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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 Rights / License: In Copyright - Non-Commercial Use Permitted This page was generated automatically upon download from the ETH Zurich Research Collection. For more information please consult the Terms of use. 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 2019 Dominik Schildknecht: Continuous dipolar moments on regular lattices: a combined Monte Carlo and group theoretical treatment, © 2019 doi: 10.3929/ethz-b-000360043 DECLARATIONOFORIGINALITY 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 nor- mal academic citation rules and that I have read and understood the in- formation 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 un- derstanding of the phenomena resulting from dipolar interactions could be advanced experimentally due to the ability to pattern arrays of nano- magnets into so-called artificial spin systems. The models describing such systems exhibit interesting physics such as continuous ground-state de- generacies, 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 ap- plied 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 con- vergence 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 de- tailed 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 differ- ent 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 de- generacy 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 in- sight 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 vier- fachen 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 gekoppel- te 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
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