormalize all directions orthogonal to the marginal operator while leaving the marginal operators unchanged [121, 127, 136] (cf. Section 3.4.2.4). However, to implement such a procedure, typically the marginal operator has to be identified first. Indeed if a marginal operator can be identified, then the dXY system is known to belong to the XYh4 universality class, and the precise determination of the critical exponents would be unnecessary.

4.2.3 Conclusion of the Universality To summarize this section, the determination of critical exponents has been attempted for two different systems using two different methods. For the Ising model, this is a simple exercise, and known results could be reproduced [120, 166]. Indeed, the FSS results obtained for the tdXY model on the square lattice with rcut = 2 agrees with previous ﬁndings for the (t)dXY model [141, 142, 156], i. e., critical exponents comparable to Ising critical exponents are found within errors. However, the attempt to improve upon the FSS result using the MCRG algorithm did not provide estimates for the critical exponents anywhere close to physical results indicating that the method did not converge. As outlined in Section 4.2.2.2, the reason why the simulation did not converge either originates from a technical issue, or it is because a marginal operator in the universality class prohibits convergence. While many different input parameters have been tested, there is still a large space open to exploration, especially concerning the choice of the interaction sets. Despite the technical possibilities, the consistent failure of the MCRG method is a strong indication that the reason for the failure is not technical, but is based on the physics of the problem. This reasoning suggests that the ﬁnite-temperature phase transition of the dXY model on the square lattice is indeed described by an XYh4 universality class and that the marginal operator prohibits any reasonable convergence of the MCRG algorithm.

5 UNDERSTANDINGTHE µ -SPINROTATION EXPERIMENTS

The results of this chapter are published in [11]: “Collective magnetism in an artificial 2D XY spin system”, N. Leo, S. Holenstein, D. Schildknecht, O. Sendetskyi, H. Luetkens, P. M. Derlet, V. Scagnoli, D. Lançon, J. R. L. Mardegan, T. Prokscha, A. Suter, Z. Salman, S. Lee, and L. J. Heyderman, Nat. Comm. (2018). In this chapter, the dipolar-coupled XY (dXY) model and its finite- temperature phase transition are revisited. This model has already been discussed in Section 4.2.2. An experimental realization of this model can be achieved with an Artificial Spin Ice (ASI). By using superparamagnetic permalloy discs, the magnetic moment will naturally lie in-plane due to the shape anisotropy of the nanomagnets. Since each of the nanomagnets carries a large dipole moment, the inter-disc interaction can be approximated to lowest order by the dipolar interaction. To probe such ASI systems, highly sensitive methods are required due to the small size of the nanomagnets and the associated fast fluctuation rates. A well-suited technique is µ-spin rotation (µSR), which uses short-lived muons (µ+) as a local probe of the magnetic fields. This technique requires, however, a theoretical model for the interpretation of the experimental results. In this chapter, the model of the dXY ASI and the associated µSR experiment is presented, which has been developed as part of Ref. [11]. In order to provide the interpretation of the experiments, this chapter is structured as follows: A short introduction to µSR is provided in Sec- tion 5.1. In order to explain the experimental data presented in Section 5.2, a model for the depolarization of the muon spins is developed in Sec- tion 5.3. This model is then compared to the experiment in Section 5.4. In Section 5.5, the chapter is concluded with some remarks about the model. Furthermore, an outlook of possible extensions to this work is provided.

5.1 the µ-spin rotation as a local magnetic field probe

The µSR technique is a local magnetic probe using short-lived muons as a local probe sensitive to small magnetic ﬁelds down to 40 µT [167] and ﬂuc-

67 68 understanding the µ-spin rotation experiments

tuations of magnetic ﬁelds with frequencies between MHz and GHz [168, 169]. This sensitivity makes µSR an ideal tool for modern experimental condensed matter physics, for example, in the determination of local magnetic ﬁelds [167, 168, 170]. Since a thorough discussion of µSR would be beyond the scope of this thesis, the interested reader is referred to textbooks such as [169]. Here, we review only the µSR techniques relevant to Ref. [11].

5.1.1 General working principle of µ-spin rotation We can consider the sketch in Fig. 5.1: In order to perform a µSR experiment, the sample is placed in a spin-polarized µ+-beam. The muons are stopped in the sample and typically reside at interstitial points of the crystal lattice [171, 172], where the spin of the muon precesses in the local ﬁeld ~B according to the Landau-Lifshitz equation [173] for muons ~ ~ ~ ∂tSµ = γµ+ Sµ × B,(5.1)

so that the magnetic component perpendicular to the spin leads to a precession.1 The muon spin precesses inside the sample according to the local magnetic ﬁeld until it decays via the decay channel

+ + µ → e + νe + ν¯µ,(5.2)

due to the weak nuclear interaction. Its mean lifetime is τµ = 2.2 µs. Since the weak interaction breaks parity [174, 175], the positron e+ is emitted preferentially along the spin direction of the muon with a theoretical asym- 1 metry between forward and backward emittance of Amax = 3 when integrated over all energies. As the muon spin ensemble will depolarize in the sample over time, the asymmetry A(t) contains information about the local magnetic ﬁelds experienced by the muons. In an experiment, the time dependence of the asymmetry is measured using two detectors, one in the direction of the initial spin direction and one in the opposite direction. Given the same

1 It should be noted that there is also a different convention for writing the Landau-Lifshitz equation, which is often applied when particles with a negative gyromagnetic ratio γ are used. There, the negative sign is often highlighted in the equation by using the absolute value of the gyromagnetic ratio and writing the minus sign explicitly. 5.1 the µ-spin rotation as a local magnetic field probe 69

~ Sµ(0) µ+ eetrfor Detector − N N + Detector for

Figure 5.1: This sketch illustrates the experimental setup of a µSR experiment. The sample is placed in a spin-polarized beam of muons, so that the muons are captured in the sample. Subsequently, the spins of the muons start to precess due to the presence of a local magnetic ﬁeld until the muon eventually decays. The positron (e+) from the decay is emitted preferentially along the spin direction and may enter either the detector for N+ or N−. efﬁciency and an equal solid angle for both detectors, the asymmetry is determined by

N+ − N− A = ,(5.3) N+ + N−

1 which is typically below the theoretical value of 3 due to experimental limitations such as detector efﬁciencies at different energies [171]. The normalized asymmetry is called the polarization function

1 N+ − N− P(t) = ,(5.4) Amax N+ + N− which is directly related to the polarization of the muon spin since ~ ~ P(t) = hSµ(t) · Sµ(0)i,(5.5) ~ where Sµ(t) is the direction of the spin of the muon at time t. Using Eqs. (5.4) and (5.5), the average time evolution of muon spins can be obtained. Hence, information about the local magnetic ﬁeld at the muon 70 understanding the µ-spin rotation experiments

site can be inferred. In the simple case that the magnetic ﬁeld is static at the muon site, the muon spin undergoes a Larmor precession giving the polarization function 2 2 ~ P(t) = cos (θ) + sin (θ) cos(γµ+ |B|t),(5.6)

−1 where γµ+ /(2π) = 135.54 MHzT is the gyromagnetic ratio of muons [176, 177], and θ is the angle between the local magnetic ﬁeld and the initial spin direction [169]. If several (magnetically) inequivalent sites are present, then Eq. (5.6) has to be averaged over these sites.

5.1.2 The µ-spin rotation experiments on Artificial Spin Ice The method described in Section 5.1.1 does not work for thin-film ASI systems, as the typical muon beams are too energetic and hence have a large penetration depth. Therefore, to apply µSR to ASI systems the energy needs to be moderated from the typical 4.2 MeV for so-called “Ari- zona” muons [178] to a few keV so that the muons get trapped in a thin- film structure. Such a low-energy beam can be provided by the low-energy muon beamline LEM at the Swiss Muon Source [168, 172, 179]. Addition- ally, a (gold) stopping layer is placed on top of the ASI system in which the muons are implanted. Therefore, the muons probe the magnetic stray field above the ASI structure. Hence, a combination of a thin stopping layer as sketched in Fig. 5.2 and low-energy muons provide a viable tool to measure the local magnetic fields associated with ASI systems such as the kagome spin ice [56]. Since, in artificial spin ice systems, the muons are now spread over many different relative positions, they experience a broad magnetic field distribution, leading to exponential damping of the muon-spin polarization [167] rather than an oscillation as expected from Eq. (5.6) for a static spatially constant field. Additionally, the samples are thermally active and, hence, dynamic fluctuations add to the depolarization of the muon spin. There- fore, a theoretical model of the spin depolarization is necessary to interpret experimental data.

5.2 experiment on dipolar xy spin systems

For the speciﬁc ASI system considered in Ref. [11], permalloy discs on a square lattice were patterned on top of a non-magnetic substrate. Sev- eral samples were fabricated with typical disc thicknesses of only a few nanometers, diameters ranging between 35 and 70 nm, and lattice period- 5.2 experiment on dipolar xy spin systems 71

~ Sµ(0) µ+ eetrfor Detector − N Stopping Layer N +

Detector for Substrate

Figure 5.2:A µSR experiment on ASI is performed using a similar geometry as Fig. 5.1, except that here the muons have smaller implantation energies and a gold capping layer is added on top of the ASI system which efficiently stops the muons. Therefore, the muons are implanted in to the non-magnetic layer and their spin precesses around the local magnetic stray field of the ASI system. icities between 55 and 100 nm. The parameters for the sample are tabulated in Table 5.1, and a Scanning Electron Microscopy (SEM) image of sample 2 (with a disc diameter of 40 nm and a lattice periodicity of 70 nm) is given in Fig. 5.3. Due to the shape anisotropy, the magnetic moment of each of the discs is well-confined in-plane. Since each of the discs carries a rather large magnetic moment, the effective interaction of the spins is dipolar, and hence the theoretical model description is based on XY spins interacting with the dipolar interaction, again denoted by dXY. This system was already introduced in Section 4.2.2 where it was established that, at low temperatures, the dXY system on the square lattice orders in a striped phase such as depicted in Figs. 4.8 and 5.4 [30]. In the µSR experiment, a control sample is crucial for disentangling different depolarization effects. The effectively non-interacting control sample is obtained by doubling the lattice periodicity of the corresponding interacting sample. The lattice periodicity for the non-interacting sample is also tabulated in Table 5.1. Since the dipolar interaction in Eq. (2.6) decays as r−3, where r is the distance between the center of the nanomagnets, the doubling of the lattice periodicity leads to an eightfold reduction of the interaction strength, which renders the samples effectively non-interacting. 72 understanding the µ-spin rotation experiments

Table 5.1: The relevant parameters are summarized for the three samples considered in [11]. Here, d is the diameter of the individual nanomagnets, and a is the lattice periodicity, which is reported for the strongly interacting sample and the non-interacting sample. These parameters lead to p percent of muons sampling time-averaged magnetic ﬁelds according to Eq. (5.11) in the strongly interacting sample. strongly interacting non-interacting Sample Set d [nm] p a [nm] a [nm] Sample Set 1 70 40% 100 200 Sample Set 2 40 60% 70 140 Sample Set 3 35 71% 55 110

Figure 5.3: SEM image of the (strongly interacting) sample 2 (see Table 5.1) used for the work in Ref. [11] (Adapted from the data repository [180]). 5.2 experiment on dipolar xy spin systems 73

Figure 5.4: The temperature selected phases of the dXY spin system on the square lattice are given by striped spin conﬁgurations along the lattice directions. (Same ﬁgure as Fig. 4.8).

0.8 ) t ( P 0.6

T = 10 K 0.4 T = 50 K T = 100 K T = 150 K 0.2 0 1 2 3 4 5 6 t [µs]

Figure 5.5: Example of the measured muon spin polarization function for a dXY ASI with lattice periodicity of 70 nm and a diameter of the permalloy discs of 40 nm (strongly interacting sample 2, cf. Table 5.1).

Examples for the time- and temperature-dependent muon-polarization function measured in the experiments on the strongly interacting sample 2 (cf. Table 5.1) are given in Fig. 5.5. The data for the interacting samples reveals that there are (at least) two exponentially damped signals plus a constant background, such that the polarization is ﬁtted with

λslowt λfastt P(t) = g0 + gslowe + gfaste ,(5.7) where the notion of fast and slow is determined by their typical time scales, −1 −1 namely λfast ∼ 0.5 µs and λslow ∼ 3 µs. Fitting of the muon data for the strongly interacting samples with Eq. (5.7) at every measured temperature yields the temperature dependence of λslow and λfast, which is presented in Fig. 5.6. The temperature dependence of the depolarization rates reveals three temperature regimes for the strongly interacting samples: At high temperatures, all the nanomagnets are ﬂuctuating, and no long-range order is established. At low 74 understanding the µ-spin rotation experiments

Set 1 Set 2 Set 3

20 ] 1

s 15 [

t s

a 10 f

0.7

0.6 ]

1 0.5

s 0.4 [

o 0.3 l s 0.2

0.1

0.0 0 20 40 60 80 100 120 140 0 20 40 60 80 100 120 140 160 0 50 100 150 200 250 T [K] T [K] T [K]

Figure 5.6: Fitting the experimental muon spin polarization data such as that provided in Fig. 5.5 with the ansatz for the polarization function Eq. (5.7) for the three different strongly interacting samples in Table 5.1 reveals the two depolarization rates λfast and λslow. The background color (blue, yellow, red) indicates the frozen-in regime, the phase where long-range order emerges and the paramagnetic phase, respectively. The striking feature is the order-parameter-like behavior of λslow.

temperatures, all the nanomagnets are static on the time scale of the muons and align along random local easy-axis anisotropies so that the muons probe a random static ﬁeld.2 The strongly interacting samples have furthermore an intermediate temperature window, where the mutual interaction of the moments leads to the emergence of long-range order. The emergence of long-range order is further established by complementary experiments [11] using the soft x-ray magnetic resonant scattering setup at the RESOXS endstation [181] of the SIM beamline [182] at the Swiss Light Source. Simultaneous to the emergence of long-range order, in Fig. 5.6, it can be observed that λfast shows a pronounced peak. Furthermore, it can be seen that λslow exhibits a temperature dependence resembling the temperature

2 This is conﬁrmed by the measurements for the non-interacting samples that also exhibit the same behavior for low temperatures, as this originates in the single nanomagnet behavior rather than the collective interaction among the nanomagnets [11]. 5.3 depolarization model of dipolar-coupled xy spin systems 75

dependence of an order parameter. While this resemblance is striking, it is not easily understood as there is no apparent reason why the exponential damping of a dynamic µSR signal should reveal the appearance of a static order parameter without the typical oscillations in the polarization function observed for long-range ordered systems [169]. Therefore, a theoretical model of the depolarization is required, in order to associate the exponential damping observed in the measurements with the order parameter of a theoretical model.

5.3 depolarization model of dipolar-coupled xy spin systems

Here, an attempt is made to model the depolarization of the muon spin ensemble due to the emerging stripe order in the experimental realization of the dXY systems. If the magnetic fields causing the depolarization would be static on the time scale of the muons, i. e., the fluctuations of the nanomagnets happen on timescales larger than 10 µs, then Eq. (5.6) would be sufficient to describe the time-evolution of a muon spin. However, the gyro- −1 magnetic ratio of permalloy is γPy/(2π) = 29.5 GHzT , i. e., much larger −1 than the gyromagnetic ratio of muons, γµ+ /(2π) = 135.54 MHzT , so that the nanomagnets typically fluctuate much faster than the muon spin. Therefore, the muons only probe a time-averaged field. The applicability of this approximation is discussed in Section 5.3.1. Subsequently, the time-averaged magnetic field is determined via the ergodicity hypothesis (see Section 2.5), so that it is sufficient to determine the thermally averaged magnetic field. Hence, the depolarization of the muon spin can be determined from the thermal average of the magnetic field generated by the array of nanomagnets. The thermal averages are determined in Section 5.3.2 using a mean-field (MF) calculation for the dXY model on the square lattice. Then, in Section 5.3.3 the prediction for the experimentally observed polarization functions is obtained by a spatial average of the muon stopping sites.3

3 It should be noted that the model that is developed in this section does not require a ﬁt to the experimental µSR data. The only input required from the muon experiment is the critical temperature Tc. Other parameters required for the theoretical model of the depolarization are determined by independent measurements. For example, the magnetization density was determined by a SQUID measurement. 76 understanding the µ-spin rotation experiments

5.3.1 Applicability of time averaging the magnetic field In order to use static quantities, such as the order parameter, to describe the (dynamic) µSR measurement, we first have to develop a criterion that determines which muons probe time-averaged quantities rather than dynamic fluctuations. This time averaging happens if the precession frequency of the muons is much smaller than the typical time scale of the fluctuations of the nanomagnets. As it will be shown in this section, this is a crude approximation, but it will prove sufficient to describe some of the experimental data discussed in Section 5.2, specifically the data presented in Figs. 5.5 and 5.6. Here, the typical fluctuation frequencies of the muons and the nanomagnets are estimated in order to obtain a criterion for whether the muons probe the ensemble average. For this, the geometry of the experiment depicted in Fig. 5.2 can again be considered: A muon at height z above the nanomagnets precesses with a typical frequency of

γµ+ MV ω = γ + |~B| ∝ ,(5.8) µ µ z3 where M is the saturation magnetization of the nanomagnets and V its volume. Analogous to the estimate for the typical time scale for the muon precession described by Eq. (5.8), the typical fluctuation frequency of the permalloy discs can be estimated: Assuming that the nanomagnets are isotropic in-plane, no internal energy barriers (such as encountered in Sec- tion 4.1) have to be overcome and, therefore, the typical fluctuation frequency of the permalloy discs is temperature independent. Then, the fluctuation frequency of the permalloy discs can be estimated to be

γPy MV ω ∝ ,(5.9) disc a3 −1 where γPy/(2π) = 29.5 GHzT [183], and a is the lattice periodicity reported in Table 5.1. Hence, muons only probe time-averaged quantities if

γµ+ MV ω 3 γ + a 3 a 3 γ µ = z = µ ⇔ Py γ MV 1 .(5.10) ωdisc Py γPy z z γµ+ a3 Speciﬁcally, in Ref. [11], the criterion for being much smaller was chosen (arbitrarily) as

a 3 1 γPy < ≈ 40. (5.11) z 5 γµ+ 5.3 depolarization model of dipolar-coupled xy spin systems 77

80 80 Set 1, a=100 nm Set 2, a= 70 nm 60 60 Set 3, a= 55 nm

40 40

20 20 Height above nanomagnets [nm] 0 0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 (unnormalized) Distribution Cumulative stopping probability

Figure 5.7: Simulated stopping depth distribution of muons at 14.3 keV implanted in an 80 nm thick gold layer on top of an (inﬁnite) silicon substrate obtained using TRIM.SP [184, 185]. In the left panel, the distribution of stopping depths is given, and in the right panel the cumulative distribution is displayed. The lines indicate where the criterion given by Eq. (5.11) is applicable, such that for muons stopping above the lines, the depolarization is well described by the time-averaged magnetic ﬁeld. The intersections of these lines with the cumulative distribution correspond to the percentage p given in Table 5.1. (TRIM.SP simulations were carried out by Dr. Naëmi Leo)

Combined with the stopping distribution shown in Fig. 5.7, this leads to the percentage of muons that only probe time-averaged quantities. The results of this section are summarized in Table 5.1, with the conclusion that the approximation is typically valid for half of the muons. For these muons, it is sufﬁcient to describe the depolarization using the static local magnetic ﬁeld, which is computed in the following sections.

5.3.2 Mean-field description of dipolar XY spins on the square lattice Depending on the lattice periodicity of the ASI system, it has been shown in Section 5.3.1 that time-averaged quantities are sufficient for 40 − 71% of the muons. As a result of the ergodicity hypothesis, rather than determining the time-averaged quantities, ensemble-averaged quantities can be considered. Here, these quantities are determined with an mean-field (MF) calculation. To perform the MF calculation, it should be remembered that, at low temperatures, the square lattice dXY system forms a striped phase as de- 78 understanding the µ-spin rotation experiments

picted in Fig. 5.4 [29, 30]. Hence, the MF calculations can be restricted to a magnetic unit cell containing four spins in the form of a 2-by-2 plaquette. Then, the spins can be rewritten as their average value plus the deviation ~ ~ ~ from the average value, i. e., Si = hSii + δSi. Hence, the MF assumption is ~ ~ conveniently written as hδSi · δSji = 0. This leads to the MF Hamiltonian ~ ~ HMF = − ∑ Sm · Mml · hSli,(5.12) Plaquettes

where the summation over plaquettes is trivial since the plaquettes are decoupled. The indices m and l denote one of the four sites in the magnetic unit cell displayed in Fig. 5.4 and hence the coupling matrix Mml describes the interaction of the mth spin in the unit cell with the lth. In contrast to the convention used in the paper [11], here M is chosen so that the Hamiltonian has an overall minus sign. Since, in Eq. (5.12), the magnetic unit cells are decoupled, the problem becomes ﬁnite-dimensional so that the partition function can be derived to be

4 ~ ~ ~ ~ Z(β, hSi) = ∏ 2πI0 β|Km| , where Km = ∑ MmlhSli,(5.13) m=1 l

where In denotes the modiﬁed Bessel functions of the ﬁrst kind [186]. Equa- ~ ~ tion (5.13) still includes hSli. However, as hSli is not an independent quan- ~ tity, i. e., it cannot be set by an external parameter, hSli has to be derived from Eq. (5.13) self-consistently. This self-consistency condition leads to the equation

~ Z ~ I1 β|Kl| ~ 1 4 ~ ~ ~ Kl hS i = d Si Si exp[−βHMF(Si, hS i)] = ,(5.14) l ( h~i) l |~ | ~ Z β, S Kl I0 β|Kl|

where the integration is over the phase space of the four XY spins in the plaquette, conveniently described by the polar angles. Equation (5.14) is equivalent to Eq. (6) in the supplemental material of Ref. [11], by writing K~ in its component form and taking the additional minus sign in Eq. (5.12) into account. Equation (5.14) can be solved graphically by plot- ting the right-hand side (S-curve) and the left-hand side (straight line) in the same ﬁgure and determining their crossing point as shown in Fig. 5.8a: ~ For T > Tc, Eq. (5.14) has only the trivial solution hSli = 0, but for temperatures T < Tc non-trivial crossings emerge. Hence, the location and number 5.3 depolarization model of dipolar-coupled xy spin systems 79

1 1 T > Tc 0.5 T = Tc 0.5 T < T

c i i i i S S 0 0 h h

−0.5 −0.5

−1 −1 −2 −1 0 1 2 0 0.5 1 hSii T/Tc (a) (b)

Figure 5.8: The main results of the discussion in Section 5.3.2 are summarized in this figure. Figure 5.8a depicts the graphical solution of Eq. (5.14), where the crossings are marked by green dots. The crossings of Fig. 5.8a are then tracked as a function of temperature, which is depicted in Fig. 5.8b. Above Tc only the trivial solution hSli = 0 exists, whereas below Tc non-zero solutions emerge and the order parameter takes a finite value. of crossings depend on the temperature. Their temperature dependence is displayed in Fig. 5.8b, which shows the typical square-root behavior of the MF order parameter around T = Tc. This MF order parameter is used in the next section to determine the static local magnetic field and, with that, the depolarization of the muon spins.

5.3.3 Depolarization of the µ-spin due to the dXY Artificial Spin Ice In order to describe the muon-spin polarization function P(t, T) for the experimentally measured dXY ASI systems in the intermediate temperature regime, we combine the results of Sections 5.3.1 and 5.3.2. Here it will be shown that the temperature dependence is described by the tempera- ~ ture dependence of the MF order parameter hSii. Subsequently, the time dependence is computed by averaging the precession of all of the muons in the stray field of the nanomagnets according to the height dependent stopping distribution shown in Fig. 5.7. The magnetic field of the nanomagnets, which causes the precession of the muons, can be described by a 80 understanding the µ-spin rotation experiments

point-dipole approximation so that a muon at position ~R precesses according to Eq. (5.1) with the ﬁeld   3(~R −~r ) ~S · (~R −~r ) µ0 MV 1 i i i ~B(~R) =  − ~S  ,(5.15) ∑ ~ 3 ~ 2 i 4π i |R −~ri| (R −~ri)

where the summation is over all sites. With the assumption that the muons ~ only probe the time-averaged field (see Section 5.3.1), Si is replaced by ~ ~ hSii so that the spin polarization of a muon at position R is described by Eq. (5.6), where θ ≡ θ(~R) and |~B| ≡ |~B(~R, T)|. Since in Eq. (5.6), the time t only appears in a product with the field magnitude |~B|, the temperature dependence of the depolarization rate has to be the same as the temperature dependence of |~B|. Furthermore, ~ ~ ~ ~ ~ ~ as B factorizes into B(R, T) = µ0 MV|hSii|~g(R), where |hSii| is the only temperature-dependent factor, it is sufficient to determine ~B(~R, T = 0) ~ and then to rescale the depolarization rates according to |hSii| to obtain the temperature-dependent depolarization rates. Finally, to obtain the spin-polarization function P(t, T), Eq. (5.6) has to be averaged over all positions of the muons. As there is no in-plane pref- erence of the muon stopping sites concerning the ASI unit cell, the lateral positions x and y are uniformly distributed. However, as shown in Fig. 5.7, the stopping depth distribution of the implanted muons follow a more complicated distribution ρ(z). Hence, the polarization function can be obtained by the distribution-weighted average

1 Z a Z a Z zmax ( ) = ( ) ( ~ ) P t, T 2 dx dy dz ρ z P t, R, T ,(5.16) a zmax 0 0 0

where zmax is the thickness of the stopping layer and ρ is the distribution of muon stopping sites along the z direction.

5.4 comparison of the model with the experimental results

The predicted polarization function P(t, T) of the muon spin according to Eq. (5.16) is qualitatively similar to the experimental data as seen in Fig. 5.9. However, in order to compare the temperature dependence of the depolarization, a similar function to Eq. (5.7) has to be ﬁtted to Eq. (5.16). Here,

λ1t λ2t P(t) = g0 + g1e + g2e (5.17) 5.4 comparison of the model with the experimental results 81

1.0 0.9 Theory 0.8 Experiment

) 0.7 t (

P 0.6 0.5 0.4 0.3 0 1 2 3 4 5 6 t [µs]

Figure 5.9: Comparison between the predicted muon-spin polarization from Eq. (5.16) and the experimental data for sample 2 at T = 10 K. is used to fit the predicted depolarization, where the depolarization rates λ1 and λ2 from the MF treatment are not equivalent to λslow and λfast from the experiment shown in Fig. 5.6. Indeed, both λ1 and λ2 share the temperature dependence of the order parameter, but only λslow shows the temperature dependence resembling an order parameter. Nevertheless, two exponential functions are required for the fit of the theoretical prediction. The second contribution λ2 is, however, not observed in the experiment, so that its appearance in the model has to be considered spurious. As g2 ≈ g1/2 and λ2 ≈ 6λ1, the faster depolarization λ2 probably emerges from muons that violate the applicability criterion Eq. (5.11), for which this simplified treatment is insufficient. As the estimate of the critical temperature using an MF is typically only providing an upper bound on the true Tc rather than a reliable estimate, the temperature scale of the depolarization rate is adjusted according to Tc determined in the experiment. In Fig. 5.10 the temperature dependence of λ1 is compared with the experimental data for λslow, where it can be seen that the temperature dependence of λ1 and λslow are similar for all three samples. Furthermore, for two of the three samples, the predicted depolarization rates take similar values as the experimental results. For the first sample with the largest lattice periodicity (a = 100 nm, cf. Table 5.1), the fast-fluctuation criterion in Eq. (5.11) is violated for a larger fraction of muons than for the other samples. As the static MF approach presented here relies on this criterion, it is expected that the least agreement between model and experiment is observed for the first sample. 82 understanding the µ-spin rotation experiments ]

1 1.0

− Theory

s 0.8 1

µ Data [ 0.6 0.4 Set 0.2 slow

λ 0.0 0 50 100 150 200 ]

1 0.7

− 0.6 s

2 0.5 µ

[ 0.4 0.3 Set 0.2

slow 0.1

λ 0.0 0 50 100 150 200

] 0.7 1

− 0.6 s

3 0.5 µ

[ 0.4 0.3 Set 0.2

slow 0.1

λ 0.0 0 50 100 150 200 T [K]

Figure 5.10: Comparison between the experimental data and the depolarization determined from the MF theory using Eqs. (5.14) to (5.17). 5.5 conclusion & outlook 83

5.5 conclusion & outlook

In this chapter, an experimental realization of the dipolar-coupled XY model on the square lattice has been considered in the form of an Ar- tificial Spin Ice system, which was measured by µ-spin rotation (µSR). Here, µSR is used as it is highly sensitive to the small magnetic fields that occur for ASI systems. However, due to the (spatial) average of muon stopping sites, many features of the polarization function are averaged out. Nevertheless, two depolarization channels could be identified: a faster depolarization with rate λfast and a slower depolarization with rate λslow. The temperature dependence of the depolarization rates revealed that the ASI system undergoes a phase transition. Specifically, the temperature dependence of λfast exhibits a peak around the critical temperature, whereas the temperature dependence of λslow resembles the temperature dependence of an order parameter. To provide an interpretation of the temperature dependence of λslow, a theoretical model was introduced in this chapter, which is based on an MF calculation for the order parameter. It could be concluded that time-averaged magnetic fields are sufficient to explain the slow muon-spin depolarization channel. As a result of the ergodicity hypothesis, time-averaged fields could be replaced by ensemble- averaged fields because the nanomagnetic moments fluctuated much faster than the muon spins. Then, the ensemble-averaged fields could be determined using an MF calculation of the order parameter. For two of the three samples, this model resulted in a reasonable agreement of the time- and temperature-dependence of the polarization function. For the third sample, the temperature-dependence is still correct, but the predicted time- dependence had the same order of magnitude as the experimental data. Furthermore, it was argued in Section 5.3.1 that the largest deviation between theory and experiment is expected for this sample as the fluctuations of the nanomagnets are slower (compared to the other samples) and the muons have a higher probability of probing non-time-averaged fields. In conclusion, it has been shown that the slow depolarization rate can be correlated with the order parameter of the phase transition of the ASI system. However, the model has some shortcomings as it predicted a second depolarization channel λ2, which was not observed in the experiment. The appearance of this depolarization channel in the model is probably spurious and originates in muons violating the applicability criterion. Hence, the second component should probably be neglected. Furthermore, the MF model could also not provide an explanation for λfast that has a peak at Tc (cf. Fig. 5.6). Most likely, the fast depolarization 84 understanding the µ-spin rotation experiments

channel seen in the experiment originates from correlated fluctuations not captured with this simplified treatment. This conclusion is further supported by the fact that, according to the criterion Eq. (5.11), the static approach does not apply to a sizable fraction of muons stopped in the gold layer. These muons can probe local fluctuations, which goes beyond the simple treatment presented here. Hence, in order to understand the rapid depolarization of the muon spin observed close to the phase transition, temporal correlations have to be taken into account, such that the stochastic Landau-Lifshitz-Gilbert equation [173, 187, 188] has to be solved for many nanomagnets. This solution could either be obtained from a simulation or by using a less restrictive approximation of the system similar to the treatment by Abragam and Anderson [189, 190]. Despite these shortcomings, a satisfactory quantitative agreement could be extracted for the time- and temperature-dependence of the polarization function. Furthermore, the model developed in this chapter could provide qualitative insight into the underlying physics of the depolarization by directly associating one depolarization rate with the order parameter of the dXY model. 6 PHASEDIAGRAMOFDIPOLAR-COUPLEDXYSPINS ONTHESQUARELATTICE

Part of the results presented in this chapter is published in Ref. [52]: “Phase diagram of dipolar-coupled XY moments on disordered square lattices”, D. Schildknecht, L. J. Heyderman, and P. M. Derlet, Phys. Rev. B (2018). Part of the data presented in this chapter is openly available at Ref. [191]. In this chapter, the dipolar-coupled XY (dXY) model on the square lattice is simulated by Monte Carlo (MC) methods in the presence of disorder. Here, the aim is to provide the phase diagram of dXY spins on the square lattice with respect to temperature and positional disorder. Up to now, only weak disorder was studied theoretically [30, 143, 192, 193], but knowledge about the behavior of the system for larger disorder is crucial for the understanding of experiments of physical realizations such as ASI systems, which are always disordered to a certain amount [10–12]. For the square lattice, as studied in Refs. [10–12], it is qualitatively known that disorder and temperature select competing magnetic phases [30], but quantitative results are limited to vacancy concentrations below 6% [143, 192, 193]. This lack of a phase diagram complicates the interpretation of the experiments, as it is not known if the results from experiments are determined by the effects of temperature or are merely a result of the disorder. Therefore, in this chapter, the quantitative phase diagram is determined, which is accomplished with parallel-tempering Monte Carlo simulations. The structure of this chapter is as follows: First, the literature is reviewed in Section 6.1, and then proper order parameters for the observed phases are determined in Section 6.2. After verifying that the order parameters work even for quite strongly disordered systems in Section 6.3, the truncated dipolar-coupled XY (tdXY) model is analyzed with Monte Carlo simulations. Results are presented for the non-disordered system in Sec- tion 6.4, for vacancy-disordered systems in Section 6.5, and for random- displacement disordered systems in Section 6.6. This chapter is concluded in Section 6.7 by pointing out a striking similarity between the two phase diagrams, which can be understood by considering the magnetic ﬂux closure in dipolar-coupled systems.

85 86 phase diagram of dipolar-coupled xy spins on the square lattice

6.1 dipolar-coupled xy spins on disordered square lattices

Here, we again consider XY spins interacting via the dipolar Hamiltonian as deﬁned in Eq. (2.6). As the spins are parametrized by their polar angle, the Hamiltonian can be simpliﬁed to Eq. (4.4). As in Section 4.2.2, a cutoff is applied to the Hamiltonian to describe the truncated dipolar-coupled XY model: D 1 1 3 H = − cos θi − θj + cos θi + θj − 2ψij ,(6.1) tdXY 2 ∑ r3 2 2 rij6rcut ij

where rij is the shorthand notation for |~rij| and ψij is the polar angle of~rij. From here on, natural units are used, so that the coupling constant D = 1 and the nearest neighbor distance a = 1. Hence, lengths are measured in units of a, and energies and temperatures are reported in units of D/a3. For the remainder of this chapter, the cutoff radius is chosen to be rcut = 2 as used in Section 4.2.2. The (t)dXY model on the square lattice has a ground state as shown in Fig. 6.1a. This ground state has a U(1) =∼ SO(2) degeneracy [29, 30], even though no such symmetry is present in the dipolar Hamiltonian (cf. Sec- tion 2.2.4). The underlying reason for this unprotected continuous ground- state degeneracy will be determined in Chapter 7. As no symmetry of the Hamiltonian protects this ground-state degeneracy, it is expected to be susceptible to perturbations such as temperature or disorder. Through an analysis of the spin waves of a nearest-neighbor tdXY model, Prakash and Henley showed that the system undergoes a so-called order-by-disorder transition if either thermal fluctuations or vacancies are introduced [30]. They showed that thermal fluctuations lead to a transition into a striped phase shown in Figs. 6.1b and 6.1c, which follows the fourfold symmetry of the lattice. Similarly, vacancy disorder leads to the so-called microvortex phase shown in Figs. 6.1d and 6.1e, which is characterized by local flux closure.1The arguments provided by Prakash and Henley for the thermal order-by-disorder transition will be discussed in more detail in Chapter 8. Since the authors of Ref. [30] only considered thermal fluctuations and disorder perturbatively, they were not able to derive a quantitative phase diagram. Therefore, soon after, Monte Carlo simulations for the (t)dXY model on the vacancy-disordered square lattice were conducted to find

1 It should be noted that Prakash and Henley [30] referred to the striped phase as the collinear phase, and to the microvortex phase as the anticollinear phase. The newer names of striped and microvortex phase were introduced later [156], but have since become the standard ter- minology for the magnetic phases of the dXY model on the square lattice. 6.1 dipolar-coupled xy spins on disordered square lattices 87

(a) Ground State Phase

π (b) Striped Phase, θ = 0 (d) Microvortex Phase, θ = 4

= π 3π (c) Striped Phase, θ 2 (e) Microvortex Phase, θ = 4

Figure 6.1: Magnetic unit cells of the three different long-range ordered phases for the (t)dXY model on the square lattice. Figure 6.1a shows the SO(2) degenerate ground state, which is parametrized by the angle π 3π θ. The striped phase arises for θ ∈ {0, 2 , π, 2 } and the microvortex π 3π 5π 7π phase for θ ∈ { 4 , 4 , 4 , 4 }. For both phases two of the four possible magnetic unit cells are shown, the other two follow from time-reversal or equivalently from θ 7→ θ + π. 88 phase diagram of dipolar-coupled xy spins on the square lattice

the phase diagram quantitatively [143, 192, 193]. These simulations used the fourth power of the spin components to determine if the spins point predominantly along diagonals or along lattice directions, which differenti- ates the microvortex phase from the striped phase. However, as the observable from Refs. [143, 192, 193] is not determined by symmetry arguments, it could only serve as an indication and not as a proper order parameter. Furthermore, the use of a simple temperature-sweep Monte Carlo algorithm in the papers [143, 192, 193] makes it likely that their system did not completely equilibrate at very low temperatures, as this algorithm is prone to get stuck in metastable states, especially for frustrated systems such as the (t)dXY model. Because of these two reasons, they could only obtain well-converged results for small vacancy densities p 6 6%. A different type of disorder has been considered by two other groups, which included disorder by displacing the lattice positions inside the plane by random vectors drawn from a Gaussian random distribution with standard deviation σ. It has been shown that magnetic excitations get increasingly localized with stronger disorder [194]. Nevertheless, the system does not enter an equilibrium spin glass phase, even for strong disorder arising from a large standard deviation σ [195]. However, no attempt was made in either of the two publications to derive a phase diagram for this type of disorder. Hence, the dXY system lacks a full quantitative phase diagram for both types of disorder. Recent experimental interest in such systems showed, however, that a quantitative understanding of the phase diagram is necessary in order to distinguish the effects of disorder and temperature [11, 12]. Therefore, the present work extends previous studies [143, 192–195] by de- riving proper order parameters for the striped phase and the microvortex phase. These are then used in parallel-tempering Monte Carlo simulations to determine the phase diagram for both types of disorder.

6.2 introducing proper order parameters

Here, in a ﬁrst step, order parameters are introduced that can distinguish the different phases shown in Fig. 6.1. As already discussed in Section 4.2.2.1, up to now, long-range order in the dXY system has been described by the order parameter deﬁned in Eq. (4.5). Here, the expression 6.2 introducing proper order parameters 89 is repeated, but the notation is changed following our paper [52]. Hence, the initial order parameter is

1 ~ yi xi M = ∑ ((−1) cos θi, (−1) sin θi) ,(6.2) N i where θi is the polar angle of the ith site, and xi and yi are the indices of the ith site along the x- and y-axis, respectively. Equation (6.2) is directly derived by “gauge transforming” the spins in the magnetic unit cell depicted in Fig. 6.1a so that the transformed spins form a ferromagnetic alignment for ground-state configurations [143]. Hence, with Eq. (6.2), it can be determined how the spins should be measured in order to obtain the order parameter M~ . Furthermore, the vector M~ is chosen to be normalized by the number of spins N so that it behaves as an order parameter density, rather than an extensive quantity. To employ the order parameter defined in Eq. (6.2) in a Monte Carlo simulation, the absolute value |M~ | has to be measured rather than the vectorial quantity. If instead the vector would be measured, the mean value would average to zero, due to the finite reorientation time of configurations in finite systems. Since M~ is derived from linear transformation of the ground-state manifold, |M~ | = 1 for all ground states so that |M~ | cannot differentiate between the microvortex phase and the striped phase. This behavior can ~ also be seen in Fig. 6.2 where√ M is shown for different configurations – specifically, M~ = (±1, ±1)/ 2 for microvortex phases, so that M~ aligns along the diagonals in this case. In contrast, M~ aligns along the x- and the y-axis for the striped phases. Hence, the microvortex phase and the striped phase can only be differentiated by the phase of M~ and not by its amplitude. ~ ~ Therefore, the polar form of M = |M|(cos θM, sin θM) can be used to disentangle the microvortex phase from the striped phase. With this, the ~ ~ 0 ~ phase of M is doubled so that the new vector M = |M|(cos 2θM, sin 2θM) is obtained. This vector M~ 0 aligns all striped phases along the x-axis and 0 ~ 0 all microvortex phases along the y-axis. Therefore, Mx = M · eˆx and 0 ~ 0 My = M · eˆy serve as valid order parameters to distinguish between the 90 phase diagram of dipolar-coupled xy spins on the square lattice

1 2 ( 2, 2)

(1, 0) (0, 0)

Figure 6.2: Illustration of M~ from Eq. (6.2) for different phases: The paramagnetic phase shows no spontaneous symmetry breaking so that M~ = 0. In contrast, the ground states form the unit circle, since |M~ | = 1 for ground-state conﬁgurations. Microvortex phases (dark blue) align along the diagonals, whereas striped phases (light blue) align along the axes.

striped phase and the microvortex phase, respectively. These values can be computed explicitly as

M2 − M2 0 ~ My x y Mx = |M| cos 2 arctan = , and (6.3a) Mx |M~ | 0 ~ My 2Mx My My = |M| sin 2 arctan = .(6.3b) Mx |M~ | 0 0 ~ Since Mx and My contain a division by |M|, the observables are unneces- sarily smeared out: Where long-range order emerges, the uncertainty in ~ 0 0 |M| affects the uncertainty in the estimators Mx and My. Therefore, the order parameters have to be further improved before they can be used to determine the detailed phase diagrams. To improve the observables, group (and speciﬁcally representation) theory can be used. In the character table of the point symmetry group of the square lattice C4v, shown in Table 6.1, one can identify the corresponding irreducible representations of the observables: The vector M~ transforms as the irreducible representation E, and hence contains all relevant informa- ~ tion since E is a faithful representation of C4v. The absolute value |M| trans- ~ forms as the trivial representation A1, such that |M| can only distinguish between long-range ordered phases and the paramagnetic phase. The or- 6.3 applicability of order parameters in disordered systems 91

Table 6.1: Character table for C4v, the point symmetry group of the square lattice. 1 C4v 2C4 C2 2σv 2σd 2 2 A1 1 1 1 1 1 x + y

A2 1 1 1 −1 −1 2 2 B1 1 −1 1 1 −1 x − y

B2 1 −1 1 −1 1 xy E 2 0 −2 0 0 (x, y)

0 0 der parameters Mx and My from Eq. (6.3) transform as the irreducible representations B1 and B2, respectively. Hence, observables transforming as B1 and B2 can differentiate between the microvortex phase and the striped phase. Therefore, the symmetry-adapted observables 00 2 2 00 Mx = Mx − My and My = 2Mx My (6.4) can be constructed. However, these observables are quadratic in Mx and My. Hence, the order parameters are obtained by taking the square root of the observables in Eq. (6.4), so that q q 2 2 Ms = |Mx − My| and Mmv = 2|Mx My| (6.5) serve as the proper order parameters. Equation (6.5) provides the proper order parameters, which can be applied to the (t)dXY model on disordered square lattices.

6.3 applicability of order parameters in disordered systems

The point symmetry group of the underlying lattice was used in the derivation of the order parameters of Eq. (6.5), but this symmetry is broken as soon as positional disorder is introduced. Hence, the order parameters constructed in Eq. (6.5) might no longer be well-defined. Nevertheless, as long as only weak disorder is considered, it is expected that the order parameters are not affected too dramatically. Yet, it should be quantified in which disorder regime the order parameters are still applicable. To define the meaning of weak disorder, one can consider that, in the def- ~ inition of the original order parameter M in Eq. (6.2), the indices xi and yi 92 phase diagram of dipolar-coupled xy spins on the square lattice

need to be well-deﬁned quantities, also for the disordered system. For the argument presented here, the focus is on random-displacement disorder as the disorder is homogeneously distributed in the lattice. For such systems, the Hamiltonian is Eq. (6.1), but the spins are placed on an approximate rather than a perfect square lattice. For the argument, we assume that the random displacements are normally distributed and uncorrelated so that the lattice positions are perturbed as ! ! ! xi xi δxi 2 7→ + , xi, yi ∈ Z, δxi, δyi ∼ N (0, σ ).(6.6) yi yi δyi

Here (xi, yi) form the initial square lattice and δxi, δyi are the Gaussian distributed random numbers centered around 0 and with a standard deviation of σ. Random-displacement disorder as introduced by Eq. (6.6) is illustrated in Fig. 6.3, where increasingly distorted square lattices are shown, which are obtained by adding random displacements to the position of the lattice sites. For this type of disorder, the order parameters are no longer well-deﬁned if two lattice sites exchange their position such that their indices are no longer a valid description of their position.2 The probabilities of two lattice positions to be at a speciﬁc location is illustrated in Fig. 6.4a. Therefore, the probability of two spins to exchange their position is the area of the shaded region, which can be computed as

1 Z ∞ Z ∞ u2 (v − 1)2 ρex(σ) = dv du exp − exp − .(6.7) 2πσ2 −∞ v 2σ2 2σ2

Therefore, the average number of defects hndefectsi can be estimated using ρex multiplied by the number of bonds on which an exchange can oc- 4 2 cur. The square lattice of size L with periodic boundary condition has 2 L bonds, as each site has four neighbors and each bond has to be counted 1 once rather than twice leading to the factor 2 . Therefore, the average number of defects in a square lattice with random displacement with standard deviation σ is

2 hndefectsi = 2L ρex(σ).(6.8)

2 This deﬁnition is essentially one-dimensional, as it only includes the exchange of an index along one axis. However, since Eq. (6.2) includes only one type of index (either xi or yi) per component, this one-dimensional description is sufﬁcient to provide a criterion for weak disorder. 6.3 applicability of order parameters in disordered systems 93

(a) Non-disordered system (σ = 0) (b) σ = 5%

(e) σ = 30%

Figure 6.3: Illustration of the random-displacement disorder. The position of each of the lattice sites is perturbed by a Gaussian with standard deviation σ from the initial position. Here, several strengths of disorder are shown, ranging from no disorder in Fig. 6.3a to strong disorder with σ = 30% shown in Fig. 6.3e. 94 phase diagram of dipolar-coupled xy spins on the square lattice

101 10−3 0 −4 10 10 48 lattice

10−1 = −5 L ex 10 ρ 10−2 10−6 10−3 10−7 2σ 2σ −4 10 Defects in an 0.14 0.16 0.18 0.2 0.22 0.24 σ (a) (b)

Figure 6.4: Illustration for the breakdown of the order parameters in terms of disorder: In Fig. 6.4a, the shaded area corresponds to ρex [cf. Eq. (6.7)]. In Fig. 6.4b, ρex is plotted as a function of disorder strength σ: Inspection of the ﬁgure shows, that no defect is expected even in the largest lattice (L = 48) for σ < 20%.

The exchange probability ρex is highly dependent on σ as can be seen in Fig. 6.4b. Here, the plot starts from σ = 13%, as ρex is close to vanishing for smaller σ. As can be seen in Fig. 6.4b, one defect per simulation is expected for the L = 48 simulation at σ ≈ 20%. Hence, it can be concluded that for σ < 20% the order parameters are well applicable. This value will be sufﬁcient for this thesis as will be seen in Section 6.6. However, in other models, the value of σ < 20% could become insufﬁcient, so that this criterion has to be validated anew. Nevertheless, it will be assumed that the order parameters are also applicable to the vacancy disorder situation discussed in Section 6.5 without providing a formal argument.

6.4 non-disordered system & thermal order-by-disorder

Before using the order parameters derived in Section 6.2 on the disordered systems, their behavior needs to be characterized. For this purpose, the non-disordered system is first simulated, which is well-understood due to previous work on the (t)dXY model on the square lattice [30, 141–145]. For the simulations of the tdXY model, which are presented in this thesis, the Hamiltonian Eq. (6.1) is used. Periodic boundary conditions are applied to the square lattice. A total of 2 · 105 lattice sweeps are used to 6.4 non-disordered system & thermal order-by-disorder 95 thermalize the system and another 1.5 · 105 lattice sweeps for the measurement of the order parameters. These measurements are recorded every 15 lattice sweeps, such that a total of 104 measurements are obtained. A fine temperature grid was used, in order to help the thermalization and to be able to resolve low-temperature features. The temperature grid consists of a linear temperature grid with 120 temperatures between T = 3 and T = 0.1 combined with a logarithmically spaced temperature grid between T = 0.1 and T = 10−6 with 100 grid points. Hence, 220 temperatures are simulated, ranging from temperatures much larger than Tc down to very small temperatures. The simulation results of the L = 16 system for the three order parameters |M~ |, Ms, and Mmv are presented in Fig. 6.5a. As seen in this figure, all order parameters rise at around T ∼ 1.0, i. e., at the position of the finite-temperature phase transition (Tc = 0.968 ± 0.01 according to the simulations in Section 4.2.2). As the system falls into a long-range ordered striped phase at finite temperatures [30], both, Ms and |M~ | show the behavior expected for order parameters. However, the behavior of Mmv is surprising, as the system is not expected to exhibit the microvortex phase in the non-disordered system [30]. The emergence of Mmv around the phase transition can be attributed to fluctuations of M~ in the phase, which are expected to be largest around the phase transition. The emergence of Mmv is further understood as a finite-size effect: The Monte Carlo results for Mmv for three different system sizes (L = 16, 32, 48) are shown in Fig. 6.5b. For temperatures around the phase transition, i. e., at T ∼ 1.0, it can be seen that the value of Mmv is more strongly suppressed for larger system sizes. However, this does not explain why Mmv for L = 16 does not decay to 0 for lower temperatures, but levels off around 0.4 as seen in the inset of Fig. 6.5a. Even worse, the value of Mmv blows up for T → 0 in larger systems as seen in Fig. 6.5b. In the inset, where the temperature axis is logarithmic, it can be observed that the rise of Mmv is continuous but very strong. This strong rise can be explained by the continuous ground-state degeneracy of (t)dXY models on the square lattice. The system has an SO(2) degenerate ground state, such that the Goldstone mode associated with this ground state restores this continuous degeneracy in the zero-temperature limit. Hence, a system spanning deformation of the striped phase towards a microvortex phase is expected for low temperatures. This homogeneous deformation is observed in Fig. 6.5c, where a spin configuration obtained 96 phase diagram of dipolar-coupled xy spins on the square lattice

by the Monte Carlo simulations at a temperature of T = 1.8 · 10−6 is shown. The system is predominantly in the striped phase, but small deviations have accumulated homogeneously so that the microvortex order parameter Mmv acquires a finite value. In the limit T = 0, the striped phase and the microvortex phase should occur equally often so that 1 Z 2π q lim hMsi = lim hMmvi = dθ 2| cos θ sin θ| ≈ 0.7628. (6.9) T→0 T→0 2π 0 This value should be the limiting behavior of all curves in Fig. 6.5b given sufficiently small temperatures and equilibration times sufficiently large to ensure thermalization. In conclusion, the order parameters Ms and Mmv defined in Eq. (6.5) serve as valid order parameters that can distinguish the striped and the microvortex phase, respectively. Furthermore, it was shown that the phase transition at finite temperature can be observed directly and clear indications of the restoration of the ground-state degeneracy at very small temperatures are present in the Monte Carlo data.

6.5 temperature versus vacancy-density phase diagram

To summarize the chapter up to this point, proper order parameters were developed in Section 6.2, and their behavior was analyzed in Sections 6.3 and 6.4. Hence, the phase diagram of the dXY model on the square lattice with respect to temperature and positional disorder can now be determined. Here, disorder is added to the tdXY system by the introduction of vacancies so that the Hamiltonian changes from Eqs. (2.6) and (6.1) to p p 0 = i j ~ · ~ − ~ · ~ · H D ∑ 3 Si Sj 3 Si rˆij Sj rˆij ,(6.10a) |~rij| rij6rcut D pi pj 1 3 = − cos θi − θj + cos θi + θj − 2ψij ,(6.10b) 2 ∑ r3 2 2 rij6rcut ij

where for the simulations, the formulation in the angles θi was used. Here, for simplicity, D = 1 as well as the nearest-neighbor distance a = 1. The pi determine the vacancy positions and are either 0 or 1, with probability p and 1 − p respectively. Here, p is the so-called vacancy den- sity3 that determines the strength of the disorder. The (t)dXY system has

3 It should be noted that this quantity was called the “dilution rate” rather than “vacancy density” in the paper [52]. 6.5 temperature versus vacancy-density phase diagram 97

0.7 1.0 1.0 0.6 0.6 0.8 0.8 0.6 0.5 0.4

0.4 0.2 0.6 0.4 0.2 v m

M 0.0 0.3 10 5 10 3 10 1 10 5 10 3 10 1 0.4 T T 0.2

|M| L = 16 0.2 0.1 L = 32 Ms L = 48 Mmv 0.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 T T (a) (b)

(c)

Figure 6.5: Monte Carlo results for the tdXY model without disorder: In Fig. 6.5a the three order parameters |M~ |, Ms, Mmv from Eqs. (6.2) and (6.5) are shown for the L = 16 lattice. In the inset, the same data is shown with a logarithmic temperature scale. In Fig. 6.5b the microvortex order parameter Mmv is shown for three different system sizes, where the inset displays the same data on a logarithmic temperature scale. Figure 6.5c shows one conﬁguration obtained at T = 1.8 · 10−6, where it can be observed that the deviation from the striped phase is homogeneously distributed throughout the whole system. 98 phase diagram of dipolar-coupled xy spins on the square lattice

been shown, to undergo an order-by-disorder transition once p becomes finite [30]. This spin-wave calculation showed that for T = 0, the phase obtained for vacancy-disordered systems is different to the thermally selected phase. Furthermore, the authors of Ref. [30] speculated about the phase diagram of the dXY model, where they estimated that the vacancy concentration above which no long-range order occurs is close to the per- dXY perc colation threshold, so that pc ∼ 1 − pc ≈ 40.7%. However, this prediction has, up to now, not been confirmed nor refuted, even though these quantitative statements are crucial when the model is to be compared with experimental systems. In experimental systems, this type of disorder arises naturally, for example due to non-magnetic impurities at the lattice positions of bulk systems or in ASI systems due to lithographic imperfections such as lift-off edges. Also theoretically the model has been studied intensively due to the order-by-disorder transition in the vacancy-disordered dXY model [30, 143, 192, 193]. However, these studies are limited to small vacancy densities of around 6% due to the lack of proper order parameters and a simulation technique suitable for highly frustrated systems. Hence, we can improve upon these results by employing the order parameters defined in Eq. (6.5) and by using the parallel-tempering Monte Carlo algorithm. Similar to the non-disordered case, the simulations use periodic boundary conditions, 3 · 105 lattice sweeps for thermalization, 7.5 · 104 lattice sweeps for the measurement of the order parameters. These measurements are taken every 15 lattice sweeps (resulting in 5 · 103 measurements). The 40 temperatures used in the parallel-tempering Monte Carlo are linearly spaced between T = 0.1 and T = 1.6, and the vacancy densities are sam- pled between p = 0 for the non-disordered system to p = 30% for a highly diluted system in steps of 1%. The disorder average is taken via 32 realizations, such that the simulations accumulated a total of 4 · 104 CPU hours. With these simulations, the temperature and vacancy-concentration dependence of the order parameters |M~ |, Ms, and Mmv can be determined. In Fig. 6.6 the order parameters are shown as a function of temperature at selected values of disorder. In Figs. 6.6g to 6.6i, it can be observed that at the highest vacancy density shown (p = 15%), all order parameters attain only small values. Furthermore, the temperature dependence does not exhibit a sharp rise upon lowering the temperature indicating that no long-range order occurs. In contrast, at the lowest vacancy density the system seems to exhibit long-range order as seen in the data for |M~ | shown in Fig. 6.6a where the 6.5 temperature versus vacancy-density phase diagram 99

value tends to 1 in the low-temperature limit. Furthermore, the data for Ms shown in Fig. 6.6b indicates that the system is for a large temperature window in the striped phase. Here, a peculiar feature is apparent: The data for |M~ | shown in Fig. 6.6a exhibits the typical system-size dependence of order parameters, namely that the numerical data for the order parameters become smaller with larger system sizes. In other words, the data for different system sizes is ordered in such a way that the smallest system size is the uppermost curve. This typical behavior is, however, not observed for Ms shown in Fig. 6.6b, where an inversion of the ordering occurs. This can indicate that at sufﬁciently low temperatures, another phase becomes dominant, and the smaller systems order at higher temperature. Since this is not observed for the order parameter |M~ |, the system stays long-range ordered. However, the phase at sufﬁciently low temperatures is no longer the striped phase but rather the microvortex phase. This expectation that there is a low-temperature microvortex phase is in agreement with previous studies [30, 143, 192, 193]. It remains an open question why this inversion point for Ms seems to be universal, i. e., the crossing between two different system sizes occurs always at the same temperature. It should be noted that a universal crossing point is expected for a Binder cumulant (cf. Section 6.5.1), but not for an order parameter, so that this behavior is unexpected. In Fig. 6.6a, it can be observed that the data for the lowest temperature does not seem to be fully thermalized for the L = 48 system4 as |M~ | exhibits a spurious drop at low temperatures. Indeed, at larger vacancy concentration of p = 6% this thermalization issue becomes more severe, as seen in Fig. 6.6d where the indication for a lack of thermalization can be observed already in the L = 32 system. This thermalization issue can be explained as a combination of several reasons: At reasonably large vacancy density p, the system is rather frustrated and the simulations are performed at low temperatures. Finally, the data for Mmv indicates that the system enters the microvortex phase around this region of the phase diagram, so that the critical slowing down in the vicinity of this phase transition further hinders the thermalization of the system. For this thesis, the thermalization issue was addressed with further simulations of the tdXY system on the L = 32 and the L = 48 square lattice at a vacancy density of 6%, where the thermalization issue is the most apparent. The simulations were performed analogously to the data presented

4 This lack of thermalization was not seen while analyzing the data for the paper [52], but it was only observed when the data was analyzed once more for this thesis. Here, we will argue that the conclusions of the paper are nevertheless unaffected by this lack of thermalization. 100 phase diagram of dipolar-coupled xy spins on the square lattice

0.8 0.8 0.4 0.6 | 0.6 s mv ~ M M | 0.4 0.4 M L=16 L=16 0.2 L=16 L=32 L=32 L=32 0.2 0.2 L=48 L=48 L=48

0.5 1 1.5 0.5 1 1.5 0.5 1 1.5 T T T (a) p = 2% (b) p = 2% (c) p = 2%

0.8 0.6 0.8 0.6 0.6 0.4 | s mv ~ M M

| 0.4 0.4 M L=16 L=16 L=16 0.2 0.2 L=32 0.2 L=32 L=32 L=48 L=48 L=48

0.5 1 1.5 0.5 1 1.5 0.5 1 1.5 T T T (d) p = 6% (e) p = 6% (f) p = 6%

L=16 0.4 L=16 L=16 L=32 0.3 L=32 0.4 L=32 L=48 0.3 L=48 L=48 | s

mv 0.2 ~ M M | 0.2 M 0.2 0.1 0.1

0.5 1 1.5 0.5 1 1.5 0.5 1 1.5 T T T (g) p = 15% (h) p = 15% (i) p = 15%

Figure 6.6: Monte Carlo results of the order parameters |M~ |, Ms, and Mmv in the tdXY system on the square lattice with vacancy disorder. Three selected values of disorder density p are shown: In Figs. 6.6a to 6.6c the data for p = 2%, in Figs. 6.6d to 6.6f the data for p = 6%, and in Figs. 6.6g to 6.6i the data for p = 15% is presented. The latter set at p = 15% suggests that no long-range order occurs. The data for |M~ | at small disorder (p = 2%, 6%) agrees with the expectation of a long-range ordered striped phase at finite temperature. Furthermore, a rise in Mmv at low temperature indicates the phase transition into a microvortex phase. The possible lack of thermalization at low temperature is discussed in Section 6.5. 6.5 temperature versus vacancy-density phase diagram 101 in the paper [52], with a varying number of thermalization sweeps. The paper used 3 · 105 lattice sweeps to thermalize the system, and here, additional data is presented for 5 · 105 and 1 · 106 thermalization sweeps. The data for the order parameter |M~ | is shown in Figs. 6.7a and 6.7b. It can be observed that for the L = 32 system, 5 · 105 lattice sweeps are sufficient for thermalization. However, even one million thermalization sweeps are clearly insufficient for the L = 48 system at the lowest temperatures. Even though some thermalization issues are present for the data used for the publication [52], the data is well thermalized at T > 0.2. Furthermore, the data for lower temperatures should be at least qualitatively correct. The Monte Carlo data for the three different order parameters Ms, Mmv, and |M~ | in simulations with system sizes L = 16, 32, and 48 are summarized in Fig. 6.8. Here, a large value of |M~ | again indicates where any type of long- range order occurs, and large values for Ms and Mmv demonstrate the emergence of the striped phase and the microvortex phase, respectively. Similar to Section 6.4, substantial finite-size effects are observed, so that a comparatively sharp transition is only visible for the largest system where L = 48. As expected, at small T and p between approximately 3 to 15%, the system falls into the microvortex phase as seen in the larger signal for Mmv. Similarly, for larger values of T, and smaller values of p, the system enters the striped phase. Finally, the signal in all order parameters vanishes if either T or p are sufficiently large so that the system does not order and hence remains paramagnetic.

6.5.1 Binder cumulant analysis & Phase diagram The order parameter data presented in Fig. 6.8 indicates the magnetic phase but, as the data originates from a finite-size simulation, it is insufficient to determine the phase diagram. To find the phase boundaries of the infinite system, the Binder cumulants can be analyzed. Examples of these Binder cumulants are shown in Fig. 6.9. In these figures, U denotes the Binder cumulant associated with |M~ |, Us with Ms, Umv with Mmv. The curves are either shown as a function of temperature at a fixed vacancy concentration (cf. Figs. 6.9a to 6.9f) or as a function of vacancy concentration at a fixed temperature (cf. Figs. 6.9g to 6.9i). As discussed in Section 3.4.3, the crossing between the Binder cumulant curves for different system sizes determine the location of the phase transition up to corrections to scaling. As for the order parameter, there is may be a thermalization issue at low temperatures where the microvortex phase emerges, which can be observed from Figs. 6.7c and 6.7d. 102 phase diagram of dipolar-coupled xy spins on the square lattice

0.9 0.9

0.8 0.8 | | ~ ~ M M 0.7 0.7 | | 5 5 Nth = 3 · 10 Nth = 3 · 10 5 5 0.6 Nth = 5 · 10 0.6 Nth = 5 · 10 6 6 Nth = 1 · 10 Nth = 1 · 10 0.5 0.5 0.1 0.2 0.3 0.4 0.1 0.2 0.3 0.4 T T (a) L = 32 (b) L = 48

0.7 0.7

0.65 0.65 U 0.6 U 0.6 5 5 Nth = 3 · 10 Nth = 3 · 10 5 5 0.55 Nth = 5 · 10 0.55 Nth = 5 · 10 6 6 Nth = 1 · 10 Nth = 1 · 10 0.5 0.5 0.1 0.2 0.3 0.4 0.1 0.2 0.3 0.4 T T (c) L = 32 (d) L = 48

Figure 6.7: Thermalization analysis for the tdXY system on the square lattice with vacancy density p = 6%. In Figs. 6.7a and 6.7b simulation results are shown for the order parameter |M~ | for two system sizes L = 32 and L = 48. In Figs. 6.7c and 6.7d, the results for the associated Binder cumulant U are shown. Besides the number of thermalization sweeps Nth, the simulations are identical to the ones used in the pa- 5 per [52]. It can be observed that the curves for Nth = 3 · 10 exhibit a drop at low temperatures, indicating that the number of thermalization sweeps used in the paper was insufficient for these input parameters. For the L = 32 system, 5 · 105 thermalization sweeps are sufficient to thermalize the system, as there is no spurious drop in either the order parameter shown in Fig. 6.7a or the Binder cumulant shown in Fig. 6.7c and the data agrees with the data of the larger thermalization 6 6 of Nth = 1 · 10 . Yet, even Nth = 10 thermalization sweeps seem to be insufficient for the L = 48 system. 6.5 temperature versus vacancy-density phase diagram 103

Ms Mmv |M| 1.0 1.5 6

1 1.0 T = 0.8 L 0.5

1.5 0.6 2

3 1.0 T =

L 0.5 0.4

1.5

8 0.2

4 1.0 T =

L 0.5

0.0 0.0 0.1 0.2 0.3 0.0 0.1 0.2 0.3 0.0 0.1 0.2 0.3 p p p

Figure 6.8: Monte Carlo results of the order parameters |M~ |, Ms, and Mmv for the tdXY model on the square lattice with a vacancy density p. Rather than single curves for selected values of disorder as presented in Fig. 6.6, here, the data for all studied values of disorder is shown as a density map. 104 phase diagram of dipolar-coupled xy spins on the square lattice

0.6 0.5 0.6 0.4 s mv U

0.5 U

0.4 U L=16 L=16 0.3 L=16 0.4 L=32 L=32 L=32 L=48 0.2 L=48 0.2 L=48 0.5 1 1.5 0.5 1 1.5 0.5 1 1.5 T T T (a) p = 6%, crossing at (b) p = 6%, crossing at (c) p = 6%, maybe a crossing at low T = 0.77 ± 0.01 T = 0.79 ± 0.01 and maybe at temperatures T = 0.17 ± 0.02

0.6 L=16 L=16 0.5 L=16 L=32 L=32 L=32 L=48 L=48 L=48 0.5 0.4 0.4 s mv U U U 0.3 0.4 0.2 0.2

0.5 1 1.5 0.5 1 1.5 0.5 1 1.5 T T T (d) p = 13%, no crossing (e) p = 13%, no crossing (f) p = 13%, no crossing

L=16 L=16 L=16 0.6 0.5 0.6 L=32 L=32 L=32 L=48 L=48 L=48 0.4 s mv U

0.5 U

0.4 U 0.3 0.4 0.2 0.2 0 0.1 0.2 0.3 0 0.1 0.2 0.3 0 0.1 0.2 0.3 p p p (g) T ≈ 0.79, crossing at (h) T ≈ 0.79, crossing at (i) T ≈ 0.79, no crossing p = (5.5 ± 0.2)% p = (6.1 ± 0.2)%

Figure 6.9: Selected Monte Carlo results of the vacancy-disordered tdXY for the Binder cumulants U, Us, and Umv, associated with the order parameters |M~ |, Ms, and Mmv, respectively. In Figs. 6.9a to 6.9c, the Binder cumulants for p = 6% are shown, which identify the microvortex phase at small temperatures (one crossing in Fig. 6.9c) and the striped phase at intermediate temperatures (two crossings in Fig. 6.9b). In Figs. 6.9d to 6.9f, no crossing can be observed for p = 13%, which indicates that the system no longer exhibits long-range order. Finally, in Figs. 6.9g to 6.9i the Binder cumulants are shown at a ﬁxed temperature T ≈ 0.79, where it can be observed that the system only orders in a striped phase but never in a microvortex phase, as there are clear crossings in Figs. 6.9g and 6.9h but none in Fig. 6.9i. As for Fig. 6.6, the data at low temperatures might not be completely thermalized. 6.5 temperature versus vacancy-density phase diagram 105

In order to determine the position of the phase boundary, the Binder cumulant data can be used with the statistical error obtained by the binning analysis implemented in ALPS [93–95]. This error information can be used in a resampling procedure, in which possible Binder cumulant curves are generated by perturbing the mean values with uncorrelated Gaussian errors according to the statistical uncertainty of the simulation. In the next step, the crossings of the effective curves for different L are analyzed by binning their location in a histogram. Finally, this histogram is fitted by one Gaussian per crossing, which provides an estimate of the phase boundary and its uncertainty. This procedure is applied for a fixed temperature T while considering the Binder cumulant as a function of the vacancy density p and vice versa for a fixed p and a variable T. The procedure is then repeated at every temperature and disorder value, for all order parameters. The application of this procedure results in the phase diagram displayed in Fig. 6.10, where the phase boundaries determined by the Binder cumulant analysis are superimposed on the density map of Mmv for L = 48, which was already shown in Fig. 6.8. The definition of the various marker styles is given in the caption of the figure. Where the Binder cumulant analysis converged, i. e., where the crossings could reliably be extracted, an excellent agreement among the Binder cumulant crossings for the different order parameters can be observed. Furthermore, a good agreement can also be observed between the Binder cumulant crossings obtained by the fixed temperature approach and the fixed vacancy-density approach. It should be noted that below T 6 0.2 the exact location of the phase boundary is not completely reliable, as the Binder cumulant crossings are affected by the lack of thermalization. This affects the microvortex phase the most, as it is almost entirely in the region where the system is ill-thermalized. However, as the phase boundary seems smooth and as there is a strong correlation among the different Binder cumulants, the microvortex phase is still expected to occur at these vacancy concentrations. Indeed, the lack of thermalization is very likely due to the critical slowing down in the vicinity of the phase transition between the striped phase and the microvortex phase. Therefore, the phase diagram for the vacancy-disordered tdXY system shown in Fig. 6.10 is quantitatively correct for T > 0.2, and it is at least qualitatively correct for T 6 0.2. The resulting phase diagram can therefore be summarized as follows: For large T and small p, the system is in the striped phase “s”. In contrast, for large p and small T, the systems enters the microvortex phase “mv”. Finally, if p or T are sufficiently large, 106 phase diagram of dipolar-coupled xy spins on the square lattice

1.4

1.2

1.0 para

0.8 T

0.6 s 0.4

0.2 mv 0.000 0.025 0.050 0.075 0.100 0.125 0.150 0.175 0.200 p

Figure 6.10: Summarized Monte Carlo results for the phase diagram of the tdXY model on the square lattice with system size L and vacancy density p. The phase diagram is obtained from the Binder cumulant analysis and is superimposed on the Monte Carlo data for Mmv in the L = 48 system. Markers correspond to |M~ | (red), Ms (orange) and Mmv (violet). Open markers correspond to Binder cumulant crossings at a fixed temperature analyzed as a function of p, while filled markers correspond to Binder cumulant crossings obtained at a fixed vacancy- density analyzed as a function of temperature. Below the line at T = 0.2, the data is affected by poor thermalization, and is therefore not completly reliable (details in Section 6.5). 6.6 temperature versus random-displacement phase diagram 107 the system no longer undergoes a phase transition and hence remains paramagnetic. This region of the phase diagram is labeled “para”. It is remarkable that the largest p where long-range order occurs is only pc(rcut = 2) ≈ 11%. This value is much lower than the previous conjecture dXY perc based on rcut = 1, which predicted pc ∼ 1 − pc = 40.7% [30]. As pc(rcut = 2) 40.7%, the disappearance of long-range order cannot be explained by percolation theory that would predict a much larger pc. Furthermore, it should be noted that pc is cutoff dependent. Indeed, for the tdXY system with rcut = 1 the system is non-frustrated and pc is expected to be close to the percolation threshold. For larger cutoff radii, percolation theory predicts a pc that is larger or equal to pc(rcut = 1) [196]. Here, however, the pc decreases with larger rcut which can be explained by the increased frustration. Hence, the estimate for pc(rcut = 2) serves only an upper bound for pc(rcut = ∞), and the physically relevant phase diagram for the non-truncated dXY model needs to be determined in future work.

6.6 temperature versus random-displacement phase diagram

As the second type of positional disorder, random displacement of lattice sites is introduced into the dXY model on the square lattice. Especially in experimental realizations such as ASI systems, this type of disorder can become relevant due to lithographic imperfections. Furthermore, it can also be deliberately added to the lattice to study the inﬂuence of disorder on ASI systems. This type of disorder has already been introduced in Section 6.3 with Eq. (6.6). To remind the reader: random-displacement disorder arises when each lattice position is perturbed by uncorrelated displacements, drawn from a Gaussian distribution with standard deviation σ. As in Section 6.5, the disordered system is simulated with the parallel- tempering Monte Carlo method. Here, σ values are chosen between 1% and 20% in steps of 1%. Otherwise, the simulation is performed with the 5 same parameters used in Section 6.5 (rcut = 2, thermalization of 3 · 10 lattice sweeps, 7.5 · 105 measurement sweeps, measurement every 15 lattice sweeps, 40 temperatures in linear steps between T = 0.1 and T = 1.6, disorder average over 32 realizations) so that the simulation accumulated approximately 6 · 104 CPU hours. 108 phase diagram of dipolar-coupled xy spins on the square lattice

Results for the order parameters at selected values of σ are presented in Fig. 6.11. Here it can be seen that, in contrast to the vacancy-disordered system, the random-displacement disordered system exhibits only at the largest disorder value signs of bad thermalization at low temperatures. The better thermalization is likely to be a result of the fact that the phase transition between the striped phase and the microvortex phase is at larger temperatures so that the associated critical slowing down occurs at larger temperatures where the Monte Carlo algorithm is still more effective. In- deed, the data for Ms in Figs. 6.11b and 6.11e indicates that the microvortex phase emerges at higher temperature than in the vacancy-disordered system. The results for all order parameters are summarized in Fig. 6.12. In comparison with Fig. 6.8, a much larger signal of the microvortex order parameter Mmv can be observed. However, the data for the vacancy-disordered system and the random-displacement disordered system should agree at p = σ = 0 the system is non-disordered. Hence, additional simulations for small values of disorder are performed. The results for the system with σ ∼ 0.1% are shown in Figs. 6.11a to 6.11c, where it can be observed that even very small disorder values are sufficient to promote the microvortex phase for considerably high temperatures. As for the vacancy-disordered system, the phase diagram can again be determined from the Binder cumulants. Specific examples for the Binder cumulants are shown in Fig. 6.13. The Binder cumulants at fixed disorder strength σ = 1% are shown in Figs. 6.13a to 6.13c, where clear crossings can be observed. In contrast, in the fixed temperature plots shown in Figs. 6.13d to 6.13i, it can be seen that the Binder cumulant crossings are rather ill-defined. The resampling procedure was applied nevertheless, in order to obtain an indication of the phase separation. Due to the ill-defined crossings, the errors are, however, much larger. This procedure yields the phase diagram shown in Fig. 6.14, where it can indeed be observed that the data points from the fixed temperature approach, i. e., data points with horizontal error bars, have a large error. In contrast to the temperature versus vacancy-density phase diagram in Fig. 6.10, an additional phase “fs” (finite-size) is observed, which is not separated from the paramagnetic phase by a phase boundary. Therefore, it is not expected that this additional phase persists in the thermodynamic limit. Nevertheless, Mmv and |M~ | acquire large values in this region of the phase diagram, much larger than expected for the paramagnetic phase. This should be considered as a finite-size effect: It can be expected that 6.6 temperature versus random-displacement phase diagram 109

1 0.8 0.8 0.8 0.6 0.6 | 0.6 s mv ~ M M 0.4 | M 0.4 0.4 L=16 L=16 L=16 L=32 0.2 L=32 L=32 0.2 0.2 L=48 L=48 L=48

0.5 1 1.5 0.5 1 1.5 0.5 1 1.5 T T T (a) σ = 0.1% (b) σ = 0.1% (c) σ = 0.1%

1 0.8 0.8 0.6 0.6 | 0.6 s mv ~ M M 0.4 | M 0.4 0.4 L=16 L=16 L=16 L=32 0.2 L=32 L=32 0.2 0.2 L=48 L=48 L=48

0.5 1 1.5 0.5 1 1.5 0.5 1 1.5 T T T (d) σ = 1% (e) σ = 1% (f) σ = 1%

0.8 0.8 0.4 0.6 0.6 | s mv ~ M M | M 0.4 0.4 L=16 0.2 L=16 L=16 0.2 L=32 L=32 0.2 L=32 L=48 L=48 L=48

0.5 1 1.5 0.5 1 1.5 0.5 1 1.5 T T T (g) σ = 8% (h) σ = 8% (i) σ = 8%

Figure 6.11: Monte Carlo results for the order parameters |M~ |, Ms, and Mmv in the tdXY on the square lattice with random displacement. Three selected values of the standard deviation σ of the random displacement disorder are shown: In Figs. 6.11a to 6.11c the data for σ = 0.1%, in Figs. 6.11d to 6.11f the data for σ = 1%, and in Figs. 6.11g to 6.11i the data for σ = 8% is presented. For σ = 0.1% and σ = 1% the data indicates at low temperatures a microvortex phase, at intermediate temperatures a striped phase and at high temperatures a paramagnetic phase. For σ = 8% the situation is less clear. The thermalization issues that were seen in the vacancy-disordered system are only weakly observed here. 110 phase diagram of dipolar-coupled xy spins on the square lattice

Ms Mmv |M| 1.0 1.5 6

1 1.0 T = 0.8 L 0.5

1.5 0.6 2

3 1.0 T =

L 0.5 0.4

1.5

8 0.2

4 1.0 T =

L 0.5

0.0 0.0 0.1 0.2 0.0 0.1 0.2 0.0 0.1 0.2

Figure 6.12: Summarized Monte Carlo results for the order parameters |M~ |, Ms, and Mmv for the tdXY model on the square lattice with system size L and random-displacement disorder with standard deviation σ. Rather than single curves for selected values of disorder as presented in Fig. 6.11, here, the data for all studied values of disorder is shown as a density map. 6.6 temperature versus random-displacement phase diagram 111

0.6 0.6 0.6 s mv U 0.5 U 0.4 U 0.4 L=16 L=16 L=16 0.4 L=32 L=32 L=32 L=48 0.2 L=48 0.2 L=48 0.5 1 1.5 0.5 1 1.5 0.5 1 1.5 T T T (a) σ = 1%, crossing at (b) σ = 1%, crossing at (c) σ = 1%, crossing at T = 0.98 ± 0.01 T = 1.00 ± 0.02 and T = 0.53 ± 0.06 T = 0.48 ± 0.07

0.6 0.6 0.6

0.5 s 0.4 0.4 mv U U U L=16 L=16 L=16 0.4 L=32 0.2 L=32 0.2 L=32 L=48 L=48 L=48 0.3 0 0.05 0.1 0.15 0.2 0 0.05 0.1 0.15 0.2 0 0.05 0.1 0.15 0.2 σ σ σ (d) T ≈ 0.52, crossing at (e) T ≈ 0.52, no clear crossing (f) T ≈ 0.52, crossing at σ = (7.3 ± 0.5)% σ = (1.8 ± 0.4)% and σ = (6.2 ± 1.3)%

0.6 0.6

0.4 s

0.5 mv U

U 0.4 U L=16 L=16 L=16 0.4 L=32 L=32 L=32 0.2 L=48 0.2 L=48 L=48 0.3 0 0.05 0.1 0.15 0.2 0 0.05 0.1 0.15 0.2 0 0.05 0.1 0.15 0.2 σ σ σ (g) T ≈ 0.91, crossing at (h) T ≈ 0.91, crossing at (i) T ≈ 0.91, no clear crossing σ = (4.5 ± 0.7)% σ = (3.8 ± 1.7)%

Figure 6.13: Selected Monte Carlo results of the tdXY model with random displacement disorder for the Binder cumulants U, Us, and Umv. In Figs. 6.13a to 6.13c, the Binder cumulants for σ = 1% are shown, where it can be observed that at low temperature the system is in the microvortex phase and at intermediate temperatures in the striped phase. In Figs. 6.13d to 6.13f and Figs. 6.13g to 6.13i, the ﬁxed temperature approach is applied at T ≈ 0.52 and T ≈ 0.90, respectively. The resampling procedure introduced in Section 6.5.1, can still yield the position of the crossings, albeit with considerably larger errors. 112 phase diagram of dipolar-coupled xy spins on the square lattice

1.4

1.2 para 1.0

0.8 s T

0.6

0.4 fs

0.2 mv

0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14

Figure 6.14: Summarized Monte Carlo results for the phase diagram of the tdXY model on the square lattice with system size L and random- displacement disorder with standard deviation σ. The phase diagram is obtained from the Binder cumulant analysis and is superimposed on the Monte Carlo data for Mmv in the L = 48 system. Markers, colors, and regions have the same deﬁnitions in Fig. 6.10. Here, the system seems to thermalize better at lower temperatures than the vacancy- disordered system discussed in Section 6.5. Therefore, the data can be considered to be more reliable. 6.7 conclusion 113 large patches of the system can still order when these patches are exposed to local disorder realizations which do not conﬂict with the magnetic order. However, these patches are separated by distortions of the lattice which isolates the patches from each other so that the magnetic order breaks up into large domains, which cannot order in the thermodynamic limit. Hence, no long-range order is expected in the thermodynamic limit for σ > σc ≈ 6%. Similar to pc, the critical value of random-displacement disorder σc is cutoff dependent. However, the emergence of the vast region “fs” could indicate that a larger cutoff radius might increase the region where long-range order can be observed. The ordered patches in this region of the phase diagram could possibly couple to each other more quickly when the cutoff radius is increased, in analogy with the percolation argument previously made for the vacancy-disordered system [30].

6.7 conclusion

In conclusion, the proper order parameters Ms and Mmv were derived using the representation theory for the point symmetry group of the square lattice. After showing that the order parameters are still applicable in weakly disordered systems, the non-disordered tdXY system was characterized using the newly found order parameters. Here, it was shown that the finite-temperature phase transition and the onset of the T = 0 phase transition to the continuous ground-state degeneracy can be observed. Us- ing extensive parallel-tempering Monte Carlo simulations of disordered systems and a Binder cumulant analysis, the phase diagram for two types of positional disorder could be obtained. Namely, vacancies were introduced into the model in Section 6.5, and the initial square lattice positions were perturbed by random displacement in Section 6.6. This led to the first successful derivation of the full phase diagram for both types of disorder. By comparison of the two phase diagrams, some quantitative differences can be observed: Random-displacement disorder is much more effective in promoting the microvortex phase than vacancy disorder is. In- deed, in Figs. 6.11a to 6.11c, it can be observed that even small random- displacement disorder is sufficient for the microvortex phase to persist to rather large temperatures. Furthermore, the occurrence of the vast finite- size correlated region in the random-displacement disordered case is remarkable. It remains an open question why the finite-size effects are seem- ingly much more significant for the random-displacement disordered situation than for the vacancy-disordered situation. 114 phase diagram of dipolar-coupled xy spins on the square lattice

Despite the differences, there is a surprising qualitative similarity between the two different phase diagrams. The zero temperature and zero disorder point is continuously degenerate, where both types of disorder promote the microvortex phase, temperature promotes the striped phase and, upon increasing either value sufficiently, the paramagnetic phase is obtained. This observation provides the motivation for the question why two very different types of disorder, the vacancy disorder which is very strong but very local, and the locally weak but homogeneous random- displacement disorder, have a similar effect. To explain this similarity, an intuitive reasoning can be given in terms of magnetic flux closure: The microvortex phase ensures the flux closure locally in the magnetic unit cell rather than on bulk length scales as in the striped phase. Therefore, the microvortex phase is more robust against positional disorder that disrupts the flux closure on large length scales. In disordered systems, the magnetostatic energy can thus be minimized by closing the flux locally, which leads to the microvortex phase. However, up to now, no formal argument has been provided for this statement so that it remains an open question if local flux closure by a magnetic phase such as the microvortex phase is always the response of dipolar-coupled systems towards positional disorder. Furthermore, it was demonstrated that even relatively small values of disorder lead to the absence of long-range order. From the simulations presented in this chapter, the critical standard deviation for random displacement is σc ≈ 6%, and the critical vacancy density is pc ≈ 11%, above which no long-range order can be observed. These values come, however, with the caveat that they are cutoff dependent, so that in order to make the statement for experimental realizations (where rcut → ∞), the simulations would need to be repeated with several different rcut, for sufficiently large system sizes while maintaining a sufficient thermalization. Subsequently, the dependencies of pc and σc on rcut need to be analyzed so that the physical value for pc(rcut = ∞) and σc(rcut = ∞) can be extracted. As these values determine if an experimental realization can form long-range order, they are crucial for the design of new experiments so that we hope that future work expands in this direction. Indeed, experiments might be able to determine the phase diagram in a much more efficient way because experimental systems inherently have rcut = ∞. In particular, both types of disorder can easily be implemented in ASI systems, which form the ideal playground to determine the phase diagram. Such an implementation, however, faces two obstacles: First, ASI 6.7 conclusion 115 systems include higher order multipole moments additional to the dipole moments that could change the phase diagram [197–199]. Second, as the microvortex phase can be constructed by a linear superposition of two striped phases, using diffraction experiments one cannot differentiate between the striped phase and the microvortex phase. However, real-space imaging techniques might be able to distinguish the two phases: If the experiment can differentiate between magnetization directions along diagonals or along axes, the phases can be distinguished. However, significant improvements to real-space imaging techniques have to be made, and the experiments have to fulfill several requirements: The ASI nanomagnets have to be sufficiently small to be thermally active, while still being large enough so that their magnetization direction can indeed be measured by real-space imaging techniques. Furthermore, experiments are even more complicated as the magnetization direction needs to be measured with a phase accuracy better than 45° and a time-resolution sufficient to measure thermally fluctuating moments. Nevertheless, recent experiments have shown that dXY ASI systems can be measured in real-space by reducing the temperature sufficiently to freeze the magnetic moments in a particular configuration [12]. As both types of disorder can be implemented in experimental systems and can be accurately measured, experiments should be able to determine the full phase diagram, without having to truncate the Hamiltonian as was done here and in the paper [52]. Hence, we hope that this work motivates the dXY ASI community to further push the boundaries of what can be achieved with real-space measurements.

7 CONTINUOUSGROUND-STATEDEGENERACYOF CLASSICALDIPOLESONREGULARLATTICES

The results presented in this chapter are published in Ref. [53]: “Continuous ground-state degeneracy of classical dipoles on regular lattices”, D. Schildknecht, M. Schütt, L. J. Heyderman, and P. M. Derlet, Phys. Rev. B (2019). As seen in previous chapters, the dXY system on the square lattice has a peculiar ground-state degeneracy. Rather than having a degeneracy which follows the finite symmetries of the Hamiltonian, the ground state is SO(2) degenerate [29]. This means that there is a continuous parameter, in this case, an angle θ which continuously deforms a ground-state without changing the energy. Such a continuous degeneracy does not only arise for dipolar-coupled XY spins on the square lattice but also for the truncated dipolar Hamilto- nian [30] or for other geometries such as the honeycomb lattice [200]. Sim- ilarly, a continuous degeneracy also arises for dipolar-coupled Heisenberg spins on the simple cubic lattice [29] or finite clusters [201]. Therefore, the degeneracy is probably not accidental but, as none of the Hamiltonians describing these systems has a continuous symmetry, the appearance of such continuous ground-state degeneracies is quite peculiar. This observation poses the interesting question of the origin of this continuous ground-state degeneracy. In this chapter, we want to answer this question based on a detailed symmetry discussion of the dipolar Hamiltonian. First, the continuous ground-state degeneracy of the dXY model on the square lattice is verified in Section 7.1 with an explicit calculation. This calculation indicates that the origin of this degeneracy lies in the symmetries of the Hamiltonian, which are discussed thoroughly in Section 7.2. In particular, the translational invariance is used in Section 7.2.1 to determine a candidate ground state. Then, the point symmetry group is analyzed in Section 7.2.2, where it is indeed shown that the degeneracy arises as a consequence of the symmetries. The emerging framework is then illustrated through some examples in Section 7.3. In particular, the dXY model on the square lattice is considered in Section 7.3.1, dipolar-coupled Heisen- berg spins on a (tetragonally distorted) cubic lattice are considered in Sec- tion 7.3.2, and finally, dXY spins on the triangular lattice are considered

117 118 continuous ground-state degeneracy of classical dipoles

θ θ

Figure 7.1: Magnetic unit cell of the ground state of the dXY model on the square lattice. According to Ref. [29], every conﬁguration described by this magnetic unit cell minimized the dipolar energy, independent of θ. Hence, the ground state is described by a manifold rather than discrete states. This ﬁgure is analogous to Fig. 6.1a.

in Section 7.3.3. As will be seen in this chapter, the ground-state degeneracy is quite fragile, so that it is necessary to go beyond the discussion presented in this chapter for the interpretation of a real system. Therefore, after a summary, we present an outlook to Chapter 8, where this issue is tackled by discussing the excitations on top of a continuously degenerate ground-state.

7.1 simple example: dipolar-coupled xy spins on the square lattice

Here, our usual example of the dXY model with moments on the sites of the square lattice is considered. The starting point of the calculation is based on XY spins on the square lattice coupled via the dipolar interaction from Eq. (2.6). In the previous chapters, the ground-state degeneracy as described by Ref. [29] has already been used, but here the degeneracy is now veriﬁed by an explicit computation of the energy. This calculation will indicate the origin of the continuous ground-state degeneracy. 7.1 dipolar-coupled xy spins on the square lattice 119

First, the ground state is introduced with the magnetic unit cell depicted in Fig. 7.1. Hence, the ground-state can be parametrized in terms of the ground-state degeneracy parameter θ as follows

y ! ~ (−1) i cos θ Si(θ) = (7.1a) (−1)xi sin θ ! ! (−1)yi 0 = cos θ + sin θ ,(7.1b) 0 (−1)xi | {z } | {z } ~S→ ~↑ i Si where xi and yi are the indices of the ith site along the x-axis and the y- axis, respectively. Here, a shorthand notation for two of the ground-state configurations is introduced: The striped phase along the x-direction is ~→ ~↑ abbreviated with Si and the striped phase along the y-direction with Si . These two states have already been shown in Figs. 6.1b and 6.1c. Equa- tion (7.1b) appears to be a trivial rewrite of the Eq. (7.1a), yet it implies that all ground-state configurations can be understood as a superposition of two orthogonal striped phases. This superposition is then inserted in the dipolar Hamiltonian in Eq. (2.6b) so that the energy of the configuration is given by

 α β  h i D 1 rijrij H ~S (θ) = δαβ − 3  i 2 ∑ ∑ |~r |3 |~r |2 i6=j α,β∈{x,y} ij ij ~→ ~↑ ~→ ~↑ · cos θSi + sin θS cos θSj + sin θS .(7.2) i α j β

In the next step, the brackets are expanded, and the different terms are ~↑~→ collected. This expansion leads to a mixed term incorporating Si Sj and ~→~→ ~↑~↑ the two terms proportional to Si Sj and Si Sj . The latter two terms sum to the ground-state energy E0, so that the energy of conﬁgurations as deﬁned in Eq. (7.1) is given by

 α β  h i 1 rijrij ↑ H ~S (θ) = E + D cos θ sin θ δαβ − 3 ~S→~S .(7.3) i 0 ∑ |~ |3  |~ |2  i,α j,β i6=j rij rij α,β∈{x,y} 120 continuous ground-state degeneracy of classical dipoles

~↑ ~→ αβ Since Si and Si are orthogonal conﬁgurations, the δ term in the Hamil- tonian vanishes. Hence, the energy of the superposition simpliﬁes to

α β h i 3D rijrij ↑ H ~S (θ) = E − sin 2θ ~S→~S .(7.4a) i 0 ∑ |~ |5 i,α j,β 2 i6=j rij α,β∈{x,y} y xj 3D (−1) i (−1) (xi − xj)(yi − yj) = E0 − sin 2θ (7.4b) 2 ∑ 2 2 5 i6=j |(xi − xj) + (yi − yj) | 2 | {z } 0

= E0,(7.4c)

where Eq. (7.1) is used to derive Eq. (7.4b) from Eq. (7.4a). Then, Eq. (7.4c) is obtained by applying the mirror symmetry which maps (x, y) 7→ (−x, y). As the term in the summation is odd under this symmetry, the sum van- ~ ishes. Hence, the energy of a conﬁguration Si(θ) is independent of θ, so that the ground state exhibits a continuous degeneracy. In Eq. (7.4), an element of the point symmetry group of the square lattice was used to show that a speciﬁc lattice summation has to vanish. However, the point symmetry group was applied ad hoc to an explicit parametrization of the ground state, so that the origin of the continuous degeneracy is not transparent. Nevertheless, the calculation presented in this section indicates that the origin of the degeneracy can be traced back to the point symmetry group, which serves as a motivation to analyze the symmetries of the system more thoroughly.

7.2 construction of the degenerate ground-states

As the calculation in Section 7.1 showed, a particular ground-state degeneracy can be veriﬁed by an explicit lattice sum. However, these calculations are not very insightful, as it is not obvious why the degeneracy arises. Hence, it is difﬁcult to predict new candidates with a similar degeneracy from such an example. Therefore, a different description of dipolar- coupled spin systems needs to be found, where the continuous degeneracy arises naturally. This description will be based on the representation theory for the symmetries of the Hamiltonian. First, the translational invariance T is used in Section 7.2.1 to introduce the method of Luttinger and Tisza (LT). Then, using the representation theory of the point symmetry group P in Sec- 7.2 construction of the degenerate ground-states 121

Table 7.1: An overview of previous studies of ground-states of dipolar- coupled spin systems on different lattices. The second column indicates if the LT method applies to the system. Lattice LT? Reference chain lattice Yes [29] rectangular lattice Yes [29] square lattice Yes [29, 30] honeycomb lattice Yes [30, 200] kagome lattice No [33, 34] cubic lattice Yes [28, 29] “fcc-kagome” lattice No [35] tion 7.2.2, a framework is developed in which the continuous degeneracy of the ground state emerges naturally.

7.2.1 Using the translational invariance: the method of Luttinger and Tisza The method of LT is based on the assumption of the existence of a magnetic unit cell and subsequent analysis of the problem by the representation theory of the broken translational invariance. Even though the first application of the LT method was on the dipolar-coupled systems, the method is generally applicable to spin systems with translational invariance [202]. Al- though the method sometimes provides unphysical ground states, it gives the correct ground state for many interesting dipolar-coupled systems as seen in Table 7.1. We will discuss the problem of unphysical solutions, once the assumptions of the LT method have been introduced. The LT method starts by assuming a magnetic unit cell of size N = nd so that the magnetic unit cell enlarges the structural unit cell in each direction by a factor of n in all d dimensions. Due to the assumption of a magnetic unit cell of size N, instead of dealing with an extensive number of ~ ~ ~ spins, only N spins S1, S2, ··· , SN inside the magnetic unit cell have to be defined to determine the configuration of the entire system. For the sake of ~ ~ ~ ~ notation, the spins are collected into a shared variable S = (S1, S2, ··· , Sn). Then, the Hamiltonian has to be minimized with respect to the Nd degrees ~ ~ of freedom posed by S, under the strong constraint |Si| = 1 for all i ∈ {1, 2, ··· , N}. This constraint can be implemented, for example, by the 122 continuous ground-state degeneracy of classical dipoles

introduction of Lagrange-multipliers [202, 203] which renders the problem non-linear. However, the LT method assumes that the strong constraint can be replaced with the weak constraint ~S 2 = N. This weak constraint can be enforced trivially, but it can lead to unphysical solutions that violate the strong constraint so that the LT method does not provide the ground state. Assuming that no Lagrange-multipliers are required, the dipolar Hamil- tonian in Eq. (2.6b) can be rewritten as the ﬁnite-dimensional quadratic form

H = −~SH~S,(7.5)

where the matrix elements of the real symmetric matrix H of dimension Nd include the required lattice summation. In Eq. (7.5), H~S can be interpreted to be the magnetic field that is induced on the lattice positions by the configuration~S, such that H can be interpreted as an effective Zeeman- interaction. This formulation is sufficient to compute the LT ground-state configuration. However, H still contains one lattice summation such as that given by Eq. (7.4) for every matrix element of H. To simplify this computation, Luttinger and Tisza proposed to work in a symmetry inspired basis that arises due to the broken translational invariance. Taking into account the magnetic unit cell, the translational invariance d d of the initial Hamiltonian [Z , see Eq. (2.8)] is broken to Zn, where Zn d is the cyclic group of size N. The representation theory for Zn provides the symmetry-adapted basis states, the so-called basic arrays, which correspond to the discrete Fourier transformations of length N. Their physical interpretation can be explained using the example of n = 2 where the magnetic unit cell is double the original size in every dimension. Then, the basic arrays form physical states such as the ferromagnetic configuration corresponding to the Fourier vector (0, 0, ··· , 0) or the antiferromagnetic configuration for (π, π, ··· , π). Since the Hamiltonian in Eq. (7.5) is translationally invariant, the different sectors of the LT matrix H do not mix. Therefore, the matrix H becomes block-diagonal in the basis of the basic arrays so that   Hferro 0 . . . 0  .   .. .. .   0 . . .  H =   .(7.6)  . .. ..   . . . 0  0 . . . 0 Hafm 7.2 construction of the degenerate ground-states 123

Hence, the only lattice summations that have to be carried out explicitly correspond to the individual block matrices Hi for i ∈ {ferro, afm, ···}. These lattice summations describe which of the blocks has the lowest energy eigenvalue. However, they have to be carried out explicitly as they change when the geometry is changed. Subsequently, the weak constraint ~ 2 is enforced by normalizing the corresponding eigenvector to Sgs = N. ~ However, at this stage it has to be verified if the strong constraint |Si| = 1 is ~ ~ fulfilled for all Si in Sgs. If the strong constraint is violated, the LT method fails to produce a physical solution, and one has to resort to different methods to find the ground-state of the system, such as introducing Lagrange- multipliers [202, 203] or using numerical methods [33–35]. If, however, the strong constraint is fulfilled, the LT method succeeds and it has found the ground state of the system.

7.2.2 Continuous ground-state degeneracy as a result of the point symmetry group Assuming that the LT method is applicable to the system under considera- tion, we know that H completely describes the ground-state conﬁguration. To simplify H, Luttinger and Tisza used the translational invariance T , which led to their method presented in Section 7.2.1. However, the dipolar Hamiltonian also supports the point symmetry group P, which can be used to simplify H even further. Since P is a symmetry of the dipolar Hamiltonian in Eq. (2.6), it is also a symmetry of the matrix H, so that group operations commute with H. However, the group elements form an abstract group, which cannot commute directly with H. Hence, a representation R of P with the dimension dim H1 is necessary to formalize the symmetry argument. By using the representation R, the symmetry condition is

[R(g), H] = R(g)H − HR(g) = 0, (7.7)

for all g ∈ P. To determine the appropriate representation R, one can consider that the Nd-dimensional vector space for ~S originated in the collection of N spin variables. Furthermore, (classical) spins transform as d- dimensional vectors, so that R = ⊕NV. Here, V is the d-dimensional vector representation (cf. Section 2.2.4), i. e., the faithful representation of P, which transforms the d-dimensional vector according to the point symmetry group.

1 Here, dim M denotes the dimension of the matrix M. Speciﬁcally for H, dim H = Nd. 124 continuous ground-state degeneracy of classical dipoles

As the LT method only applied a basis change, nothing about the symmetry sector changed. Hence, the representation for each block matrix of H = ⊕iHi is V, so that the symmetry condition for each block matrix is

[V(g), Hi] = 0, (7.8) for all g ∈ P. This symmetry condition has to hold for all block matrices Hi, but speciﬁcally also for the ground-state block matrix Hgs. Now, Schur’s lemma can be applied, which states that if a matrix Hgs commutes with a representation V(g) for all group elements g, and if V is irreducible, then Hgs has to be a scalar multiple of the identity. Formally ) ∀g ∈ P : [V(g), Hgs] = 0 ⇒ Hgs = hgs1,(7.9) V is irreducible

where hgs ∈ R because H, and therefore also Hgs, is real and symmetric. If the preconditions for Schur’s lemma are fulﬁlled, ddeg = d mutually orthogonal ground states are constructed, since Hgs is d-dimensional. For situations where V is reducible, Schur’s lemma cannot be applied directly. Here, the block matrix Hgs decays into smaller block matrices. Group theory cannot provide information about which subblock describes the lower energy conﬁguration so that the lattice summations have to be carried out for Hgs explicitly. The subblock further decays into smaller blocks, where the dimension of the lowest energy subblock is denoted with db. Then, Schur’s lemma is applicable to this subblock, so that ddeg = db mutually orthogonal ground states are found. ~S The mutually orthogonal states, which are enumerated by 1,...,ddeg , can be used to construct linear superpositions of the states. The energy of such a superposition is given by

   †   ddeg ddeg ddeg Dipol ~ ~ ~ H  ∑ αiSi =  ∑ αiSi H  ∑ αjSj (7.10a) i=1 i=1 j=1     ddeg ddeg ~ † ~ =  ∑ αiSi   ∑ αjHSj (7.10b) i=1 j=1

ddeg ddeg ddeg ~ †~ 2 2 = ∑ ∑ αiαjhgs Si Sj = Nhgs ∑ |αi| = Nhgs~a . i=1 j=1 |{z} i=1 Nδij (7.10c) 7.3 application of our method 125

Hence, as long as ~a 2 = 1, i. e., the states ~S are correctly normalized, the energy stays invariant. Therefore, a degeneracy on the (ddeg − 1)-dimensional sphere is found. This also includes the case where ddeg = 1, so that the degeneracy is equivalent to the 0-sphere, which is isomorphic to Z2. In this situation, no continuous ground-state degeneracy has been found. Otherwise, a continuous ground-state degeneracy in the form of a finite- dimensional sphere is found. Specifically, we have established that for LT systems, the vector representation V is crucial in the determination of a continuous ground-state degeneracy. If V is irreducible, a degeneracy equivalent to the (d − 1)- dimensional sphere is found. In contrast, if V is reducible, the dimension of the degeneracy is reduced. However, for all systems where the LT method provides a ground state, the degeneracy is described by a sphere in some dimension ddeg. The findings of this procedure are summarized by the flow diagram shown in Fig. 7.2. In conclusion, the fundamental reason for the degeneracies is that there are mutually orthogonal ground-state configurations obtained by the point symmetry group and that the Hamiltonian is quadratic in the spin variables so that all linear superpositions of the mutually orthogonal ground states have the same energy.

7.3 application of our method

In Section 7.2, it was shown how the continuous ground-state degeneracy emerges in dipolar-coupled spin systems, under the assumption that the LT method is applicable. However, the proof was somewhat technical, so that we want to illustrate the method by applying it to some examples. Namely, in Section 7.3.1, the dipolar-coupled XY model on the square lattice is considered as it is our standard example. In Section 7.3.2, dipolar- coupled Heisenberg spins are considered on the (tetragonally distorted) cubic lattice, which nicely illustrates the difference between d and db. Fi- nally, in Section 7.3.3, dXY spins on the triangular lattice are analyzed, which will demonstrate the limitations of the LT method and therefore our method.

7.3.1 Dipolar-coupled XY spins on the square lattice Here, the dXY model on the square lattice is considered, so that the same results as in Section 7.1 and Ref. [29] should be obtained. The system has the symmetries (cf. Section 2.2.4)

• Time-reversal Z2, 126 continuous ground-state degeneracy of classical dipoles

Classical dipolar- coupled spin system

Luttinger- no No general Tisza method statement works? possible

yes

no Decays to yes Is V irre- only 1D ducible? irreps?

yes no

Low energy no db > 1?

yes

Continuous Continuous No degeneracy in degeneracy in continuous dimension d dimension db degeneracy

Figure 7.2: The main finding of this chapter can be summarized by this flow diagram. For any dipolar-coupled spin system, for which a ground state can be determined by the LT method, the vector representation V plays an important role in the determination of the degeneracy. If V is irreducible, the degeneracy is of dimension d. If V is reducible, the dimension of the degeneracy depends on specific details of the lattice. However, ddeg is always smaller than d if V is reducible. 7.3 application of our method 127

∼ 2 • Translational invariance Tsq = Z × Z = Z ,

• Point symmetry group of the square lattice C4v. Now, Fig. 7.2 can be used to determine the ground-state degeneracy. Hence, in a first step, the LT method is applied to the system. For this method, we assume a 2 × 2 magnetic unit cell [29], which breaks the translational invariance down to a subgroup. The basic arrays are therefore given by the square roots of unity along each direction. As there are 22 = 4 sites in the magnetic unit cell, there are equivalently 4 basic arrays, which can be described by their Fourier vector. These basic arrays are depicted in Fig. 7.3. The energy of each of these basic arrays has to be computed by explicit lattice summation. These calculations are, however, simpler than the one in Section 7.1, since only a single state has to be considered, rather than a family of states parametrized by an angle θ. These calculations show ~ dXY that the striped configuration Sgs depicted in Fig. 7.3c has the lowest ~ dXY energy. As Sgs fulfills the strong constraint, it can be concluded that the LT method worked successfully. Following the flow diagram, the next step is to determine if V is irreducible on the point symmetry group P = C4v. The standard algorithm to decompose representations is based on the grand orthogonality theorem and can be found in textbooks on group theory such as Ref. [204, Ch. 3] so that we keep the discussion for this example brief. Generally, the decomposition of a representation is accomplished with the characters of the representation and the character table of the underlying group. The character table for C4v, the point symmetry group of the square lattice, is provided in Table 7.2. The characters of V can be determined directly from the definition of the character of a representation:

χV (g) = TrV(g),(7.11) for g in the point symmetry group P. To simplify the computation of the characters of V, we use that the character function is constant on a conjugacy class. Hence, the character table is typically stated with only one group element per conjugacy class. A conjugacy class for an element g is deﬁned as

{g} = { hgh−1 | h ∈ G},(7.12) which formally forms an equivalence class on G. For the example of C4v, 1 the ﬁve conjugacy classes are { }, {C4}, {C2}, {σh}, and {σd}. 128 continuous ground-state degeneracy of classical dipoles

(a) (0, 0) (b) (0, π)

Figure 7.3: The four basic arrays for the LT method implemented on the 2 × 2 square lattice with their corresponding Fourier vectors. Dotted lines indicate the lattice, and the magnetic unit cell is highlighted in gray. The basic array in Fig. 7.3a describes a ferromagnetic configuration, that in Fig. 7.3d describes an antiferromagnetic configuration, and those in Figs. 7.3b and 7.3c describe two inequivalent striped configurations.

Table 7.2: Character table of C4v, the point symmetry group of the square lattice. The character table can be found in textbooks on group theory. The characters for V are determined in Eq. (7.13). The reduction is V ≡ E, and hence V is irreducible on C4v. 1 C4v 2C4 C2 2σh 2σd

A1 1 1 1 1 1

A2 1 1 1 −1 −1

B1 1 −1 1 1 −1

B2 1 −1 1 −1 1 E 2 0 −2 0 0 V 2 0 −2 0 0 ≡ E 7.3 application of our method 129

Hence, the characters of V can be computed as ! 1 0 χV (1) = TrV(1) = Tr = +2, (7.13a) 0 1 ! 0 −1 χV (C4) = TrV(C4) = Tr = 0, (7.13b) 1 0 ! −1 0 χV (C2) = TrV(C2) = Tr = −2, (7.13c) 0 −1 ! 1 0 χV (σh) = TrV(σx) = Tr = 0, (7.13d) 0 −1 ! 0 1 χV (σd) = TrV(σxy) = Tr = 0, (7.13e) 1 0 which can be used to reduce V into its irreducible representations. The characters found in Eq. (7.13) are added to the relevant character table given in Table 7.2 as the lowest row. This can be used to determine a set of integers ni so that

χV (g) = ∑ niΓi(g),(7.14) i for all g ∈ P, where Γi enumerates the irreducible representations of P. In general, the reduction can be carried out using the grand orthogonality theorem [204]. However, here it is particularly easy to guess the unique solution V ≡ E so that V is irreducible. As V is irreducible, there are ddeg = d = 2 orthogonal basic arrays, ~↑ ~→ corresponding to the configurations Si and Si defined in Eq. (7.1b). Due to the calculation presented in Eq. (7.10), the superposition has the same energy, as long as the normalization condition is fulfilled. Hence, the two- dimensional degeneracy with the topology of the unit circle S1 found in Section 7.1 is recovered. This conclusion is in agreement with our explicit calculation in Section 7.1 and literature [29]. However, now the origin of the continuous degeneracy is made completely transparent through symmetry arguments.

7.3.2 Dipolar-coupled Heisenberg spins on the (distorted) cubic lattice As a second example, dipolar-coupled Heisenberg spins on the (distorted) cubic lattice are considered, as this allows us to discuss all the branches 130 continuous ground-state degeneracy of classical dipoles

of the ﬂow diagram displayed in Fig. 7.2 in a single example. First, the perfect cubic lattice with lattice parameter a is considered. This system has the symmetries

• Time-reversal Z2, ∼ 3 • Translational invariance Tc = Z × Z × Z = Z ,

• Point symmetry group of the cubic lattice Oh. For the perfect cubic lattice, the LT method starts by assuming a magnetic unit cell of size 23 [28, 29]. The basic arrays emerging for this problem are depicted in Fig. 7.4. As this system is three-dimensional, i. e., at the critical dimension for the dipolar-interaction to become long-range, some care has to be taken for the ferromagnetic configuration, as the energy of this configuration becomes sample-shape dependent [64]. However, as Lut- tinger and Tisza showed, the striped configuration depicted in Fig. 7.4g is the lowest energy configuration, independent of the sample shape [28]. 2 Then, the vector representation V is reduced on P = Oh. The characters can again be found by explicitly constructing the transformation matrices as in Section 7.3.1. Alternatively, the characters can also be determined by considering how many coordinates stay invariant and how many coordinates map to their opposite direction. Either way can be used to determine to the characters of V, which are then used to reduce V over Oh in Ta- ble 7.3. There it can be seen that V ≡ T1u, so that V is irreducible. Hence, the system exhibits a three-dimensional ground-state degeneracy, which is equivalent to a 2-sphere. This degeneracy has been found previously by Belobrov et al. [29] from an explicit calculation of the ground-state energy. In a second step, the system is distorted tetragonally. Therefore, if all lattice parameters of the initial cubic lattice are a, then one lattice parameter in the tetragonally distorted system is changed to c 6= a. This reduces the point symmetry group from Oh to D4h, but the other symmetry groups stay invariant under this distortion. The point symmetry group is now smaller so that V is no longer irreducible in D4h as it can be seen in Table 7.4 that no single row coincides with the row for the characters of V. Instead, V ≡ D4h D4h D4h D4h A2u ⊕ Eu , where Eu is two-dimensional and A2u is one-dimensional. Their dimension can be read off the character table as the dimension of a 1 representation Γ is dim Γ = χΓ( ).

2 It should be noted that, in contrast to Section 7.3.1, here, V denotes the three-dimensional vector representation, as the system consists of Heisenberg spins rather than XY spins. 7.3 application of our method 131

(a) (0, 0, 0) (b) (0, 0, π) (c) (0, π, 0)

(d) (0, π, π) (e) (π, 0, 0) (f) (π, 0, π)

(g) (π, π, 0) (h) (π, π, π)

Figure 7.4: The 23 = 8 basic arrays for the 2 × 2 × 2 cubic lattice with their corresponding Fourier vectors. The lattice is indicated by a dotted line, and the magnetic unit cell is highlighted in a light gray. The explicit lattice summations show that, for the dipolar case, the conﬁguration depicted in Fig. 7.4g minimizes the energy. 132 continuous ground-state degeneracy of classical dipoles

Table 7.3: Character table of Oh, the point symmetry group of the cubic lattice and the reduction of the three-dimensional vector representation V. The reduction leads to V ≡ T1u, i. e., V is irreducible. 1 2 Oh 8C3 6C2 6C4 3C4 i 6S4 8S6 3σh 6σd A1g 1 1 1 1 1 1 1 1 1 1

A2g 1 1 −1 −1 1 1 −1 1 1 −1

Eg 2 −1 0 0 2 2 0 −1 2 0

T1g 3 0 −1 1 −1 3 1 0 −1 −1

T2g 3 0 1 −1 −1 3 −1 0 −1 1

A1u 1 1 1 1 1 −1 −1 −1 −1 −1

A2u 1 1 −1 −1 1 −1 1 −1 −1 1 Eu 2 −1 0 0 2 −2 0 1 −2 0

T1u 3 0 −1 1 −1 −3 −1 0 1 1

T2u 3 0 1 −1 −1 −3 1 0 1 −1

V 3 0 −1 1 −1 −3 −1 0 1 1 ≡ T1u

Depending on the distortion, the one-dimensional representation or the two-dimensional representation is lower in energy. In general, the lattice summation provides the answer. However, for this system, it becomes clear that if c > a, i. e., when the cubic lattice is stretched in the third dimension, the system consists of weakly coupled square-lattice layers. Therefore, the dominant interaction is within the layers, so that the two-dimensional rep- D4h resentation Eu is lower in energy. Hence, the c > a distorted system shows an SO(2) degeneracy, analogous to the dXY system on the square lattice. In contrast, if c < a, the system is compressed along the third axis. Hence, the system consists of weakly coupled chains of dipoles. Therefore, the lower-energy block corresponds to the one-dimensional representation D4h A2u . Therefore, db = 1 and no continuous degeneracy is found. Hence, it can be concluded that dipolar-coupled Heisenberg spins on the (distorted) cubic lattice display a rich behavior in terms of the degeneracy. The cubic lattice shows an SO(3) degeneracy. For the tetragonally distorted system, an SO(2) degeneracy is found for a tensile deformation, whereas a discrete Z2 degeneracy is found for a compressive deformation. 7.3 application of our method 133

Table 7.4: Character table of D4h, the point symmetry group of the tetragonal lattice and the reduction of the three-dimensional vector representation V. Here, V does not correspond to a single line of the character table, so that V is reducible. The reduction can be accomplished using the grand orthogonality theorem (see, e. g. Ref. [204]), which leads to D4h D4h the reduction V ≡ A2u ⊕ Eu . 1 0 00 D4h 2C4 C2 2C2 2C2 i 2S4 σh 2σv 2σd A1g 1 1 1 1 1 1 1 1 1 1

A2g 1 1 1 −1 −1 1 1 1 −1 −1

B1g 1 −1 1 1 −1 1 −1 1 1 −1

B2g 1 −1 1 −1 1 1 −1 1 −1 1

Eg 2 0 −2 0 0 2 0 −2 0 0

A1u 1 1 1 1 1 −1 −1 −1 −1 −1

A2u 1 1 1 −1 −1 −1 −1 −1 1 1

B1u 1 −1 1 1 −1 −1 1 −1 −1 1

B2u 1 −1 1 −1 1 −1 1 −1 1 −1 Eu 2 0 −2 0 0 −2 0 2 0 0

V 3 1 −1 −1 −1 −3 −1 1 1 1 ≡ A2u ⊕ Eu 134 continuous ground-state degeneracy of classical dipoles

Figure 7.5: Ground-state of dipolar-coupled XY spins on the triangular lattice [205]. The magnetic unit cell coincides with the structural unit cell and is highlighted in light gray.

7.3.3 Dipolar-coupled XY spins on the triangular lattice Finally, the dXY model on the triangular lattice is considered. In contrast to the previous examples, where a truncation of the Hamiltonian did not change the ground state, here it will be observed that the ground state crucially depends on any truncation of the dipolar Hamiltonian. First, we again summarize the symmetries: The Hamiltonian is still time- reversal symmetric (described by Z2) and the translational invariance Ttri is given by the mapping ! ! 1 b 1 Ttri : ~ri 7→~ri −~t, where ~t = a + √ with a, b ∈ Z, 0 2 3 (7.15)

so that the translational invariance is isomorphic to Z × Z = Z2. Finally, the point symmetry group P of the triangular lattice is C6v. The ﬁrst step in the application of the framework illustrated in Fig. 7.2 consists of applying the LT method and examining if a ground state of the system can be found by this method. For the non-truncated Hamiltonian, the system is known to exhibit a ferromagnetic ground state [205], so that the magnetic unit cell coincides with the structural unit cell as illustrated in Fig. 7.5. In this situation, the strong constraint is equivalent to the weak constraint and the LT method works trivially. If the dipolar Hamiltonian is, however, truncated, then the ground state is no longer ferromagnetic but rather a striped conﬁguration emerges as the ground state [66]. Such ground states cannot be described by an LT basic array, as they would require the superposition of several basic arrays rather than merely one so that the LT method is no longer applicable. 7.4 conclusion 135

Table 7.5: Character table of C6v, the point symmetry group of the triangular lattice, and the reduction of the two-dimensional vector representation V. As V ≡ E1, the vector representation is irreducible. 1 C6v 2C6 2C3 C2 3σv 3σd

A1 1 1 1 1 1 1

A2 1 1 1 1 −1 −1

B1 1 −1 1 −1 1 −1

B2 1 −1 1 −1 −1 1

E1 2 1 −1 −2 0 0

E2 2 −1 −1 2 0 0

V 2 1 −1 −2 0 0 ≡ E1

Therefore, only the non-truncated dipolar Hamiltonian is considered for this geometry. The next step according to Fig. 7.2, reduces the vector representation V over the point symmetry group P. The reduction of V is carried out with the character table Table 7.5, where it can be observed that V is irreducible, since V ≡ E1. Therefore, a continuous ground-state degeneracy described by SO(2) has been found, in agreement with previous literature [206].

7.4 conclusion

In this chapter, the origin of the continuous ground-state degeneracy in many dipolar-coupled systems has been elucidated. For systems where the LT method is applicable, the vector representation V mainly determines if there is a continuous degeneracy in the ground state. If V is irreducible, the degeneracy is d-dimensional. If it is reducible, a continuous degeneracy can emerge, but with a lower dimension. The procedure to determine the degeneracy is summarized in the flow diagram in Fig. 7.2. This procedure has subsequently been applied to three examples in Section 7.3, which showed that previously known results [28–30, 206] were unified and generalized. Here, the focus was on systems where the ground-state can be constructed by the LT method. For systems where the LT method is not applicable, the situation is more complicated, as seen in specific examples such as the kagome lattice [33, 34] or the “fcc-kagome” lattice [35]. Never- 136 continuous ground-state degeneracy of classical dipoles

theless, also for these systems, a symmetry analysis might provide some insight into the origin of a continuous degeneracy, or its absence. Also, we showed that the ground state shows a unusual degeneracy in certain cases. However, as such a degeneracy is not protected by any symmetry of the Hamiltonian, it is not expected that the degeneracy survives perturbations. Indeed, the system should restore the ﬁnite symmetry of the Hamiltonian once thermal ﬂuctuations or disorder are included, in analogy to the order-by-disorder transition discovered for the dXY model on the square lattice [30] (cf. Chapter 6). As the continuous degeneracy only arises at a single point in the phase space, it is complicated to access experimentally or numerically.3 Therefore, this ground-state discussion is extended to a symmetry-guided analysis of excitations in the next chapter.

3 Though not impossible as shown in Chapter 6. 8 SPINWAVESANDORDER-BY-DISORDERFOR CLASSICALDIPOLESONREGULARLATTICES

In the last chapter, we have seen that the ground-state of many dipolar- coupled spin systems exhibits a continuous degeneracy that is not protected by a symmetry of the Hamiltonian. Because of this, excitations or disorder readily disrupt it. Indeed, for several examples, it has been shown that the system undergoes a so-called order-by-disorder transition [207–209] where thermal fluctuations induce a transition that restores the finite symmetry of the dipolar system [30, 52, 143, 192, 193, 210–212]. In this chapter, the first steps towards a generalization to other lattices are given for systems where the continuous ground-state degeneracy emerges by the mechanism discussed in Chapter 7. For this purpose, a linear spin-wave calculation is employed, augmented with insights from group theory to analyze the effects of symmetries on the excitation spectrum. This chapter is separated into three main sections, and each of these sections has a more detailed overview of its contents in the beginning. Hence, only a coarse overview of this chapter is presented here. In the first section, Section 8.1, two spin-wave calculations for particular systems are given. This discussion leads to valuable insights for the generalization to more general lattices accomplished in Section 8.2. The results are then illustrated with some examples in Section 8.3. Finally, a summary and an outlook of this chapter are provided in Section 8.4.

8.1 spin waves in dipolar-coupled spin systems for simple lattices

In this section, a non-interacting spin-wave calculation is carried out for two examples: In Section 8.1.1, dipolar-coupled XY (dXY) spins on the linear chain are considered where, from the dispersion relation, it can be observed that the ground state and the excitations do not share the same magnetic unit cell. The second example is presented in Section 8.1.2 and consists of dXY spins on the square lattice, where it will be shown how the lowest-energy excitation can be identiﬁed by symmetry arguments.

137 138 spin waves for classical dipoles on regular lattices

(a) Spins parallel to the chain direction

(b) Spins orthogonal to the chain direction

Figure 8.1: The two ground-state candidates of the dXY model on the linear chain according to the “head-to-tail” rule illustrated in Fig. 2.2. The calculation in Eq. (8.2) shows that the conﬁguration depicted in Fig. 8.1a has lower dipolar energy than the conﬁguration depicted in Fig. 8.1b.

8.1.1 Spin waves for dXY spins on the chain lattice First, we consider dXY spins on a linear chain lattice. The reasons to choose this model are twofold: On the one hand, it admits a simple experimental implementation [213], by placing ASI nanomagnets with XY anisotropy like the ones used in Refs. [10–12] on a linear chain lattice. On the other hand, this system admits an exact solution for the entire excitation spectrum of non-interacting spin waves [214]. In contrast to the exact solution found in this model, all later examples will only admit an asymptotic solution in terms of a series expansion. Hence, this example is discussed in some detail, as it will provide the foundation for the following examples. In Section 8.1.1.1, the ground state of dXY spins on the chain lattice is constructed by the method of Luttinger and Tisza (LT) and the method from Chapter 7. Then, the dispersion relation of non-interacting spin waves is computed in Section 8.1.1.2.

8.1.1.1 Ground state of dipolar-coupled XY spins on the linear chain Here, the ground state of the dXY model on the linear chain is determined by the LT method presented in Chapter 7. Since the system is one- dimensional, all irreducible representations of the point symmetry group are one-dimensional as well. Hence, no continuous degeneracy can arise, according to the arguments provided in Chapter 7. The geometry is deﬁned as follows: The XY spins lie in the xy-plane, the linear chain is along the x-direction, and the nearest-neighbor distance is a = 1. Therefore, the LT states are described by

~ ~ i ~ ~ i Si,1 = eˆx, Si,2 = (−1) eˆx, Si,3 = eˆy, Si,4 = (−1) eˆy,(8.1) 8.1 spin waves for dipoles on simple lattices 139

~ ~ where i is the index along the chain. The configurations Si,2 and Si,3 can be excluded as ground states since they do not fulfill the “head-to-tail” rule illustrated in Fig. 2.2. ~ ~ The remaining two configurations, Si,1 and Si,4 depicted in Figs. 8.1a and 8.1b, respectively, have to be considered more closely. The energy densities of these configurations can be computed with the dipolar Hamilto- nian in Eq. (2.6) as E 1 D −2 1 = H[~S ] = = −2Dζ(3) ≈ −2.40D, and (8.2a) i,1 ∑ | |3 N N 2 i6=0 i E 1 D (−1)i 3D 4 = H[~S ] = = − ζ(3) ≈ −0.90D,(8.2b) i,4 ∑ | |3 N N 2 i6=0 i 4 ~ ~ so that the ground-state configuration is S0 = Si,1 = eˆx. This configuration is depicted in Fig. 8.1a and corresponds to a ferromagnetic spin alignment along the chain.

8.1.1.2 Dispersion relation of non-interacting spin waves In a second step, excitations are introduced, perturbing the ground-state ~ conﬁguration Si,1. In contrast to domain-wall excitations in systems with discrete degrees of freedom[5], the natural description of excitations for spin systems with continuous degrees of freedom are spin waves. Here, spin waves are introduced using position-dependent rotations of the ground-state spin. The convenient parametrization of such rotations relies on the Lie group that generates the spin degrees of freedom and the associated Lie algebra. Speciﬁcally, XY spins are described by the Lie group SO(2) with the associated Lie algebra so(2). The connection between SO(2) and so(2), and more generally between any simply-connected (matrix) Lie group and the corresponding Lie algebra, is formally given by a matrix exponential. As the Lie algebra so(2) has only one so-called generator G, any group element R ∈ SO(2) of the Lie group can be parametrized by a single angle φ via a matrix exponential as ! 0 −1 R(φ) = exp(φG), where G = .(8.3) +1 0 A spin-wave excitation can then be parametrized by a position-dependent φ so that the spin at position x is described by ~ ~ ~ S(x) = R[φ(x)]S0(x) = exp[φ(x)G]S0(x),(8.4) 140 spin waves for classical dipoles on regular lattices

~ where S0 ≡ eˆx is the ground-state spin conﬁguration. With the assumption that the rotation angles φ(x) are small, the matrix exponential may be expanded up to quadratic order:

h 2 2i ~S(x) ≈ 1 + φ(x)G + φ(x) G eˆx,(8.5)

which can be used to determine the dispersion relation of spin waves in this system. Here, the expansion is carried out up to quadratic order (instead of linear order) in order to determine the energy gap of the dispersion relation. By using Eq. (8.5), the energy of a spin-wave-perturbed conﬁguration can be computed as

h i† h i 1 21 1 21 H = ∑ + φ(x)G − φ(x) eˆx Hx−y + φ(y)G − φ(y) eˆx , x6=y (8.6)

where Hx−y is used for brevity and is described by Eq. (2.6b). Explicit use of the generator G as deﬁned in Eq. (8.3) and the dipolar Hamiltonian in Eq. (2.6b) leads to

D 1 H = −2 + φ(x)φ(y) + 2[φ(x)2 + φ(y)2] + O(φ3) ,(8.7) ∑ | − |3 2 x6=y x y

where the φ-independent term yields the ground-state energy E0 already computed in Section 8.1.1.1. To determine the excitation spectrum of spin waves, i. e., periodic de- formations of the ground state, the rotation angles φ should be Fourier R transformed. Hence, φ(x) = BZ dq exp(iqx)φ(q) where the argument of φ indicates the real-space or momentum-space function. Using this expression in Eq. (8.7) leads to the spin-wave Hamiltonian. Introducing the center of mass coordinates, y + x d d d = y − x, m = , ⇔ x = m − , y = m + ,(8.8) 2 2 2 8.1 spin waves for dipoles on simple lattices 141

leads to a further simpliﬁcation of the spin-wave Hamiltonian due to the translational invariance, so that

ω(q) z }| { Z π ∞ cos (qd) + 8 = + ( ) ( ) H E0 dq φ q φ q DV ∑ 3 ,(8.9a) −π d=1 d DV h i ω(q) = Li (eiq) + Li (e−iq) + 8ζ(3) 2 3 3 h iq i = DV Re Li3(e ) + 4ζ(3) .(8.9b)

Here, ω(q) is the dispersion relation of a spin wave with wave vector q, which can be written in a closed form by the trilogarithm Li3 [215]. The dispersion relation is shown in Fig. 8.2. In this ﬁgure, it can be observed that the spin-wave dispersion relation is gapped, i. e., ω(q) > 0 for all q, which is expected for a system without a continuous degeneracy. Furthermore, the minima of the dispersion relation are around q = ±π, so that the low-energy excitations of the system are staggered rotations rather than a homogeneous rotation. Hence, the system still adheres to the “head-to-tail” rule, and the lowest energy excitation exhibits a similar antiferromagnetic pattern to that of the second lowest-energy LT state shown in Fig. 8.1b. If the calculation were to be carried out in magnetic unit cell with a magnetic lattice parameter that is twice that of the structural unit cell, as suggested by the second ground-state candidate, two spin-wave modes would have been obtained in a smaller Brillouin zone. The result of such a procedure is equivalent of backfolding the dispersion relation ω(q) from Eq. (8.9b) at the new Brillouin-zone boundary. Then, the dispersion relation resembles a more typical dispersion relation where ω(q) is minimal at q = 0. However, the lower energy mode corresponds to an optical mode rather than an acoustic mode.1 In conclusion, the excitations behave as if there would be a unit cell containing two moments, rather than the unit cell with one moment necessary to describe the ground state.

1 The notion of acoustic versus optic mode is taken from the language of phonons. An acoustic phonon describes the coherent movement of atoms, which are in phase, so that if one atom moves to the right so do its neighbors. In contrast, optic phonons describe the movement which is maximally out of phase, so that if an atom moves to the right, its nearest neighbors move to the left. Here, acoustic would mean that, if a spin rotates clockwise, so do its neighbors and in contrast optic means that if a spin rotates clockwise, its neighbors rotate counterclockwise. 142 spin waves for classical dipoles on regular lattices

DV 5 / ) q ( ω

−π π π π − 2 0 2 1 q/ a

Figure 8.2: Dispersion relation ω(q) from Eq. (8.9b) for spin waves in a dXY spin chain obtained using linear spin-wave theory. It can be observed that the dispersion relation is gapped, i. e. ω(q) > 0 for all q, as expected. Remarkably, the minimal energy excitations occur at the Brillouin-zone boundary rather than at q = 0, which means that the lowest energy excitations are antiferromagnetic perturbations of a ferromagnetic ground state. This can be understood in an enlarged magnetic unit cell, which is equivalent to backfolding the dispersion relation. The backfolded dispersion relation is indicated by the dotted line so that the minima now occurs at the Brillouin-zone center. The lowest energy excitations are, however, in the language of phonons, optical modes rather than acoustic modes. 8.1 spin waves for dipoles on simple lattices 143

8.1.1.3 Analytic behavior of the dispersion relation In the following sections, we will heavily rely on the Taylor expansion of the dispersion relation. However, already for the dispersion relation of the chain lattice in Eq. (8.9), it is not evident that the Taylor expansion exists. Therefore, we discuss the (non-)analytic behavior of the dispersion relation in this section. The trilogarithm Li3 itself is not holomorphic or even meromorphic on C,2 so that it is not guaranteed that the Taylor expansion is well-behaved. Indeed, at q = 0, the dispersion relation ω exhibits the non-analytic behavior of the trilogarithm. There, all derivatives of order higher or equal to 2 diverge. Such a divergence indicates that the thermodynamic limit of the summation in Eq. (8.9a), i. e., the summation up to ∞, is ill-defined for q = 0. Since, the summation is, however, truncated for finite systems, this means that ferromagnetic perturbations at q = 0 crucially depend on the system size and the boundaries of the system. In contrast, the low-energy excitations around q = ±π, in the original Brillouin zone, possess a well-behaved expansion in terms of a ζ-function regularization [217]. Specifically, the derivatives of ω(q) are

 ∂n ∞ cos(qr) 0, n odd = n ∑ 3 (8.10) ∂q r n/2 n−2 d=1 q=π (−1) (2 − 1)ζ(3 − n), n even.

Equation (8.10) can only exhibit a divergence if the ζ function diverges. However, the only divergence of the ζ-function on the real axis R occurs at 1, i. e., at the second derivative (n = 2) of the dispersion relation. There, the factor (2n−2 − 1) = 0, so that the divergence is a removable singularity and the low-energy excitations can be treated by a Taylor expansion in q. To summarize this section, the dispersion relation for spin waves for dXY spins on the chain lattice was computed. The expression has already been found before (see, e. g., Ref. [214]); however, some important observations could be made. Namely, the dispersion relation is gapped as expected for this system. Furthermore, the magnetic unit cell of the excitations appears

2 Holomorphic functions are functions that are complex-differentiable on all of C, whereas meromorphic functions are complex-differentiable functions everywhere but at well- separated poles in C. (Formally, well-separated means that the poles do not form a dense set of points in C). Holomorphic and meromorphic functions are typically considered well- behaved, as they can be described by series expansions. Non-meromorphic functions, such as the trilogarithm, however, can exhibit complicated behavior. For a more comprehensive discussion, the reader is referred to textbooks on complex analysis such as Ref. [216]. 144 spin waves for classical dipoles on regular lattices

to be larger than the magnetic unit cell of the ground state. This observation can be traced back to the “head-to-tail” rule, which favors a staggered rotation of a ferromagnetic conﬁguration.

8.1.2 Spin waves for dXY spins on the square lattice In Section 8.1.1, the spin-wave dispersion relation was determined for the dXY chain. However, the spin-wave dispersion relation is gapped for this system, because it has a trivial ground state. In contrast, the dXY model on the square lattice has a continuous ground-state degeneracy, which is fragile according to the ground-state discussion presented in Chapter 7. Indeed, it has been shown previously that the system undergoes an order- by-disorder transition with temperature [30, 143, 210, 218]. Here, we rederive the literature results in a slightly different way. Anal- ogous to previous work [30], a linear spin-wave calculation is employed in Section 8.1.2.1. However, instead of treating the entire spin-wave spectrum at once, symmetry arguments are used in Section 8.1.2.2 to identify the low-energy sector responsible for the order-by-disorder transition.

8.1.2.1 Dispersion relation of spin waves for dipolar-coupled XY spins on the square lattice First, the notation for the ground state of dXY spins on the square lattice has to be introduced formally: The ground-state parametrization is shown in Fig. 8.3, where the sites inside the magnetic unit cell are numbered 1 to 4. In contrast to the previous versions of this ﬁgure (cf. Figs. 6.1a and 7.1), all angles are measured as polar angles, i. e., starting from the x-axis, in order to emphasize the sense of rotation. The ground-state angles in the magnetic unit cell are

~θ = (θ, π − θ, π + θ, −θ),(8.11)

where θ is the angle deﬁning the ground state. Each of the spins has an XY-anisotropy, as for the example of the chain in Section 8.1.1. However, in contrast to the chain example, four spin-wave modes will be obtained, as the system has a 2 × 2 magnetic unit cell. Specif- ically, a spin-wave excitation, transforming the angle of each spin, can be written as ~ h ~ i ~ ~ SK,k = exp nk(RK +~rk)G Sk,0(θ),(8.12) 8.1 spin waves for dipoles on simple lattices 145

π + θ 2 3 π − θ

θ 4 1 −θ

Figure 8.3: Continuous ground state of the square lattice. This ﬁgure is analogous to Figs. 6.1a and 7.1. However, here the angles are always measured starting from the x-axis so that the sense of rotation becomes clear.

~ where SK,k is the kth spin in the Kth magnetic unit cell. The ground-state spin direction of the kth spin inside any magnetic unit cell is denoted with ~ S0,k. Therefore, the spin waves can be parametrized with the generator G ~ of the Lie algebra so(2) and the rotation angle nk(RK +~rk) that is applied ~ to the spin SK,k. Similar to the expansion of Eq. (8.5) in Section 8.1.1.2, the matrix exponential in Eq. (8.12) is expanded in small rotation angles nk. Here, the expansion is truncated at linear order, which is equivalent to neglecting the interaction between spin waves. In this approximation, h i ~ 1 ~ ~ ~ SK,k ≈ + nk(RK +~rk)G Sk,0(θ).(8.13)

This approximation can be used in the dipolar Hamiltonian to obtain the energy and therefore the dispersion relation of spin waves. Three types of terms arise during this procedure: Terms independent of the rotation angle nk that generate the ground-state energy, terms linear in nk that vanish due to translational invariance, and terms quadratic in nk that constitute the dispersion relation as seen in Section 8.1.1.2. Speciﬁcally, it can be shown that h i† h i sw ~ † ~ ~ ~ HdXY = E0 + ∑ Sk,0(θ) nk(RK +~rk)G HjJkK nj(RJ +~rj)G Sj,0(θ), jJkK (8.14) 146 spin waves for classical dipoles on regular lattices

which can be treated analogously to Eq. (8.9) in Section 8.1.1.2. Hence, the analogous equation Z sw 2 † HdXY = d q~n(~q) M(~q, θ)~n(~q),(8.15a) BZ h i ~ ~ † † ~ M(~q, θ) = ∑ exp i~q · R +~rkj S0,k(θ) G Hk−jGS0,j(θ),(8.15b) ~R

where a shorthand notation for ~n ≡ (n1, n2, n3, n4) has been introduced. As ~n is a vector, M is not yet a dispersion but rather a dispersion matrix. The eigenvalues of this matrix correspond to the dispersion of the individual spin-wave modes. In this situation, in previous studies [30, 143, 210, 218] M was computed explicitly and the entire spin-wave spectrum was determined. For example, Prakash and Henley [30] computed the dispersion matrix3 M using a nearest-neighbor approximation as   3 aν 0 bµ   1 aν 3 bµ 0  M(~q, θ) =   ,(8.16a) 2    0 bµ 3 aν bµ 0 aν 3 2 with aν = [1 + cos (θ)] cos(qx) (8.16b) 2 and bµ = [1 + sin (θ)] cos(qy),(8.16c)

where qx,y = ~q · eˆx,y is x-component and y-component of the momentum, respectively. Equation (8.16) includes all spin-wave modes of the 2 × 2 plaquette. Here, however, a similar approach to that used in Chapter 7 is taken, ﬁrst determining a symmetry-adapted basis to simplify the problem. In this basis set, the low-energy branch of M can be found so that, for all calculations in the low-temperature limit, it is sufﬁcient to restrict M to this symmetry sector.

8.1.2.2 Symmetry analysis of the dispersion matrix As in Section 7.2.1, the computation is well deﬁned but rather tedious in practice. However, as the spin-wave Hamiltonian in Eq. (8.15) has been

3 Here, the notation is different to that used in the publication of Prakash and Henley [30]. They denoted M with Aq, and the ground-state angle φ with θ. Furthermore, they did not use natural variables, so that their matrix included the dipolar interaction strength, which they denoted with J. 8.1 spin waves for dipoles on simple lattices 147 derived from the dipolar Hamiltonian, we know that it is symmetric with respect to P. Therefore, the representation theory can be used to deﬁne a symmetry-adapted basis set for ~n so that M becomes block-diagonal. To construct the symmetry-adapted basis, one can consider that the vector ~n describes how the ith site inside a magnetic unit cell is rotated. When a group element g ∈ P is applied to this vector, the components of ~n are permuted according to the permutation representation [219]. However, as ~n corresponds to rotation angles, each component itself does not transform as a scalar, but rather as a pseudo-scalar, i. e., the rotation angles change sign upon mirroring. The representation corresponding to this transformation behavior will be denoted by the spin-wave representation Σ. The transformation behavior represented by Σ is illustrated for some examples in Table 8.1. In order to construct the symmetry-adapted basis for ~n, Σ has to be de- composed into its irreducible representations. The decomposition is again based on the characters of the representation. The characters can be determined by explicitly writing down the transformation matrices for one element per conjugacy class, as we did with Eq. (7.13). Here, the characters are determined in Table 8.1. Hence, the representation can be reduced to Σ ≡ A2 ⊕ B1 ⊕ E, as seen in Table 8.2. For each of the representations, the symmetry-adapted basis can be found by a well-known algorithm such as shown, for example, in Ref. [204]. Mathematically speaking, let Γ be an irreducible representation of Σ, and let ~n0 be a so-called trial vector in the space of spin-wave rotations in a plaquette. Then a symmetry-adapted state ~nΓ, which transforms as Γ, can be found using

~nΓ = ∑ χΓ(g)Σ(g)~n0,(8.17) g∈P where χΓ is the character function of Γ. This algorithm leads to the states ~ ( ) nA2 ∝ 1, 1, 1, 1 ,(8.18a) ~ ( − − ) nB1 ∝ 1, 1, 1, 1 ,(8.18b) (1) (2) ~nE ∝ (1, 0, −1, 0), and ~nE ∝ (0, 1, 0, −1),(8.18c) where the representation E has two symmetry-adapted states since the representation is two-dimensional. These states also have a physical interpretation. Namely, each of these ~ states transforms a plaquette of spins as shown in Table 8.3. The vector nA2 148 spin waves for classical dipoles on regular lattices

defined in Eq. (8.18a) applies a rotation of all states in the same direction. This vector corresponds to the usual low-energy spin-wave excitation for regular (anti-)ferromagnets. However, in Fig. 8.3 it can be observed that, for a ground-state configuration to stay invariant, the spins have to be rotated in opposite senses to give an alternating checkerboard pattern due to the “head-to-tail” rule. The corresponding symmetry-adapted vector is ~ nB1 , defined in Eq. (8.18b). The representation B1 of this symmetry-adapted vector is called the Goldstone mode representation. Here, the primary interest is the description of the order-by-disorder transition that breaks the SO(2) ground-state degeneracy at infinitesimal temperature. It is therefore only necessary to focus on the small energy spin waves. Since the only gapless mode is the Goldstone mode describing the SO(2) ground state, it is sufficient to consider the Goldstone representation symmetry sector of M. Hence, for the low energy excitations, Eq. (8.15) can be restricted to the symmetry sector B1: Z sw = 2 ~ (~)† (~ )~ (~) HdXY d q nB1 q M q, θ nB1 q (8.19a) BZ Z ⇒ eff = 2 | (~)|2 (~ ) HdXY d q nB1 q ω q, θ ,(8.19b) BZ (~ ) where nB1 is now merely a scalar. Then, ω q, θ , the dispersion of the Gold- stone mode, can be computed as a single lattice summation. It can be observed that the dispersion relation has a ground-state angle θ-independent part and a θ-dependent part:

ω(~q, θ) = ωiso(~q) + cos(2θ)ωan(~q),(8.20a) ~·~ 0 eiq r ωiso(~q) = x2 ((−1)x − 2(−1)y) + y2 ((−1)y − 2(−1)x) , ∑ |~r|5 (x,y) 2 (8.20b) ~·~ 0 eiq r ωan(~q) = x2 ((−1)x + 2(−1)y) − y2 ((−1)y + 2(−1)x) , ∑ |~r|5 (x,y) 2 (8.20c)

where ~r = (x, y), and the primed summation denotes that the summation excludes the point (x, y) = (0, 0). This dispersion relation is shown in Fig. 8.4 for a truncation of 50 lattice sites. Speciﬁcally, in Fig. 8.4c the disper- π iso sion is shown for θ = 4 so that ω ≡ ω . In this ﬁgure, it can be observed that for very small |~q|, the dispersion is isotropic, i. e., ωiso(~q) ≡ ωiso(|~q|) 8.1 spin waves for dipoles on simple lattices 149

Table 8.1: Construction of the characters χΣ of the spin-wave representation Σ for XY spins on the square lattice, under the assumption of a 2 × 2 magnetic unit cell.

g Graphical representation Σ(g) χΣ(g)   1 0 0 0 n2 n3 n2 n3   0 1 0 0 1 1   4 ⇒   n1 n4 n1 n4 0 0 1 0 0 0 0 1   0 0 0 1 n2 n3 n3 n4   1 0 0 0 C C4   0 4 ⇒   n1 n4 n2 n1 0 1 0 0 0 0 1 0   0 0 1 0 n2 n3 n4 n1   0 0 0 1 C C2   0 2 ⇒   n1 n4 n3 n2 1 0 0 0 0 1 0 0   0 −1 0 0 n2 n3 −n1 −n4   −1 0 0 0  σ σh ⇒x   0 n n − −  0 0 0 −1 1 4 n2 n3   0 0 −1 0   −1 0 0 0 n2 n3 −n4 −n3    0 0 0 −1 σxy − σd ⇒   2 n n − −  0 0 −1 0  1 4 n1 n2   0 −1 0 0 150 spin waves for classical dipoles on regular lattices

Table 8.2: Character table of the point symmetry group of the square lattice C4v and the reduction of Σ for this lattice. 1 C4v 2C4 C2 2σh 2σd

A1 1 1 1 1 1

A2 1 1 1 −1 −1

B1 1 −1 1 1 −1

B2 1 −1 1 −1 1 E 2 0 −2 0 0

Σ 4 0 0 0 −2 ≡ A2 ⊕ B1 ⊕ E

Table 8.3: Interpretation of the symmetry-adapted spin-wave modes found in Eq. (8.18). For example, A2 corresponds to a homogeneous rotation of all sites in the same direction, whereas B1 transforms the sites in a checkerboard pattern. Irreducible Graphical representation Eq. representation

n n

A2 Eq. (8.18a) n n

−n n

B1 Eq. (8.18b) n −n

0 −n

n 0

E Eq. (8.18c)

−n 0

0 n 8.2 generalization of spin waves to other lattices 151 and that for larger |~q|, the dispersion relation starts to become slightly anisotropic. In contrast to the dispersion relation of the chain lattice, Eqs. (8.20b) and (8.20c) cannot be evaluated analytically. However, as only the low- energy excitations for small |~q| are relevant, it is sufficient to expand Eq. (8.20) to the lowest order in ~q. This expansion leads to the quadratic dispersion relation ! c − c cos(2θ) 0 ω(~q, θ) = ~q † 1 2 ~q + O(q4) (8.21a) 0 c1 + c2 cos(2θ) 2 2 4 = c1q + c2 cos(2θ) cos(2φq)q + O(q ) (8.21b) where q = |~q| is the length of the vector ~q, and φq is the polar angle of ~q. The coefficients c1 and c2 can be extracted by fitting the numerical data presented in Fig. 8.4. Evaluating the dipolar interaction in Eqs. (8.20b) 2 2 2 2 and (8.20c) for a truncation of x + y 6 rcut = 50 leads to an estimate of c1 ≈ 0.943 and c2 ≈ 0.802. The dispersion in Eq. (8.20) and its approximation in Eq. (8.21) share some interesting features. This spin-wave mode is a Goldstone mode, and as such, the dispersion is gapless, i. e., ω(~q → 0) = 0. Furthermore, it π 3π can be observed that for θ ∈ {0, 2 , π, 2 }, the dispersion relation shows a pronounced soft direction (cf. Figs. 8.4a and 8.4b) due to ωan that softens π 3π 5π 7π one direction and stiffens the other. However, for θ ∈ { 4 , 4 , 4 , 4 } the dispersion is ω ≡ ωiso so that for small |~q| the dispersion is isotropic as can be seen in Fig. 8.4c.

8.2 generalization of spin waves to other lattices

In Section 8.1, it was shown for particular lattices how the excitation spectrum could be determined. Similar calculations have been carried out before in order to describe the order-by-disorder transition in dipolar- coupled spin systems on the square lattice or the honeycomb lattice [30, 143, 210], by computing the free energy. However, in Chapter 7, it was shown that many fundamental properties of such systems can be traced back to the point symmetry group. Hence, in this section, we attempt to generalize the calculations shown in Section 8.1 by employing representation theory implementing the same methods as used in Chapter 7. The generalization of the order-by-disorder transition is accomplished by realizing that we only need to describe the lowest-energy excitations in the low-temperature limit. Therefore, in Section 8.2.1, we ﬁrst derive an 152 spin waves for classical dipoles on regular lattices

(q) (q) 11 11 10 10 9 9 2 2 8 8 7 7

y 6 y 6

q 0 q 0 5 5 4 4 3 3 2 2 2 2 1 1 0 0 0 0 2 2 2 2 qx qx

π (a) θ = 0 (b) θ = 2

(q) 11 10 9 2 8 7

y 6

q 0 5 4 3 2 2 1 0 0 2 2 qx

π (c) θ = 4

Figure 8.4: Dispersion relation ω(~q, θ) for spin waves in the dXY model on the square lattice obtained by evaluating Eq. (8.20) with a truncation of |~r| 6 rcut = 50 applied to the summation. The contour lines are located 1 1 at ω(~q) = 10 , 2 , 1, 2 in white, light gray, dark gray, and black, respectively. It can be observed that, for θ = π/4, the dispersion relation is symmetric and nearly isotropic for small |~q|, whereas, for θ = 0 and θ = π/2, the spin-wave dispersion has one soft mode and one stiff mode. 8.2 generalization of spin waves to other lattices 153 effective low-energy Hamiltonian by describing the dispersion relation of the Goldstone mode in the ~q → 0 limit. Subsequently, the free energy is computed in Section 8.2.2, which determines the magnetic phase that is selected during the thermal order-by-disorder transition.

8.2.1 Dispersion relation Here, the results obtained for specific lattices in Section 8.1 are generalized to any lattice where our method from Chapter 7 is applicable and provides a continuous ground-state degeneracy. In such a system, a Gold- stone mode emerges as a consequence of the continuous ground-state degeneracy. Specifically, the Goldstone mode is a spin-wave mode that continuously transforms one ground state into another in the long-wavelength limit, so that ω(~q → 0) → 0. In this setting, the system is described by the magnetic unit cell with N sites and a degeneracy parameter, which determines the configuration within the ground-state manifold. Hence, the ground-state is characterized by ~ ~ 0 ~ 0 S(Φ) = S1 (Φ), ··· , SN(Φ) ,(8.22) ~ where S0,j(Φ) is the ground-state configuration of the jth site inside the magnetic unit cell described by the degeneracy parameter Φ, which is an angle in d = 2 and a solid angle in d = 3. The framework of this calculation is analogous to the treatment of spin waves on the square lattice discussed in Section 8.1.2, namely, a position- dependent rotation that is parametrized using the underlying Lie algebra is applied to each spin. Here, the discussion is generalized to include other degrees of freedom beyond XY spins, so that the Lie algebra can be changed from so(2) describing XY spins to so(3) for Heisenberg spins. For such systems, more than one generator G is required per site to describe general rotations. Hence, the rotation angles obtain a new index, describing the rotation axis. Then, the perturbed spin reads in the non-interacting spin-wave approximation as " # ~ ~ 1 ~ ~ 0 Sj(R, Φ) = + ∑ nj,k R +~rj Gk Sj (Φ),(8.23) k where Gk are the generators of the Lie algebra and nj,k the rotation applied to the jth spin along the kth axis. The offset ~rj of the jth site inside the magnetic unit cell is not necessary in the argument of nj,k. However, as 154 spin waves for classical dipoles on regular lattices

seen in Section 8.1.2, it is convenient to include the offset in the argument for explicit computations of the dispersion matrices. Equation (8.23) is then used in the dipolar Hamiltonian so that the effective Hamiltonian becomes

all modes H = H0 + Hsw (Φ),(8.24a) Z all modes d † Hsw (Φ) = d q~n(~q) M(~q, Φ)~n(~q),(8.24b) BZ h i 0 ~ ~ 0 † † ~ 0 [M (~q, Φ)]jj0kk0 = ∑ exp i~q · (R +~rjj0 ) Sj (Φ) Gk Hjj0 Gk0 Sj0 (Φ),(8.24c) ~R where the primed summation again denotes that the self-energy term 0 ~ (j = j and R = 0) is not included in the summation. Here, H0 is the Hamiltonian describing the ground state, which is Φ-independent and is from now on neglected. The quadratic form in Eq. (8.24b) arises due to the linearization in Eq. (8.23) and due to translational invariance so that the linear contributions vanish. Finally, M has four indices: Here, j and j0 denote the interaction between spin waves on the sites j and j0 in the magnetic unit cell. The indices k and k0 denote the different rotation axes, which have to be introduced for spins associated with Lie algebras that contain more than one generator such as Heisenberg spins. As in Section 8.1.2, M can be analyzed in terms of its transformation properties with respect to the point symmetry group P. Here, M acts on the vector space of rotations ~n = (n1,1, ··· , nN,K), where the ﬁrst index denotes the site inside the magnetic unit cell, and the second index denotes the rotation axes. Therefore, the spin-wave representation is given by Σ = τ ⊗ ρ, where τ is the permutation representation of the magnetic unit cell and ρ is the representation determining the transformation properties of the rotations. For XY spins, ρ denotes the transformation property of a pseudo-scalar. For Heisenberg spins, however, the transformation property is slightly more complicated as the rotation axis can additionally be permuted. Formally, the representation Σ acts on a vector of rotation angles ~n according to

Σ(g)~n = (··· , sρ(g)nτ(g)◦j, ρ(g)◦k, ··· ),(8.25)

where g is an element of the point symmetry group and sρ(g) = ±1 determines the possible change in the sense of rotation due to g since the angles transform as pseudo-scalars rather than scalars. As shown in Section 8.1.2, the next step is to decompose Σ into irreducible representations in order to find the Goldstone representation G 8.2 generalization of spin waves to other lattices 155 that is associated with the ground-state degeneracy and the Goldstone mode. Intuitively, the Goldstone mode transforms the ground-state degeneracy parameter Φ in Eq. (8.22) to Φ + δΦ on large length scales. Formally, the Goldstone mode can be identified by finding a symmetry-adapted ~nG so that Σ(g)~nG ∼ G(g)~nG and † ~nG M(~q → 0, Φ)~nG = 0. (8.26) The existence of this gapless mode is guaranteed as a result of the Gold- stone theorem [220, 221], even if spin-wave interactions are included. As the other spin-wave modes are gapped, the Goldstone mode is the only relevant spin-wave mode to describe the low-energy physics and, with that, the order-by-disorder transition at low temperatures. In analogy to the example for the square lattice, we generalize the result from Eq. (8.20a) to

ω(~q, Φ) = ωiso(~q) + f (Φ)ωan(~q),(8.27) where the dispersion is separated into the isotropic part ωiso and the anisotropic part ωan. In Eq. (8.27), the influence of the specific ground-state configuration on the dispersion relation is described by f (Φ). The isotropic part describes the recovery of the continuous ground-state degeneracy for ~q → 0 and further ensures that every excitation costs energy. In contrast, the anisotropic dispersion relation relates the lattice symmetries to the dispersion relation. Since ω(~q, Φ) has to transform with P, so does f (Φ)ωan(~q). Hence, the following ansatz can be made for small |~q|: ( ) an(~ ) = ( ) ( ) (|~|) f Φ ω q, Φ ∑ fΓi Φ fΓj Ω~q ωΓiΓj q ,(8.28) Γi,Γj P where Γi are irreducible representations of , and fΓi is the associated angular harmonic. Furthermore, the wave vector is written in polar coordinates described by the length |~q| and the (solid-)angle Ω~q. Since ω transforms trivially under P, so does f (Φ)ωan and therefore terms where Γi 6= Γj vanish. Furthermore, only the Goldstone mode is necessary to describe the low-energy physics, so that only the Goldstone mode representation remains from the summation in Eq. (8.28). Hence,

an an f (Φ)ω (~q) = fG (Φ) fG (Ω~q)ω˜ (|~q|),(8.29) where ω˜ an has up to a scalar factor c the same asymptotic behavior as ωiso to maintain the stability of the system (ω˜ an ≡ c · ωiso for small |~q|). 156 spin waves for classical dipoles on regular lattices

For the sake of simplicity, a Debye-like approximation4 is employed, so that the dispersion is quadratic up to an ultraviolet-cutoff (UV cutoff)5 Λ [222]. The Hamiltonian, therefore, reads Z eff d Hsw = d q~nG (−~q) ·~nG (~q)ω(~q, Φ),(8.30a) |~q|<Λ 2 with ω(~q, Φ) = ω0|~q| [1 + ε fG (Φ) fG (Ω~q)],(8.30b) in analogy to Eq. (8.21b) for the square lattice. Normally, in the Debye approximation, the UV cutoff Λ is chosen so that the total number of states is conserved. Here, however, we do not need to determine Λ because it is not required to describe the order-by-disorder transition (see Section 8.2.2). It seems feasible to generalize the following discussion to cases where Eq. (8.30) does not hold, but the derivations become much more complex. However, using the approximation, which confines the dispersion relation to regions where it is quadratic is quite restrictive and, in principle, also uncontrolled.6 Nevertheless, as the main focus of this chapter lies on small energy excitations around ~q = 0, Eq. (8.30) should be sufficient to capture the order-by-disorder phenomena correctly. In this region, the main features of the dispersion are determined by the symmetry-adapted angular harmonics f . For many purposes in representation theory, it is sufficient to work with angular harmonics f which are only determined up to a prefactor. However, in this chapter, they are incorporated not only taking their symmetry into account but also using them in quantitative statements such as Eq. (8.30b), so that they need to be properly normalized. As the symmetry-adapted angular harmonics f appear as scalar products integrated over the entire sphere, the natural normalization condition is Z Z 2 1 dΩ [ fG (Ω)i] = , so that dΩ fG (Ω) · fG (Ω) = 1, (8.31) S dG S

4 The Debye approximation was ﬁrst used in the determination of the heat capacity of acoustic phonons where the dispersion relation is assumed to be linear up to the Debye temperature [222]. The quadratic dispersion assumed here resembles more the dispersion obtained from the treatment of magnons in a ferromagnet [223]. 5 Historically, the word ultraviolet was used to refer to electromagnetic waves in the ultraviolet range, which were problematic in the context of black-body radiation when they were not treated as quantum-mechanical photons. Here, however, we use the word ultraviolet to refer to the short wavelength regime of all excitations with a wave-like character. 6 Here, uncontrolled means that, in contrast to well-controlled methods such as a Taylor- approximation, there is no error bound on this approximation. This implies that rather than a quadratic dispersion relation other terms could appear in the radial part of the dispersion relation, such as linear or logarithmic terms. 8.2 generalization of spin waves to other lattices 157

where dG = ddeg is the dimension of G (and therefore the dimension associated with the degeneracy), and S is the (dG − 1)-sphere.

8.2.2 Free energy The effect of thermal fluctuations on the continuous ground-state degeneracy can be determined by the free energy as this thermodynamic potential is minimized for systems in thermal equilibrium. As discussed in Section 2.5, the free energy can be obtained from the canonical partition function, which can be determined from the Hamiltonian. Here, it is sufficient to describe the system by the effective Hamiltonian given in Eq. (8.30), rather than the full Hamiltonian, as the order-by-disorder transition occurs at infinitesimal temperature. To compute the canonical partition function, all microscopic degrees of freedom have to be integrated out. Specifically, in Eq. (8.30a), the angles ~n(~q) have to be integrated out for every ~q. The integration is therefore a path integral: Z h eff i Z = D~n(~q) exp −βHsw[~n(~q)] (8.32a) Z Z = D~n(~q) exp −β ddq~n(−~q)~n(~q)ω(~q, Φ) ,(8.32b) |~q|<Λ

eff where the effective low-energy Hamiltonian Hsw from Eq. (8.30) is used. The free energy can then be obtained from the canonical partition function by the equation in Table 2.1. Namely,

1 F = − log Z (8.33a) β 1 Z Z = − log D~n(~q) exp −β ddq~n(−~q)~n(~q)ω(~q, Φ) (8.33b) β |~q|<Λ

where the free energy depends on the temperature via β, and on the ground-state angle Φ via the dispersion relation as given in Eq. (8.30b). To simplify Eq. (8.33), we use the fact that the exponential contains only a single integration over ~q.7 Therefore, the system behaves according to a

7 This is due to the approximation that the spin-waves are non-interacting. In contrast, if spin- wave interactions would be included, by expanding the matrix exponential in Eq. (8.23) to higher order, the exponential in Eq. (8.33) would generally contain several integrations over different ~q. 158 spin waves for classical dipoles on regular lattices

theory of free particles so that the integration over ~n and the integration over ~q can be exchanged. Therefore,

log π Z 1 Z F(β, Φ) = − ddq + ddq log βω(~q, Φ),(8.34) 2β |~q|<Λ 2β |~q|<Λ | {z } F˜0(β)

by evaluation of the Gaussian integral over ~n in Eq. (8.33b). Further, using the factorization in Eq. (8.30b), the free energy can be simpliﬁed to

Z Λ Z 1 d−1 2 F(β, Φ) = F˜0(T) + dq q dΩ~q log βω0q 2β 0 Sd−1 Z Λ Z 1 d−1 + dq q dΩ~q log[1 + ε fG (Φ) fG (Ω~q)],(8.35) 2β 0 Sd−1

where we call the ﬁrst integral the radial integral Frad and the second integral the angular integral Fang. Here, it is crucial to restrict the integration to small q as, otherwise, the UV catastrophe8 inherent to classical systems emerges. For the dispersion relation given in Eq. (8.30), the UV cutoff is given by the Λ of the Debye approximation. In contrast, if the Debye approximation is not used, the UV cutoff preventing the UV catastrophe is the Brillouin zone boundary BZ.

8.2.2.1 The radial integral and the breakdown of the classical treatment Here, we will observe that Eq. (8.35) is intrinsically ﬂawed, being a classical treatment in the T → 0 limit. In order to show this, the radial integral Frad can be considered:

Z Λ Z 1 d−1 2 Frad = dq q dΩ~q log βω0q = 2β 0 Sd−1 TΛd d ω Λ2 − vol Sd−1 1 − log 0 ,(8.36) d2 2 T

8 The UV catastrophe denotes the phenomenon that, if there is no lower bound on the wavelength of harmonic modes, then the system would carry an inﬁnite energy. This is due to the equipartition theorem that states that each mode of the harmonic oscillators carries an average energy of kBT. If there is no lower limit on the wavelength, there is an inﬁnite number of modes. If a UV cutoff is introduced, either by a cutoff Λ or by using the Brillouin zone BZ, this UV divergence disappears. In a quantum-mechanical treatment, this divergence issue is resolved by treating the excitations as indistinguishable bosons following Bose-Einstein distribution. 8.2 generalization of spin waves to other lattices 159

which can be combined with F˜0(T) to give

TΛd log π 1 1 ω Λ2 F = F˜ + F = − vol Sd−1 + − log 0 . 0 0 rad d 2 d 2 T (8.37)

The breakdown of the classical limit can be illustrated by the entropy:

∂F S = − 0 (8.38a) ∂T Λd ∂ log π 1 1 ω Λ2 = vol Sd−1 T + − log 0 (8.38b) d ∂T 2 d 2 T Λd 1 1 log π 1 ω Λ2 = vol Sd−1 + + − log 0 ,(8.38c) d 2 d 2 2 T which can become negative at low temperatures. This is a typical problem in classical systems. The free energy in Eq. (8.37) has the same temperature dependence as the free energy of an ideal gas that exhibits the same problem at low temperatures. Furthermore, this issue also arose in previous treatments of dipolar-coupled systems, but it has largely been ignored [30, 143, 210]. In order to resolve this negative entropy, the system would have to be treated quantum mechanically. The simplest approach to include quantum mechanics is to treat excitations as bosons so that, rather than Boltzmann statistics, Bose-Einstein statistics would have to be used in Eq. (8.32). In- deed, if Bose-Einstein statistics would be used to compute the partition function, then a similar angular integral appears in the calculation, so that the calculation presented in the next section applies for either statistics. However, introducing the excitations as quantum objects, without treating the underlying spins quantum mechanically could lead to wrong conclusions as the quantum ﬂuctuations are not necessarily adequately captured. Indeed, to properly treat the system, a large-S expansion would have to be conducted. This is, however, beyond the scope of this thesis, and as in previous literature [30, 143], we simply ignore the problematic term.

8.2.2.2 The angular contribution and the order-by-disorder transition The order-by-disorder transition associated with the thermal ﬂuctuations arises from the second, angular integral in Eq. (8.35), which contains the dependence of the thermal free energy on the ground-state parameter Φ. As the ground state of the system is stable, ω(~q) > 0 for all ~q 6= 0, so that 160 spin waves for classical dipoles on regular lattices

ε fG (Φ) fG (Ω~q) < 1. Hence, the logarithm can be expressed by its Taylor series around 1 for small ε fG (Φ) fG (Ω~q). Therefore, this integral reads as

d Z ∞ k−1 Λ (−1) k Fang(T, Φ) = dΩ~q [ε fG (Φ) fG (Ω~q)] ,(8.39) d−1 ∑ 2dβ S k=1 k where the next step is to exchange the integration and the summation. As fG is an angular harmonic, the odd powers vanish. Then to lowest order in ε (and therefore also in fG )

h 2 2 4 i Fang(T, Φ) ∝ T 1 − ε fG (Φ) + O(ε ) ,(8.40)

where higher order terms do not change the position of the minima of Fang(T, Φ) as only even powers of fG appear.

8.2.2.3 Physical interpretation: Order-by-disorder transition For low temperatures, the free energy can be summarized as

2 F(T, Φ) = F0(T) − cT fG (Φ),(8.41) where c > 0. Since the system minimizes the free energy, it maximizes 2 fG (Φ) with respect to Φ for any ﬁnite temperature T. This leads to the order-by-disorder transition that breaks the continuous ground-state degeneracy. Before the result in Eq. (8.41) is applied to some simple systems in Sec- tion 8.3, the limitations of this derivation should be pointed out. Some approximations were applied to derive Eq. (8.41): The Hamiltonian was only derived in the Debye approximation, and it was assumed that a completely classical treatment is sufﬁcient, even in the T → 0 limit. Both assumptions, even though uncontrolled, are not expected to lead to deviations from our main results, namely that the thermally selected phase max- 2 imizes fG (Φ) and that the transition occurs at T = 0. Indeed, in Section 8.3 it will be observed that our results are in agreement with previous literature where analytical methods [30, 143, 210] and simulations [65, 143, 192, 211, 212, 224] were used for different systems. Nevertheless, our results are based on some uncontrolled approximations so that the predictive power of Eq. (8.41) is somewhat limited. Finally, it should be mentioned that, from Eq. (8.41), it follows that the order-by-disorder transition at T = 0 is not a proper phase transition, but rather that it is merely a crossover phenomenon as the Φ-dependent contribution to the free energy does not diverge in any derivative. However, this 8.3 application of the dipolar spin-wave theory 161 is probably an artifact of the linearized spin-wave treatment and it would only be possible to clarify if the transition is a proper phase transition or not by including interactions between spin waves.

8.3 application of the dipolar spin-wave theory

In Section 8.2.2, it was shown how thermal excitations break the continuous ground-state degeneracy of dipolar-coupled spin systems. However, Eq. (8.41) is formulated in terms of the symmetry-adapted angular harmonic fG so that interpretation of Eq. (8.41) is not entirely transparent. Therefore, as in Chapter 7, we illustrate the application of Eq. (8.41) to some specific systems to gain a physical insight into the typical thermal order-by-disorder transitions. In the first example, the results presented in in Section 8.1.2 on dXY spins on the square lattice are extended to determine the magnetic phase at a finite but small temperature in Section 8.3.1. Here, it is shown that our theory is in agreement with previous literature [30, 52, 143, 192, 193, 210] and the results presented Chapter 6. Dipolar-coupled Heisenberg spins on the simple cubic lattice serve as the second example in Section 8.3.2, where the numerical finding of a nematic ordering at finite temperatures [211, 212] can be confirmed.

8.3.1 Dipolar-coupled XY spins on the square lattice Here, the discussion concerning the dXY system on the square lattice presented in Section 8.1.2 is continued and brought into the generalized spin- wave context derived in Section 8.2. The discussion here is kept short as most of the necessary calculations for the application of Eq. (8.41) have already been carried out in Section 8.1.2. The ground state of the dXY system on the square lattice can be computed using the method of Luttinger and Tisza and our method presented in Chapter 7. The system exhibits an SO(2) degeneracy with the magnetic unit cell depicted in Fig. 8.3. In Section 8.1.2.1, it was observed that the spin-wave representation Σ is four-dimensional and decomposes into

Σ ≡ A2 ⊕ B1 ⊕ E,(8.42) according to Table 8.2. The symmetry-adapted spin-wave modes were constructed and it was observed that the Goldstone representation G = B1. 162 spin waves for classical dipoles on regular lattices

Now, the relevant symmetry-adapted angular harmonic has to be determined, which can be derived from the symmetry-adapted polynomial already seen in Chapter 6:

2 2 fG (θ) ∝ x − y ∝ cos(2θ).(8.43)

Therefore, in the case of dXY spins on the square lattice the free energy derived in Eq. (8.41) now becomes

2 F = F0(T) − cT cos (2θ),(8.44)

so that the thermal order-by-disorder transition leads to a phase that maximizes cos2(2θ). This phase is the striped phase so that our prediction is in agreement with previous literature [30, 143, 192, 210] and the simulations shown in Chapter 6.

8.3.2 Dipolar-coupled Heisenberg spins on the simple cubic lattice As a higher dimensional example, the effect of temperature can be determined for dipolar-coupled Heisenberg spins on the simple cubic lattice. In Chapter 7, it was shown that this system exhibits an SO(3) ground-state degeneracy. Particular ground states are depicted in Fig. 8.5. Speciﬁcally, the conﬁg- urations that are shown form the orthogonal basis set

~ yi+zi ~ xi+zi ~ xi+yi Xi = (−1) eˆx, Yi = (−1) eˆy, Zi = (−1) eˆz,(8.45)

so that the complete ground-state manifold can be obtained by a linear combination as follows: ~ ~ ~ ~ Si = αxXi + αyYi + αzZi,(8.46a) 2 2 2 where 1 = αx + αy + αz,(8.46b)

according to Chapter 7. As Eq. (8.46b) describes the sphere in three dimensions, it is convenient to express ~α ≡ (αx, αy, αz) in terms of polar angles:

αx = cos φ sin θ, αy = sin φ sin θ, αz = cos θ,(8.47)

so that Eq. (8.46b) is trivially fulfilled. To find the thermally selected magnetic phase, the Goldstone representation G has to be determined. One way to find G is to construct the spin-wave representation Σ and to find the symmetry-adapted spin-wave 8.3 application of the dipolar spin-wave theory 163

~ ~ ~ ~ (a) S = Xi (b) S = Yi

~ ~ (c) S = Zi

Figure 8.5: The three basic arrays described by Eq. (8.45) that are found by the LT-method combined with our method derived in Chapter 7. 164 spin waves for classical dipoles on regular lattices

modes. Here, Σ = τ ⊗ ρ, where τ is the 8-dimensional permutation representation of the sites inside the magnetic unit cell, and ρ is the 3-dimensional representation that permutes (up to a sign) the generators of the SO(3) rotations that describe the Heisenberg spins. Hence, Σ is 24-dimensional. It would be tedious to derive the transformation matrices in the 24-dimensional space. However, as

χτ⊗ρ(g) = χτ(g)χρ(g),(8.48) the characters can be constructed more efﬁciently. The construction of the characters of Σ is carried out in Table 8.4. Hence, we can determine the decomposition of Σ into irreducible representations as

Σ = τ ⊗ ρ ≡ A2g ⊕ Eg ⊕ 2T1g ⊕ T2g ⊕ A1u ⊕ Eu ⊕ T1u ⊕ 2T2u.(8.49) Similar to the calculation in in Section 8.1.2.2, the symmetry-adapted spin- wave modes could be determined for each of the irreducible representations. However, to construct the symmetry-adapted spin-wave modes, all the transformation matrices have to be known. Hence, this construction, even though possible, is very tedious in practice. There is, nevertheless, also an easier way to ﬁnd the representation G. As the representation G describes the Goldstone mode of an SO(3) degenerate state, G has to be two-dimensional because there are two angles that can be changed by the Goldstone mode. As seen in Table 8.4, there are only two 9 two-dimensional representations, namely Eg and Eu. To ﬁnd out which of the two representations describes G, the transformation properties of the

Goldstone mode for a symmetry element g ∈ P where χEg (g) 6= χEu (g) can be considered. A suitable symmetry-group element is, for example, the ~ ~ inversion i. As Sx,y,z = Sx+1,y+1,z+1, the inversion has to leave the angles invariant and therefore χG (i) = 2. Therefore, the Goldstone representation is determined to be G = Eg, which can also be derived from the decomposition of ρ into irreducible representations followed by a determination of the spin-wave modes. In the next step, the symmetry-adapted angular harmonic should be found. For Eg, this function is commonly given as

2 2 2! 2 2 ! ˜ 2z − x − y 2 cos θ − sin θ fEg = ∝ ,(8.50) x2 − y2 sin2 θ(cos2 φ − sin2 φ) 1 9 To remind the reader, the dimension of a representation Γ is deﬁned as dim Γ = χΓ( ). Therefore, the dimension of a representation can easily be found in the character table as the entry in the ﬁrst column (the column headed by 1) and the row corresponding to the representation Γ. 8.3 application of the dipolar spin-wave theory 165

Table 8.4: Character table of Oh, the point symmetry group of the cubic lattice and characters for τ, ρ, and Σ = τ ⊗ ρ. The reduction of Σ that follows from this character table is given in Eq. (8.49). 1 2 Oh 8C3 6C2 6C4 3C4 i 6S4 8S6 3σh 6σd A1g 1 1 1 1 1 1 1 1 1 1

A2g 1 1 −1 −1 1 1 −1 1 1 −1

Eg 2 −1 0 0 2 2 0 −1 2 0

T1g 3 0 −1 1 −1 3 1 0 −1 −1

T2g 3 0 1 −1 −1 3 −1 0 −1 1

A1u 1 1 1 1 1 −1 −1 −1 −1 −1

A2u 1 1 −1 −1 1 −1 1 −1 −1 1 Eu 2 −1 0 0 2 −2 0 1 −2 0

T1u 3 0 −1 1 −1 −3 −1 0 1 1

T2u 3 0 1 −1 −1 −3 1 0 1 −1 τ 8 2 0 0 0 0 0 0 0 4 ρ 3 0 −1 1 −1 3 1 0 −1 −1 Σ 24 0 0 0 0 0 0 0 0 −4 166 spin waves for classical dipoles on regular lattices

Figure 8.6: The function f 2 is shown as a spherical plot. Namely, a large ra- Eg dial value corresponds to a large value of the function. It can, therefore, be observed that the function f 2 is maximal along the axes. Eg

where the Cartesian form is typically found in textbooks accompanying the character tables. Here, however, the polar form is preferable, since we want to determine the ground-state angle Φ = (φ, θ) that minimizes the free energy given in Eq. (8.41). It should be noted that Eq. (8.50) has to be normalized so that √ ! 1 5 1 + 3 cos(2θ) fEg (φ, θ) = √ √ ,(8.51) 4π 4 2 3 sin2(θ) cos(2φ)

and therefore 1 5 f 2 (φ, θ) = [12 sin4(θ) cos2(2φ) + (1 + 3 cos(2θ))2].(8.52) Eg 4π 16 8.4 conclusion 167

In Fig. 8.6, f 2 (φ, θ) is shown, where it becomes evident that f 2 is maxi- Eg Eg mized at  (φ , 0) ,  0  0, π ,  2  h i  π , π , 2 ( ) = 2 2 arg max fEg φ, θ (8.53) (φ,θ)  π  π, 2 ,   3π π  2 , 2 ,   (φ0, π) , where φ is arbitrary. This means that f 2 (φ, θ) is maximized along the 0 Eg axes. Since the free energy given in Eq. (8.41) is for the special case of dipolar- coupled Heisenberg spins given by ( ) = ( ) − 2 ( ) F T, φ, θ F0 T cT fEg φ, θ ,(8.54) ﬁnite temperatures lead to an order-by-disorder transition selecting the angles that are described by Eq. (8.53). This corresponds to the magnetic phase where the spins align along the axes of the lattice. This phase is very similar to the (thermally selected) striped phase of the dXY model on the square lattice and therefore this phase is also called the striped phase. Examples for the magnetic unit cell of this striped phase are depicted in Fig. 8.5. In conclusion, here it was determined that thermal excitations break the continuous ground-state degeneracy of dipolar-coupled Heisenberg spins on the simple cubic lattice. The thermally selected phase is a striped phase, which is very similar to the striped phase of the dXY model on the square lattice that was discussed in Section 8.3.1. Our results for dipolar-coupled Heisenberg spins on the cubic lattice are in agreement with previous literature, where it was observed that the system exhibits an order-by-disorder transition to the striped phase at ﬁnite temperatures [211, 212].

8.4 conclusion

In this chapter, the ground-state discussion from Chapter 7 was extended including thermal excitations given by spin waves, which were treated by a linearized calculation so that the spin waves were effectively non- interacting. In the ﬁrst part of this chapter, non-interacting spin waves 168 spin waves for classical dipoles on regular lattices

were considered in dipolar-coupled spin systems on two simple lattices. The first example was the dXY model on the linear chain, where we observed that the magnetic unit cell for excitations is twice as large as the magnetic unit cell for the ground state. As a second example, dXY spins on the square lattice were considered for which the dispersion relation was obtained without the need to truncate the Hamiltonian. In this example, we also used the symmetries for the first time to identify the Goldstone mode of the system. This symmetry discussion was then generalized to gain an insight into systems with a continuous ground-state degeneracy according to the construction of Chapter 7. With the calculation for the spin waves in the square lattice readily generalizable, the dispersion relation could be obtained for a general system for small |~q|. This led to the prediction of the magnetic phase that is selected due to the thermal order-by-disorder transition. In- deed, the prediction is rather general, so that it is expected that all dipolar systems with a continuous degeneracy, originating from the mechanism discussed in Chapter 7, exhibit this type of order-by-disorder transition. The results of this general discussion were illustrated explicitly on two examples: The first example was the dXY system on the square lattice where the predictions agree with previous literature [30, 143, 192, 210], namely that the system exhibits an order-by-disorder transition to a striped phase. The second example consisted of dipolar-coupled Heisenberg spins, placed on the simple cubic lattice where a similar behavior was observed. Here, thermal fluctuations also led to a striped phase, in agreement with previous literature [211, 212]. The calculations presented in this chapter could be extended in various ways: In Chapter 6 and in previous literature [30, 143, 192–194], it was observed that the inclusion of disorder also leads to an order-by-disorder transition. Indeed, all previously studied examples exhibit an order-by- disorder transition to a competing magnetic order. Therefore, it would be interesting in the future to expand our general arguments to also consider the effects of disorder on the magnetic configuration of the system. Furthermore, the nature of the order-by-disorder transition remains unknown. The non-interacting spin-wave treatment leads to the free energy in Section 8.2.2, where the transition appears to be merely a crossover. This behavior is, however, likely to be an artifact of the linearized treatment. The inclusion of spin-wave interactions could hence provide new physical insight into the nature of the transition. 8.4 conclusion 169

Finally, in Section 8.2.2.1, it was observed that the classical free energy leads to a negative entropy in the low-temperature regime. In this chapter, we have ignored this fact as was done in previous literature [30, 143, 210]. In order to resolve the negative entropy issue, a proper quantum- mechanical treatment would have to be applied in the form of a large-S expansion. A simpler, but incomplete, solution to this issue can also be obtained by only quantizing the excitations. For this solution, the spin waves would be described by magnons, obeying Bose-Einstein statistics.

9 CONCLUSION&OUTLOOK

In this thesis, the primary goal was to gain a deeper understanding of the magnetic phases and the phase transitions of dipolar-coupled spin systems with continuous degrees of freedom. Such systems are fascinating in many respects. Historically, dipoles ap- peared as the fundamental building blocks of magnetism. From the theoretical standpoint, the dipolar Hamiltonian exhibits unusual symmetries for a spin Hamiltonian leading to intriguing physics. For the understanding of experimental systems, the dipolar interaction is successfully used to describe systems ranging from pyrochlores to Artificial Spin Ice (ASI). Because of the fascinating theoretical physics, as well as the applicability to contemporary experiments, this thesis tried to answer relevant questions related to the dipolar interaction with various methods. Monte Carlo simulations were used to determine the phase diagram of dipolar- coupled XY (dXY) spins on the square lattice with respect to disorder [52] and to model the time-dependence of nanomagnetic logic gates quantitatively [51]. Monte Carlo simulations were also used in combination with finite-size scaling (FSS) and the Monte Carlo renormalization group (MCRG) method in an attempt to determine the universality class of the finite-temperature phase transition of the dXY model on the square lattice. In contrast, analytic mean-field calculations were used to describe the recent µ-spin rotation (µSR) experiments on ASI systems [11]. Finally, a qualitative yet profound insight was obtained using a detailed symmetry discussion, which showed that many previously known features generalize to a broad class of systems [53]. This thesis, therefore, describes many properties of dipolar-coupled spin systems, but at the same time raises new questions and opens new research directions. In this last chapter, these directions are presented after the main results of this thesis are recapitulated.

171 172 conclusion & outlook

9.1 universality class of dipolar-coupled xy spins on the square lattice

The (t)dXY model on the square lattice exhibits a finite-temperature phase transition between the paramagnetic phase and a striped phase. The universality class of this transition is under debate since previous literature showed that the transition of the nearest-neighbor truncated dipolar- coupled XY (tdXY) model undergoes a Berezinskii–Kosterlitz–Thouless transition [65] but that for a larger truncation radius, or no truncation at all, the transition belongs either to the XY-model with a four-fold anisotropy (XYh4) universality class [143–145] or the Ising universality class [30, 141, 142, 156]. These two universality classes can be hard to differentiate, as the marginal XYh4 universality class shares the critical exponents with the Ising universality class in a particular limit. Indeed, the dXY model on the square lattice is known to exhibit critical exponents that are in agreement with Ising critical exponents up to numerical errors [30, 141, 142, 156]. However, the symmetry would indeed suggest the XYh4 universality class. Therefore, it is still unclear if the critical exponents are precisely Ising or if they correspond to a different universality class. In this thesis, an attempt to resolve this ongoing debate was presented, which was based on the high-precision determination of the critical exponents. First, however, the well-established FSS method was used, which provided a crude but reliable estimate of the critical exponents. Subse- quently MCRG simulations were employed to refine the estimates. Both methods have been applied successfully to the Ising model in two dimensions, where the critical exponents could be determined reliably to a high precision with less than 1% deviation from the true critical exponents. In contrast, for the tdXY model, we were only able to extract critical exponents with FSS, as the MCRG simulations did not yield any physical values for the critical exponents. The FSS estimates are in agreement with Ising critical exponents, but the precision of these estimates is insufficient to be conclusive. As the MCRG results did not converge at all, no sufficiently precise determination of critical exponents could be obtained to be conclusive about the universality class of the transition. The lack of convergence of the MCRG method can have two origins, technical or physical. Technical limitations could arise from insufficient renormalization procedures or not incorporating the relevant interactions. However, the lack of convergence with various input parameters suggests a more profound physical reason. The most probable cause for the lack of convergence is the strong influence of a marginal operator. It is known that 9.2 µ-spin rotation experiments of dipolar-coupled xy spins 173 the MCRG method can exhibit non-convergence in systems with marginal operators. In the future, this issue of non-convergence could, however, be alleviated by special renormalization procedures that leave the marginal operator invariant [121, 127, 136]. Therefore, in order to prove that the (t)dXY model belongs to the XYh4 universality class with the MCRG algorithm, an appropriate renormalization scheme would have to be found that leaves the marginal operator invariant. Furthermore, in the future, methods other than the MCRG method could also be considered for the determination of the universality class of dXY spins on the square lattice. Using a more precise FSS analysis of larger lattices, it might be possible to determine the critical exponents with sufficient precision in order to rule out Ising behavior. Similarly, the system could be studied experimentally by performing a scattering experiment on ASI systems, analogous to previous work on the artificial square ice [42, 225].

9.2 µ-spin rotation experiments of dipolar-coupled xy spins

Experiments on ASI systems resembling the dXY model on the square lattice have already been carried out [10–12]. Specifically, in Ref. [11], µSR measurements were conducted to observe the phase transition in the ASI system. This experimental technique is highly sensitive to local magnetic fields and fluctuations, and is therefore well suited to study ASI systems. However, the interpretation of the experiments typically requires a detailed model of the muon-spin dynamics. As part of this thesis, the µSR experiments and, specifically the muon- spin depolarization were modeled. The model showed that the slow depolarization rate λslow, measured in the experiment, can be associated with the order parameter of the dXY model. This association is valid in the approximation that the muons only probe the static magnetic field and not the fluctuations. For the geometry of Ref. [11], this approximation is valid for roughly 50% of the muons, according to an estimate based on the fluctuation rates of the nanomagnets and the muons. For the muons that only probe the static field, the spin-rotation could be determined from the order parameter, which was calculated using the mean-field approximation. The combination of the depolarization model and the mean-field model provided a prediction for the depolarization of the muon-spin ensemble, which is in good agreement with the experimental data. Hence, it can be 174 conclusion & outlook

concluded that the λslow component of the depolarization can give a measure of the order parameter in the dXY model. The model described in this thesis, however, does not provide an interpretation of the fast depolarization rate λfast that was measured in the experiment and exhibited a dominant peak at the phase transition. Here, it was shown that the fast depolarization cannot originate from the static magnetic field, but has to originate from the fluctuations. As such, all methods that neglect the physical time evolution are unable to describe the fast depolarization. Hence, for future work to extend the model to include the fast depolarization, the microscopic dynamics has to be determined by solving the stochastic Landau-Lifshitz-Gilbert equation. The solution can either be obtained in some approximate scheme or by a micromagnetic simulation at finite temperature. While a simulation is not difficult concep- tually, due to the large fluctuations close to the phase transition, the simulations need to be repeated many times to obtain meaningful averages. Additionally, even if the simulations can reproduce the results of the µSR experiments, it is not clear whether it is possible to identify a direct corre- spondence between thermodynamic observables and measured quantities. Nevertheless, the implementation of a micromagnetic simulation would be useful in the future as it would help in the design of experiments.

9.3 phase diagram of dipolar-coupled xy spins

Experimental realizations always include some disorder, such as vacancies or the locations of the magnetic moments slightly away from the lattice sites. Therefore, assuming no disorder in the underlying model can be an oversimplification. Indeed, the (t)dXY model on the square lattice is known to exhibit a (vacancy) disorder-driven order-by-disorder transition breaking the SO(2) degenerate ground-state [30, 143]. As the disorder-selected microvortex phase competes with the temperature-selected striped phase, a rich phase diagram is known to emerge. However, the predicted phase diagram was only obtained for perturbatively small vacancy disorder. Other types of disorder or sufficiently strong vacancy disorder were not successfully treated up to now, mainly due to a lack of proper order parameters. In this thesis, and specifically in the associated publication [52], the quantitative phase diagram for a tdXY model was determined by introducing proper order parameters and performing large scale parallel-tempering Monte Carlo simulations. From these simulations, the full phase diagram was obtained for vacancy disorder and random-displacement disorder. Sur- 9.4 symmetry analysis of dipolar-coupled spin systems 175 prisingly, the critical values of disorder above which no long-range order emerges are much lower than previously predicted. For a truncation of rcut = 2 (in units of the nearest-neighbor distance), the critical vacancy concentration is only pc ≈ 11%, and the critical standard deviation for the random-displacement disorder is only σc ≈ 6%. As these values are cutoff dependent, they cannot be compared directly with the dXY model where no cutoff is applied. In order to extract the phase diagram of the non- truncated dXY system, future simulations would have to be carried out for increasing cutoff until the phase diagram is converged, or by implementing the true long-range interaction using techniques such as Ewald summation [226–228]. The phase diagram could also be determined experimentally, by designing ASI systems with a well-defined disorder and subsequent measurement using a method able to differentiate the two competing phases. The main conclusion that can be drawn of the two phase diagrams, which was derived in this thesis, is that they are rather similar. Both types of disorder favor the microvortex phase, which adheres to local flux closure. Furthermore, in both cases, sufficient temperature leads to the formation of a striped phase. Intuitively, disorder should disrupt the flux closure at bulk length scales and therefore strongly affects the striped phase. Fol- lowing this argument, disorder would always promote local flux closure. In the future, this favoring of microvortex-like phases in the presence of disorder could be further probed by studying different geometries such as the cubic lattice or the triangular lattice [28, 30, 206, 211], or by introducing different types of perturbations like higher order multipoles [197– 199]. Eventually, a formal proof of the connection between the disordered phases and magnetic flux closure would be highly desirable.

9.4 symmetry analysis of dipolar-coupled spin systems

While much of the work presented in this thesis was concerned with quantitative statements that can be compared with experiments, the generalization in the ﬁnal two chapters of this thesis was more concerned with a better understanding of fundamental principles. Previous literature and the other chapters revealed much of the physics for speciﬁc dipolar-coupled systems such as the (t)dXY model on the square lattice [30, 52, 143, 192, 193], the (t)dXY model on the honeycomb lattice [30, 200], or the dipolar- coupled Heisenberg spin system on the cubic lattice [28, 29, 201, 211, 212]. However, even though these systems showed similar behavior in many re- 176 conclusion & outlook

gards, irrespective of dimension, geometry, or a truncation of the dipolar Hamiltonian, the deeper reason for this behavior remained elusive. The work presented in the final chapters of this thesis could shed some light on the origin of the similarities for many systems. Namely, that for all dipolar-coupled spin systems where the Luttinger and Tisza (LT) method is applicable, the appearance of the continuous ground-state degeneracy, exhibited by many dipolar-coupled spin systems, could be traced back to the symmetries of the lattice. Additionally, based on group theory and spin- wave theory, the effect of thermal excitations on the ground state could be predicted, with the conclusion that any finite temperature is sufficient to break the continuous ground-state degeneracy of any dipolar-coupled spin systems. As in previous literature [30, 143, 210], we only treated the classical limit, which is somewhat unphysical as it neglects the appearance of a negative entropy. Instead, in future work, the system should rather be treated in a proper quantum mechanical setting, by quantizing the spin-waves and considering them as indistinguishable magnons, described by the Bose- Einstein statistics. However, this still does not adequately treat the quantum limit so that the most comprehensive approach to this problem is a large-S expansion, for example carried out using the Holstein-Primakoff approximation. Either way, quantum fluctuations could induce an order- by-disorder transition, as has been seen before for other systems [229]. Therefore, it would certainly be interesting to generalize the results presented in this thesis to a quantum mechanical setting. Furthermore, in this thesis systems were only treated where the LT method is applicable. While this method applies to many dipolar-coupled spin systems, dipolar systems on lattices with large frustration typically do not admit an LT solution for their ground state. Most systems of interest to the ASI community, like the honeycomb lattice or the square lattice, can indeed be understood by the LT method. However, systems of interest to the bulk spin-ice community, such as dipolar systems with continuous spins on the kagome lattice [33, 34] or the pyrochlore lattice [230, 231], can not. For such systems, the results of this thesis do not apply, as the critical precondition of the applicability of the LT method fails. Yet, also in these systems, the point symmetry group of the lattice is a symmetry of the Hamiltonian. Therefore, a thorough analysis of the symmetries in such systems in future work should provide some deeper insight into the appearance of a continuous ground-state degeneracy [35] or its absence [33, 34]. 9.5 concluding remarks 177

9.5 concluding remarks

In conclusion, this thesis was concerned with several topics, including applications such as nanomagnetic logic gates, the interpretation of experimental µSR results and the abstract theoretical work describing the ground- state degeneracy. The results for the different topics were obtained by various methods, including Monte Carlo simulations, mean-ﬁeld calculations, the use of group and representation theory, and the use of spin-wave theory. With these methods, interesting physics such as the universality of phase transitions and the effects of temperature and disorder in magnetic systems could be explored. This led to a better overall understanding of the dipolar-interaction and its effects on classical spin systems. However, also new questions were raised, which were not easily accessible with the methods presented here, which we hope are soon addressed by future research. Therefore, this provides an important step towards a better understanding of dipolar-coupled spin systems and physics in general.

A ALGORITHMS a.1 metropolis-rosenbluth-rosenbluth-teller-teller algorithm

Here, T is the temperature, E(S) the energy of the system in state S measured in the same units as temperature (by setting kB to 1), rand(I) denotes a (pseudo-)random variable in the interval I. Furthermore, Neq is the number of times an update should be performed to achieve equilibration (burn-in time), Nmeas is the number of times the observables should be measured, and Ndc is the number of updates applied to the system before the observables are measured again (in order to decorrelate the measurements).

Algorithm 1 Metropolis-Rosenbluth-Rosenbluth-Teller-Teller (Metropolis algorithm)

1: for i := 0 . . . Neq do . Burn in / Thermalization 2: Eold ← E(S) 0 3: Propose S . For example single spin-ﬂip 0 4: Enew ← E(S ) 5: if rand([0, 1]) < exp[(Eold − Enew)/T] then 0 6: S ← S 7: for i := 0 . . . Nmeas do . Measurements 8: for j := 0 . . . Ndc do . Do several updates 9: Eold ← E(S) 0 10: Propose S 0 11: Enew ← E(S ) 12: if rand([0, 1]) < exp[(Eold − Enew)/T] then 0 13: S ← S 14: for O ∈ {Observables} do . Loop over observables 15: Measure & Store O(S)

179 180 algorithms

a.2 rejection-free kinetic monte carlo algorithm

To study the time evolution of a system S with N discrete states from a starting time t0 up to the ﬁnal time t f the so-called rejection-free kinetic Monte Carlo (kMC) can be applied. For every state i in the system the transition rates to the state j are known, denoted here by Γij. For forbid- den transitions the rate is 0. Then the rejection-free kinetic Monte Carlo algorithm is given in Algorithm 2.

Algorithm 2 Rejection-free kinetic Monte Carlo algorithm

1: Initialize system is the initial state i ← i0 2: t ← t0 3: while t < t f do . While time passes 4: Γi ← 0 5: for j ∈ {1, 2, . . . , N} do . Accumulate total transition rate 6: Γi ← Γi + Γij Γij 7: Choose a state j with probability . Γi 8: i ← j . Carry out the update 0 9: u ← rand((0, 1]) . Random number for time update −1 1 10: δt ← Γi log u0 . Generate time update 11: t ← t + δt . Update time B MCRGINTERACTIONSET

In TableB. 1 a new interaction set is provided, which was developed for this thesis in order to ﬁnd the critical exponents of the Ising model and the dXY model. The notation for the indices is now given as subscript (x, y), where x is the index along the x-axis and analogous for y. The interactions start at the origin and are then parametrized by an offset. In the simulation every site was used as an origin by a trivial shift of the interaction.

Table B.1: Interaction set used for Figs. 4.6 and 4.11. For XY spins, the x- component of all the spins was used for simplicity. Hence, the vector notation is suppressed for this table. No. of n = 2 n = 4 n = 6 spins

hS0,0S1,0i hS0,0S0,1S1,0S1,1i hS0,0S0,1S0,2S1,0S1,1S1,2i

hS0,0S2,0i hS0,0S−1,0S0,1S1,0i hS0,0S1,0S2,0S0,1S1,1S2,1i

hS0,0S3,0i hS0,0S−1,0S0,1S1,1i hS0,0S1,0S0,1S1,1S2,0S0,2i

hS0,0S4,0i hS0,0S0,1S0,2S1,0i hS0,0S−1,0S0,1S−1,1S−2,0S0,2i

hS0,0S5,0i

hS0,0S6,0i

hS0,0S1,1i

hS0,0S2,1i

hS0,0S3,1i

hS0,0S4,1i

hS0,0S5,1i

hS0,0S2,2i

hS0,0S3,2i

hS0,0S4,2i

181

ACKNOWLEDGMENTS

While pursuing a PhD, I assume everyone, but certainly me, needs some help along the way. This includes guidance on the scientific questions, ad- vice on the required methods, and help with presenting the results. How- ever, it has not all been about work and I was able to spend time off-work enjoying the friendship with many people who made my life so much more enjoyable. I am incredibly grateful for the last few years both in terms of my scientific supervision and in terms of the acquired friendships. Thanks to all of you. First, I need to thank Peter Derlet, as it is merely due to your constant guidance during all phases of my PhD that I could achieved my profes- sional goals. Under your supervision, I learned how to think critical about a problem and you always pushed me to strive for a deeper insight into the fundamental questions. By no means less, I need to thank Laura Heyderman: Indeed, your supervision has been critical to all my scientific achievements. You have taught me how to accomplish my goals more in an organized way rather than in a random walk. Also, I have to say a special thank you for all the time you have spent teaching me the how to write up my results in a clear and understandable manner. I also want to thank Jan Vermant, Frédéric Mila and Nicola Spaldin who have agreed on being part of my examination committee. I am very thankful for the help of Naëmi Leo: You helped me getting started and staying motivated throughout my PhD. Indeed, our collaboration on the muon paper was my first exposure to the exciting topic of artificial spin systems. Furthermore, you pulled me into the interesting nanomagnetic logic project, where I further need to thank Hanu Arava for letting me be part of this research activity. Of course, I also need to thank all the other co-authors of these two papers, without your help, these papers could not have been published. Specifically, I need to thank all the experimental collaborators who spent quite some time explaining their experiments such that even I could understand them: Thank you very much Oles Sendetskyi, Stephan Holenstein, Hubertus Luetkens, Diane Lançon, Valerio Scagnoli, and in general the MESOSYS group at PSI. A special thanks goes to those of you, who read specific chapters of this thesis and helped me improve the descriptions of the experiments. 183 184 acknowledgments

Not only on the experimental side I had wonderful collaborators, but I am truly grateful for the learning opportunity from the collaboration with Michael Schütt. I need to thank you for your help with the group theory project and in particular for your excellent supervision of the spin-wave calculations. Additional thanks goes to Christopher Mudry and Markus Müller who gave me the opportunity to give seminars, where I could learn a lot from their critical questions. In general, I want to thank CMT for all the lunchtime discussions, be it on physics, politics or merely a silly discussion about a recent TV show. A special thanks goes to my ofﬁce mates over the years, Andrea Scaramucci, Kevin Tham, and Manuel Grimm, for enduring my colorful language: I truly apologize. In particular, I need to thank Manuel Grimm for becoming a very close friend of mine. Thank you for the beers and light moments, but also for being around in the troublesome times of a PhD. I truly enjoy our friendship. Ebenso möchte ich meinen Kollegen aus der Betriebsfeuerwehr des PSI danken. Die Feuerwehrarbeit war etwas ganz anderes als mein Doktorat, und trotzdem konnte ich auch in der Feuerwehr enorm viel lernen unter der Führung von Roger Schneider. Im Speziellen möchte ich mich bei bei meinen Feuerwehrkollegen Florian Schläpfer, Heiko Kromer, Helge Brands und Goran Marinkovic bedanken, mit welchen ich auch manch eine Dis- kussion über die Feuerwehr hinaus geführt habe. Schliesslich möchte ich meinen Freunden abseits des PSI danken, welche mich immer wieder daran erinnert haben, dass es auch ein ganz norma- les Leben ausserhalb des PSI gibt. Meinen alten Studienfreunden Silvan Hunziker und Anina Leuch. Meinen Boulderkollegen, im Speziellen Jonas Burkhardt und Sven Schutzbach. Meiner Tante, Brigitte Manser. Meinen guten Freunden in meiner alten Heimat rund um Gossau, im Speziellen Fabian Diethelm, Rebecca Gamma, Raphael Rietmann, und Michael Wag- ner. Letztendlich möchte ich der Familie Inauen danken, die mich als einen der Ihren aufgenommen haben: Herzlichen Dank, Susanne Inauen, Roland Donner, Adrian Inauen, Stefanie Kunz, Philipp Inauen und Pascale Jacob. BIBLIOGRAPHY

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Articles in peer-reviewed journals:

• “Collective magnetism in an artiﬁcial 2D XY spin system” N. Leo, S. Holenstein, D. Schildknecht, O. Sendetskyi, H. Luetkens, P. M. Derlet, V. Scagnoli, D. Lançon, J. R. L. Mardegan, T. Prokscha, A. Suter, Z. Salman, S. Lee, and L. J. Heyderman Nat. Comm. Vol. 9, 2018 (doi: 10.1038/s41467-018-05216-2)

• “Phase diagram of dipolar-coupled XY moments on disordered square lattices” D. Schildknecht, L. J. Heyderman, and P. M. Derlet Phys. Rev. B Vol. 98, 2018 (doi: 10.1103/PhysRevB.98.064420)

• “Engineering Relaxation Pathways in Building Blocks of Artiﬁcial Spin Ice for Computation” H. Arava, N. Leo, D. Schildknecht, J. Cui, J. Vijayakumar, P. M. Derlet, A. Kleibert, and L. J. Heyderman Phys. Rev. Appl. Vol. 11, 2019 (doi: 10.1103/PhysRevApplied.11.05408 6)

• “Continuous ground-state degeneracy of classical dipoles on regular lattices” D. Schildknecht, M. Schütt, L. J. Heyderman, and P. M. Derlet Phys. Rev. B Vol. 100, 2019 (doi: 10.1103/PhysRevB.100.014426)

203

NOTATION

glossary Heisenberg spin A spin on the sphere, parametrized by two angles. XY spin A spin on the unit circle, parametrized by one angle. Ising spin A spin taking the two values ±1. Ising-like spin Similar to an Ising spin, however, different spins are not restricted to the same easy axis. Ising model Classical spin model with Ising spins coupled by a nearest-neighbor interaction as in Eq. (2.5). degree of freedom Single variable, parametrizing one dimension of the phase space. order-by-disorder The notation that inclusion of excitations by thermal/quantum fluctuations or disorder leads to order by a spontaneous symmetry breaking ALPS Library for Monte Carlo Simulation, developed at ETH Zürich (Algorithms and Libraries for Physics Simulations) parallel-tempering A type of Monte Carlo algorithm, used to ther- Monte Carlo malize even strongly frustrated systems. (Details in Section 3.3) list of abbreviations ASI Artificial Spin Ice BKT Berezinskii–Kosterlitz–Thouless dXY dipolar-coupled XY FSS finite-size scaling kMC kinetic Monte Carlo LT Luttinger and Tisza MC Monte Carlo

205 206 notation

MCRG Monte Carlo renormalization group MF mean-ﬁeld µSR µ-spin rotation PEEM Photoemission Electron microscopy RG Renormalization Group SEM Scanning Electron Microscopy tdXY truncated dipolar-coupled XY (t)dXY (truncated) dipolar-coupled XY, when it is irrele- vant if tdXY or dXY is meant XYh4 XY-model with a four-fold anisotropy

frequently used symbols E Energy H Hamiltonian β inverse Temperature T Temperature ~S normalized moment, spin Sd d-sphere, embedabble in d + 1-dimensions z micro-canonincal partition function Z canonical partition function Z grand-canonincal partition function U Internal energy F (Helmholtz) Free energy Ω Grand canonical potential N Number of sites/spins/particles L System size along one direction

rcut Cutoff radius, applied to the dipolar Hamiltonian Λ UV cutoff, applied to regularize otherwise diver- gent integrals

eˆa Unit vector along the axis a ∈ {x, y, z, ···} P Point symmetry group of a lattice T Translational-invariance symmetry group LISTOFFIGURES

2.1 Sketch of a highly frustrated triangle of Ising-spins with an antiferromagnetic interaction. The lower left spin is set to point up without loss of generality. Next, the lower right spin has to point down in order to minimize its energy. Finally, the last spin can either minimize its interaction with the left or the right spin, but not with both simultaneously, so that this spin is frustrated...... 17 2.2 Illustration of the “head-to-tail” rule. Dipoles behave like these dogs, i. e., if the spins can align with their bond, then they prefer a parallel alignment. If the spins are forced to be orthogonal to their connecting vector, then the spins prefer to align antiferromagnetically. (Courtesy of Dr. Naëmi Leo) ...... 18 3.1 Sketch illustrating the working principle of parallel-tempering Monte Carlo. The blue boxes illustrate independent simulations at a fixed temperature T = Ti, where the configurations have energy Ei. Every once in a while the simulations are allowed to exchange their temperatures according to Eq. (3.12). Here, ∆E = Ei+1 − Ei and ∆β = βi+1 − βi ...... 32 3.2 Illustration of two rescaling procedures for Ising spins. In Fig. 3.2b, the sum of the spins in the lower left and the upper right block is 0. Here, the lower left spin is applied as the tie-breaker...... 37 4.1 Sample designed by Hanu Arava [51]. In Fig. 4.1a, the design schematic is shown, illustrating how the original gate (for d = 0) is modified by a displacement d. In Fig. 4.1b an SEM image of one of the structures with d = 0 is shown. (Courtesy of Hanu Arava). . . . 44 4.2 The nanomagnetic system depicted in Fig. 4.1a is initialized in the configuration shown in Fig. 4.2a. The magnetic field is then removed, and the system evolves for three hours at 460 K. After the three hours, the probability of ending up in G1,2 as defined in Figs. 4.2b and 4.2c is measured. This procedure results in the displacement-dependent probabilities shown in Fig. 4.2d...... 46 4.3 A simple sketch, explaining the process from a state i to a state j. Here, ∆Ei→j < 0, so that the final state is lower in energy. Fur- thermore, Eb is the single particle activation energy, i. e., the energy barrier an isolated nanomagnet has to overcome to reorient its magnetization...... 48 207 208 listoffigures

4.4 Monte Carlo (MC) results for the Ising model on the square lattice with size L: (a) order parameter, (b) susceptibility, (c) heat capacity, (d) Binder cumulant, with a zoom-in of the relevant region in (e). The estimate of the critical temperature read off from the Binder cumulant agrees with the exact value, Tc ≈ 2.27...... 52 4.5 FSS collapse of the Monte Carlo data for the two-dimensional Ising model shown in Fig. 4.4 using the critical exponents from Table 4.1. As expected, the data admits for an excellent scaling collapse, especially for the larger lattices, where corrections to scaling are smaller. . 53 4.6 MCRG results for the Ising model on the square lattice, with a detuning from the critical temperature. Points where the MCRG procedure did not converge, are not shown. Here, the exact critical exponents are indicated at the exact critical temperature with a black cross. The blue data points correspond to MCRG estimates for the critical exponents using Wilson’s interaction set [89], and the orange data points using Swendsen’s interaction set [119]. Additionally, a new interaction set was used for the even sector which is plotted with green data points. This interaction set is tabulated in TableB. 1. The comparison of the MCRG estimates to the exact values should be made at T = Tc ≈ 2.269...... 56 4.7 Illustration of the behavior of the critical exponent β in the XYh4 model. According to [58, 59], β is in the interval [1/8, 0.23]. The critical exponents of the dXY model on the square lattice is specu- 1 lated to lie close to or at the point β = 8 ...... 58 4.8 The magnetic order of the (t)dXY spin system on the square lattice, at a ﬁnite but low temperature, is given by one of four striped con- ﬁgurations along the lattice directions. Two are depicted here, the other two follow by time reversal...... 59 4.9 Monte Carlo (MC) results for the tdXY model on the square lattice with size L and a cutoff radius of rcut = 2: (a) order parameter, (b) associated susceptibility, (c) Binder cumulant, with a zoom-in of the relevant region in (d). The estimate for the critical temperature is Tc = 0.968 ± 0.01 determined by the Binder cumulant...... 61 4.10 FSS collapse of the Monte Carlo data for the tdXY model on the square lattice. Here, Ising critical exponents are used as well as our 3 best estimate for the critical temperature Tc/(D/a ) = 0.968 ± 0.01. The non-scaled Monte Carlo data is shown in Fig. 4.9...... 62 list of figures 209

4.11 MCRG results for the tdXY model on the square lattice, with a detuning in the critical temperature. Points where the MCRG procedure did not converge, are not shown. Especially for η in the first iteration (shown in Fig. 4.11a), the procedure did not seem to converge at all and only one point is given. In contrast, the data for ν is randomly scattered. It is expected that the values for ν are close 1 to Ising critical values, and the values of η should be exactly 4 for both possible universality classes of the dXY system [59]. The Ising critical exponents are indicated with a dashed line. From the figures it is therefore evident that no convergence can be observed...... 64 5.1 This sketch illustrates the experimental setup of a µSR experiment. The sample is placed in a spin-polarized beam of muons, so that the muons are captured in the sample. Subsequently, the spins of the muons start to precess due to the presence of a local magnetic field until the muon eventually decays. The positron (e+) from the decay is emitted preferentially along the spin direction and may enter either the detector for N+ or N−...... 69 5.2 A µSR experiment on ASI is performed using a similar geometry as Fig. 5.1, except that here the muons have smaller implantation energies and a gold capping layer is added on top of the ASI system which efficiently stops the muons. Therefore, the muons are implanted in to the non-magnetic layer and their spin precesses around the local magnetic stray field of the ASI system...... 71 5.3 Scanning Electron Microscopy (SEM) image of the (strongly interacting) sample 2 (see Table 5.1) used for the work in Ref. [11] (Adapted from the data repository [180])...... 72 5.4 The temperature selected phases of the dXY spin system on the square lattice are given by striped spin configurations along the lattice directions. (Same figure as Fig. 4.8)...... 73 5.5 Example of the measured muon spin polarization function for a dXY ASI with lattice periodicity of 70 nm and a diameter of the permalloy discs of 40 nm (strongly interacting sample 2, cf. Table 5.1). 73 210 listoffigures

5.6 Fitting the experimental muon spin polarization data such as that provided in Fig. 5.5 with the ansatz for the polarization function Eq. (5.7) for the three different strongly interacting samples in Ta- ble 5.1 reveals the two depolarization rates λfast and λslow. The background color (blue, yellow, red) indicates the frozen-in regime, the phase where long-range order emerges and the paramagnetic phase, respectively. The striking feature is the order-parameter-like behavior of λslow...... 74 5.7 Simulated stopping depth distribution of muons at 14.3 keV implanted in an 80 nm thick gold layer on top of an (infinite) silicon substrate obtained using TRIM.SP [184, 185]. In the left panel, the distribution of stopping depths is given, and in the right panel the cumulative distribution is displayed. The lines indicate where the criterion given by Eq. (5.11) is applicable, such that for muons stopping above the lines, the depolarization is well described by the time-averaged magnetic field. The intersections of these lines with the cumulative distribution correspond to the percentage p given in Table 5.1. (TRIM.SP simulations were carried out by Dr. Naëmi Leo) . 77 5.8 The main results of the discussion in Section 5.3.2 are summarized in this figure. Figure 5.8a depicts the graphical solution of Eq. (5.14), where the crossings are marked by green dots. The crossings of Fig. 5.8a are then tracked as a function of temperature, which is depicted in Fig. 5.8b. Above Tc only the trivial solution hSli = 0 exists, whereas below Tc non-zero solutions emerge and the order parameter takes a finite value...... 79 5.9 Comparison between the predicted muon-spin polarization from Eq. (5.16) and the experimental data for sample 2 at T = 10 K...... 81 5.10 Comparison between the experimental data and the depolarization determined from the mean-field (MF) theory using Eqs. (5.14) to (5.17)...... 82 6.1 Magnetic unit cells of the three different long-range ordered phases for the (t)dXY model on the square lattice. Figure 6.1a shows the SO(2) degenerate ground state, which is parametrized by the angle π 3π θ. The striped phase arises for θ ∈ {0, 2 , π, 2 } and the microvortex π 3π 5π 7π phase for θ ∈ { 4 , 4 , 4 , 4 }. For both phases two of the four possible magnetic unit cells are shown, the other two follow from time-reversal or equivalently from θ 7→ θ + π...... 87 list of figures 211

6.2 Illustration of M~ from Eq. (6.2) for different phases: The paramagnetic phase shows no spontaneous symmetry breaking so that M~ = 0. In contrast, the ground states form the unit circle, since |M~ | = 1 for ground-state configurations. Microvortex phases (dark blue) align along the diagonals, whereas striped phases (light blue) align along the axes...... 90 6.3 Illustration of the random-displacement disorder. The position of each of the lattice sites is perturbed by a Gaussian with standard deviation σ from the initial position. Here, several strengths of disorder are shown, ranging from no disorder in Fig. 6.3a to strong disorder with σ = 30% shown in Fig. 6.3e...... 93 6.4 Illustration for the breakdown of the order parameters in terms of disorder: In Fig. 6.4a, the shaded area corresponds to ρex [cf. Eq. (6.7)]. In Fig. 6.4b, ρex is plotted as a function of disorder strength σ: Inspection of the figure shows, that no defect is expected even in the largest lattice (L = 48) for σ < 20%...... 94 6.5 Monte Carlo results for the tdXY model without disorder: In Fig. 6.5a the three order parameters |M~ |, Ms, Mmv from Eqs. (6.2) and (6.5) are shown for the L = 16 lattice. In the inset, the same data is shown with a logarithmic temperature scale. In Fig. 6.5b the microvortex order parameter Mmv is shown for three different system sizes, where the inset displays the same data on a logarithmic temperature scale. Figure 6.5c shows one configuration obtained at T = 1.8 · 10−6, where it can be observed that the deviation from the striped phase is homogeneously distributed throughout the whole system...... 97 6.6 Monte Carlo results of the order parameters |M~ |, Ms, and Mmv in the tdXY system on the square lattice with vacancy disorder. Three selected values of disorder density p are shown: In Figs. 6.6a to 6.6c the data for p = 2%, in Figs. 6.6d to 6.6f the data for p = 6%, and in Figs. 6.6g to 6.6i the data for p = 15% is presented. The latter set at p = 15% suggests that no long-range order occurs. The data for |M~ | at small disorder (p = 2%, 6%) agrees with the expectation of a long-range ordered striped phase at finite temperature. Further- more, a rise in Mmv at low temperature indicates the phase transition into a microvortex phase. The possible lack of thermalization at low temperature is discussed in Section 6.5...... 100 212 listoffigures

6.7 Thermalization analysis for the tdXY system on the square lattice with vacancy density p = 6%. In Figs. 6.7a and 6.7b simulation results are shown for the order parameter |M~ | for two system sizes L = 32 and L = 48. In Figs. 6.7c and 6.7d, the results for the associated Binder cumulant U are shown. Besides the number of thermalization sweeps Nth, the simulations are identical to the ones used in 5 the paper [52]. It can be observed that the curves for Nth = 3 · 10 exhibit a drop at low temperatures, indicating that the number of thermalization sweeps used in the paper was insufficient for these input parameters. For the L = 32 system, 5 · 105 thermalization sweeps are sufficient to thermalize the system, as there is no spurious drop in either the order parameter shown in Fig. 6.7a or the Binder cumulant shown in Fig. 6.7c and the data agrees with the data of the 6 6 larger thermalization of Nth = 1 · 10 . Yet, even Nth = 10 thermalization sweeps seem to be insufficient for the L = 48 system...... 102 6.8 Monte Carlo results of the order parameters |M~ |, Ms, and Mmv for the tdXY model on the square lattice with a vacancy density p. Rather than single curves for selected values of disorder as presented in Fig. 6.6, here, the data for all studied values of disorder is shown as a density map...... 103 6.9 Selected Monte Carlo results of the vacancy-disordered tdXY for the Binder cumulants U, Us, and Umv, associated with the order parameters |M~ |, Ms, and Mmv, respectively. In Figs. 6.9a to 6.9c, the Binder cumulants for p = 6% are shown, which identify the microvortex phase at small temperatures (one crossing in Fig. 6.9c) and the striped phase at intermediate temperatures (two crossings in Fig. 6.9b). In Figs. 6.9d to 6.9f, no crossing can be observed for p = 13%, which indicates that the system no longer exhibits long-range order. Finally, in Figs. 6.9g to 6.9i the Binder cumulants are shown at a fixed temperature T ≈ 0.79, where it can be observed that the system only orders in a striped phase but never in a microvortex phase, as there are clear crossings in Figs. 6.9g and 6.9h but none in Fig. 6.9i. As for Fig. 6.6, the data at low temperatures might not be completely thermalized.104 list of figures 213

6.10 Summarized Monte Carlo results for the phase diagram of the tdXY model on the square lattice with system size L and vacancy density p. The phase diagram is obtained from the Binder cumulant analysis and is superimposed on the Monte Carlo data for Mmv in the L = 48 system. Markers correspond to |M~ | (red), Ms (orange) and Mmv (violet). Open markers correspond to Binder cumulant crossings at a fixed temperature analyzed as a function of p, while filled markers correspond to Binder cumulant crossings obtained at a fixed vacancy-density analyzed as a function of temperature. Be- low the line at T = 0.2, the data is affected by poor thermalization, and is therefore not completly reliable (details in Section 6.5)...... 106 6.11 Monte Carlo results for the order parameters |M~ |, Ms, and Mmv in the tdXY on the square lattice with random displacement. Three selected values of the standard deviation σ of the random displacement disorder are shown: In Figs. 6.11a to 6.11c the data for σ = 0.1%, in Figs. 6.11d to 6.11f the data for σ = 1%, and in Figs. 6.11g to 6.11i the data for σ = 8% is presented. For σ = 0.1% and σ = 1% the data indicates at low temperatures a microvortex phase, at intermediate temperatures a striped phase and at high temperatures a paramagnetic phase. For σ = 8% the situation is less clear. The thermalization issues that were seen in the vacancy-disordered system are only weakly observed here...... 109 6.12 Summarized Monte Carlo results for the order parameters |M~ |, Ms, and Mmv for the tdXY model on the square lattice with system size L and random-displacement disorder with standard deviation σ. Rather than single curves for selected values of disorder as presented in Fig. 6.11, here, the data for all studied values of disorder is shown as a density map...... 110 6.13 Selected Monte Carlo results of the tdXY model with random displacement disorder for the Binder cumulants U, Us, and Umv. In Figs. 6.13a to 6.13c, the Binder cumulants for σ = 1% are shown, where it can be observed that at low temperature the system is in the microvortex phase and at intermediate temperatures in the striped phase. In Figs. 6.13d to 6.13f and Figs. 6.13g to 6.13i, the fixed temperature approach is applied at T ≈ 0.52 and T ≈ 0.90, respectively. The resampling procedure introduced in Section 6.5.1, can still yield the position of the crossings, albeit with considerably larger errors...... 111 214 listoffigures

6.14 Summarized Monte Carlo results for the phase diagram of the tdXY model on the square lattice with system size L and random- displacement disorder with standard deviation σ. The phase diagram is obtained from the Binder cumulant analysis and is superimposed on the Monte Carlo data for Mmv in the L = 48 system. Markers, colors, and regions have the same definitions in Fig. 6.10. Here, the system seems to thermalize better at lower temperatures than the vacancy-disordered system discussed in Section 6.5. Therefore, the data can be considered to be more reliable. 112 7.1 Magnetic unit cell of the ground state of the dXY model on the square lattice. According to Ref. [29], every configuration described by this magnetic unit cell minimized the dipolar energy, independent of θ. Hence, the ground state is described by a manifold rather than discrete states. This figure is analogous to Fig. 6.1a...... 118 7.2 The main finding of this chapter can be summarized by this flow diagram. For any dipolar-coupled spin system, for which a ground state can be determined by the LT method, the vector representation V plays an important role in the determination of the degeneracy. If V is irreducible, the degeneracy is of dimension d. If V is reducible, the dimension of the degeneracy depends on specific details of the lattice. However, ddeg is always smaller than d if V is reducible. . . . . 126 7.3 The four basic arrays for the LT method implemented on the 2 × 2 square lattice with their corresponding Fourier vectors. Dotted lines indicate the lattice, and the magnetic unit cell is highlighted in gray. The basic array in Fig. 7.3a describes a ferromagnetic configuration, that in Fig. 7.3d describes an antiferromagnetic configuration, and those in Figs. 7.3b and 7.3c describe two inequivalent striped configurations...... 128 7.4 The 23 = 8 basic arrays for the 2 × 2 × 2 cubic lattice with their corresponding Fourier vectors. The lattice is indicated by a dotted line, and the magnetic unit cell is highlighted in a light gray. The explicit lattice summations show that, for the dipolar case, the configuration depicted in Fig. 7.4g minimizes the energy...... 131 7.5 Ground-state of dipolar-coupled XY spins on the triangular lattice [205]. The magnetic unit cell coincides with the structural unit cell and is highlighted in light gray...... 134 list of figures 215

8.1 The two ground-state candidates of the dXY model on the linear chain according to the “head-to-tail” rule illustrated in Fig. 2.2. The calculation in Eq. (8.2) shows that the configuration depicted in Fig. 8.1a has lower dipolar energy than the configuration depicted in Fig. 8.1b...... 138 8.2 Dispersion relation ω(q) from Eq. (8.9b) for spin waves in a dXY spin chain obtained using linear spin-wave theory. It can be observed that the dispersion relation is gapped, i. e. ω(q) > 0 for all q, as expected. Remarkably, the minimal energy excitations occur at the Brillouin-zone boundary rather than at q = 0, which means that the lowest energy excitations are antiferromagnetic perturbations of a ferromagnetic ground state. This can be understood in an enlarged magnetic unit cell, which is equivalent to backfolding the dispersion relation. The backfolded dispersion relation is indicated by the dotted line so that the minima now occurs at the Brillouin-zone center. The lowest energy excitations are, however, in the language of phonons, optical modes rather than acoustic modes. 142 8.3 Continuous ground state of the square lattice. This figure is analogous to Figs. 6.1a and 7.1. However, here the angles are always measured starting from the x-axis so that the sense of rotation becomes clear...... 145 8.4 Dispersion relation ω(~q, θ) for spin waves in the dXY model on the square lattice obtained by evaluating Eq. (8.20) with a truncation of |~r| 6 rcut = 50 applied to the summation. The contour lines are 1 1 located at ω(~q) = 10 , 2 , 1, 2 in white, light gray, dark gray, and black, respectively. It can be observed that, for θ = π/4, the dispersion relation is symmetric and nearly isotropic for small |~q|, whereas, for θ = 0 and θ = π/2, the spin-wave dispersion has one soft mode and one stiff mode...... 152 8.5 The three basic arrays described by Eq. (8.45) that are found by the LT-method combined with our method derived in Chapter 7...... 163 8.6 The function f 2 is shown as a spherical plot. Namely, a large radial Eg value corresponds to a large value of the function. It can, therefore, be observed that the function f 2 is maximal along the axes...... 166 Eg

LISTOFTABLES

2.1 Summary of the different partition functions and their corresponding thermodynamic potentials. Here ω is a phase-space element and β is the inverse temperature...... 21 2.2 Definition of some thermodynamic observables and their critical behavior around a phase transition upon changing the external parameter p, where p is often the temperature. Here, F is the free energy and hφ is the field conjugate to the order parameter. All quantities are considered in the limit hφ → 0. Furthermore, the order parameter is extended to φ 7→ φ(~x) so that the order parameter is hφ(~x)i...... 24 3.1 An overview of previous MCRG studies of classical spin systems in various dimensions...... 35 4.1 Summary of the critical exponents (defined in Table 2.2) and the critical temperature for the Ising model with nearest-neighbor coupling J on the square lattice...... 51 5.1 The relevant parameters are summarized for the three samples considered in [11]. Here, d is the diameter of the individual nanomagnets, and a is the lattice periodicity, which is reported for the strongly interacting sample and the non-interacting sample. These parameters lead to p percent of muons sampling time-averaged magnetic fields according to Eq. (5.11) in the strongly interacting sample...... 72 6.1 Character table for C4v, the point symmetry group of the square lattice. 91 7.1 An overview of previous studies of ground-states of dipolar- coupled spin systems on different lattices. The second column indicates if the LT method applies to the system...... 121 7.2 Character table of C4v, the point symmetry group of the square lattice. The character table can be found in textbooks on group theory. The characters for V are determined in Eq. (7.13). The reduction is V ≡ E, and hence V is irreducible on C4v...... 128 7.3 Character table of Oh, the point symmetry group of the cubic lattice and the reduction of the three-dimensional vector representation V. The reduction leads to V ≡ T1u, i. e., V is irreducible...... 132

217 218 listoftables

7.4 Character table of D4h, the point symmetry group of the tetragonal lattice and the reduction of the three-dimensional vector representation V. Here, V does not correspond to a single line of the character table, so that V is reducible. The reduction can be accomplished using the grand orthogonality theorem (see, e. g. Ref. [204]), which D4h D4h leads to the reduction V ≡ A2u ⊕ Eu ...... 133 7.5 Character table of C6v, the point symmetry group of the triangular lattice, and the reduction of the two-dimensional vector representation V. As V ≡ E1, the vector representation is irreducible...... 135 8.1 Construction of the characters χΣ of the spin-wave representation Σ for XY spins on the square lattice, under the assumption of a 2 × 2 magnetic unit cell...... 149 8.2 Character table of the point symmetry group of the square lattice C4v and the reduction of Σ for this lattice...... 150 8.3 Interpretation of the symmetry-adapted spin-wave modes found in Eq. (8.18). For example, A2 corresponds to a homogeneous rotation of all sites in the same direction, whereas B1 transforms the sites in a checkerboard pattern...... 150 8.4 Character table of Oh, the point symmetry group of the cubic lattice and characters for τ, ρ, and Σ = τ ⊗ ρ. The reduction of Σ that follows from this character table is given in Eq. (8.49)...... 165 B.1 Interaction set used for Figs. 4.6 and 4.11. For XY spins, the x- component of all the spins was used for simplicity. Hence, the vector notation is suppressed for this table...... 181 CURRICULUMVITÆ personal data

Name Dominik Schildknecht Date of Birth September 23, 1992 Citizen of Waldkirch SG, Switzerland education

now – ETH Zurich September 2014 Zürich, Switzerland Degree applied for: Doctor of Sciences April 2016 – ETH Zurich September 2014 Zürich, Switzerland Final degree: Master of Science ETH in Physics October 2014 – ETH Zurich September 2011 Zürich, Switzerland Final degree: Bachelor of Science ETH in Physics July 2011 – Kantonsschule am Burggraben August 2007 St. Gallen, Switzerland Final degree: Matura employment

June 2016 – Ph.D. Student Paul Scherrer Institute, Villigen, Switzerland and ETH Zurich, Zurich, Switzerland teaching experience

Fall 2018 Substitute Lecturer “Introduction to numerical Modeling”, EPFL, Lausanne, Switzerland

219 220 curriculumvitæ

Fall 2017 Teaching Assistant “Einfühurung in die Material- wissenschaften”, ETHZ, Zurich, Switzerland Summer 2015 Teaching Assistant “Prüfungsvorbereitungskurs: Numerische Methoden”, ETHZ, Zurich, Switzer- land Spring 2015 Teaching Assistant “Numerische Methoden”, ETHZ, Zurich, Switzerland Summer 2014 Teaching Assistant “Prüfungsvorbereitungskurs: Numerische Methoden”, ETHZ, Zurich, Switzer- land Spring 2014 Teaching Assistant “Numerische Methoden”, ETHZ, Zurich, Switzerland Spring 2013 Teaching Assistant “Numerische Methoden”, ETHZ, Zurich, Switzerland