Tensor Network State Methods and Applications for Strongly Correlated Quantum Many-Body Systems

Home , Tensor network

Bildquelle: Universität Innsbruck

Dissertation zur Erlangung des akademischen Grades Doctor of Philosophy (Ph. D.)

eingereicht von Michael Rader, M. Sc.

betreut durch Univ.-Prof. Dr. Andreas M. Läuchli

an der Fakultät für Mathematik, Informatik und Physik der Universität Innsbruck

April 2020

Zusammenfassung

Quanten-Vielteilchen-Systeme sind faszinierend: Aufgrund starker Korrelationen, die in die- sen Systemen entstehen können, sind sie für eine Vielzahl an Phänomenen verantwortlich, darunter Hochtemperatur-Supraleitung, den fraktionalen Quanten-Hall-Effekt und Quanten- Spin-Flüssigkeiten. Die numerische Behandlung stark korrelierter Systeme ist aufgrund ih- rer Vielteilchen-Natur und der Hilbertraum-Dimension, die exponentiell mit der Systemgröße wächst, extrem herausfordernd. Tensor-Netzwerk-Zustände sind eine umfangreiche Familie von variationellen Wellenfunktionen, die in der Physik der kondensierten Materie verwendet werden, um dieser Herausforderung zu begegnen. Das allgemeine Ziel dieser Dissertation ist es, Tensor-Netzwerk-Algorithmen auf dem neuesten Stand der Technik für ein- und zweidi- mensionale Systeme zu implementieren, diese sowohl konzeptionell als auch auf technischer Ebene zu verbessern und auf konkrete physikalische Systeme anzuwenden. In dieser Dissertation wird der Tensor-Netzwerk-Formalismus eingeführt und eine ausführ- liche Anleitung zu rechnergestützten Techniken gegeben. Besonderes Augenmerk wird dabei auf die Implementierung von Tensor-Netzwerk-Operationen mithilfe der Programmiersprache Python und enorme Geschwindigkeitszuwächse, die bei Verwendung von Graphikprozessoren für Tensor-Netzwerk-Kontraktionen erreicht werden können, gelegt. MPSs und PEPSs – zwei konkrete Instanzen von Tensor-Netzwerk-Zuständen – werden eingeführt und mehrere zuge- hörige Algorithmen vorgestellt, wobei der Fokus auf nichttrivialen Einheitszellen liegt. Das Grundzustands-Phasendiagramm von Ketten von Rydberg-Atomen mit langreichweiti- gen Van-der-Waals-Wechselwirkungen, wie sie in aktuellen Experimenten als Rydberg-Quan- ten-Simulatoren realisiert werden, wird mithilfe des Dichtematrix-Renormierungsgruppen-Al- gorithmus untersucht. Präzise Phasengrenzen werden ermittelt und zusätzlich zu bekannten kristallinen und ungeordneten Phasen mit Anregungslücken wird eine ausgedehnte kritische Phase mit zentraler Ladung 푐 = 1 gefunden – eine sogenannte „gleitende“ Phase. PEPSs können so konstruiert werden, dass sie unendlich große Korrelationslängen aufwei- sen. Allerdings wird gezeigt, dass die energetisch besten Zustände, die mithilfe von Optimie- rungsverfahren auf dem neuesten Stand der Technik erhalten werden, nur über endliche Korre- lationslängen verfügen. Für diese Untersuchung werden der konform invariante quantenkriti- sche Punkt des (2+1)D Ising-Modells mit transversem Feld und zwei Instanzen mit spontan ge- brochenen kontinuierlichen Symmetrien mit Goldstone-Moden ohne Anregungslücken – das 푆 = 1/2 antiferromagnetische Heisenberg- und XY-Modell – herangezogen. Mittels feldtheore- tischer Einsichten wird ein mächtiger Werkzeugsatz eingeführt, mithilfe dessen Observablen in den Grenzfall unendlich großer Korrelationslängen extrapoliert werden können. Diese Fort- schritte erlauben es, Lorentz-invariante Modelle ohne Anregungslücke zu untersuchen, was von großer Wichtigkeit ist – für Anwendungen, die von kondensierter Materie bis Hochener- giephysik reichen.

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Abstract

Quantum many-body systems are fascinating: Due to strong correlations that can emerge in these systems, they give rise to a rich landscape of phenomena, including high-temperature superconductivity, fractional quantum Hall physics, and quantum spin liquids. Unfortunately, the numerical treatment of such strongly correlated systems is extremely challenging because of their true many-body nature and the dimension of the Hilbert space, that grows exponentially with the system size. Tensor network states are a vast family of variational ansatz wave functions, which are used in condensed matter physics to face this challenge. The overall objective of this thesis is to implement state-of-the-art tensor network algorithms for one- and two-dimensional quantum systems, improve them both on a conceptual and a technical level, and to apply them to concrete physical systems. This thesis introduces the tensor network formalism and gives a comprehensive guide to corresponding computational techniques. Particular attention is drawn to implementations of several tensor network operations using the Python programming language and to enormous speedups that can be achieved by utilising graphics processing units for tensor network contractions. Matrix product states and projected entangled-pair states, which are two specific instances of tensor network states, are introduced and several associated algorithms are presented with a focus on nontrivial unit cells. The ground state phase diagram of chains of Rydberg atoms with long-range van der Waals interactions, as they are realised in recent experiments implementing Rydberg quantum simulators, are studied using the density matrix renormalisation group algorithm. Accurate phase boundaries are reported and in addition to the known, gapped crystalline and disordered phases, an extended critical phase with central charge 푐 = 1 is found – a so-called floating phase. The obtained results enable immediate experimental realisations and investigations of these floating phases. Projected entangled-pair states can be constructed to have infinite correlation lengths. How- ever, it is shown that the energetically best states obtained for several gapless models, using state-of-the-art optimisation techniques, only display finite correlation lenghts. For this study, the conformally invariant quantum critical point of the (2 + 1)D transverse-field Ising model and two instances of spontaneously broken continuous symmetries with gapless Goldstone modes – the 푆 = 1/2 antiferromagnetic Heisenberg and XY model – are considered. By incorporating field theoretical insights, a powerful finite correlation length scaling framework is established, resulting in formulae that enable extrapolations of observables to infinite correlation lengths. These advances allow for studying Lorentz-invariant, gapless models, which is of great importance for applications ranging from condensed matter to high-energy physics.

Acknowledgements

First, I want to thank my supervisor, Andreas Läuchli, for giving me the opportunity to join his research group and for accepting me as a PhD student. I am very grateful for his guidance through the field of condensed matter physics, but also for giving me the freedom tomakemy own explorations. I am very thankful for the nice colleagues, who made my time in the research group so enjoyable. I would particularly like to thank Christoph Pernul, Michael Schuler, and Alexander Wietek for always lending a helping hand and assisting me at bringing my own thoughts into the right order. My special thank goes to Thomas Lang, not only for all the helpful and often quite lengthy discussions and for standing me in our office, but also for helping me to improve the readability of this thesis. For four months I was visiting the research groups of Frank Verstraete and Jutho Haegeman in Ghent and I want to express my sincere gratitude for their overwhelming hospitality as well as for all the enlightening moments I experienced thanks to them. I also want to thank the people I met in Ghent for immediately including me into the research group and especially Laurens Vanderstraeten and Bram Vanhecke for always sharing their valuable insights with me. My thank goes to Frank Pollmann for giving me the opportunity to stay for two months with his research group in Garching and for his advice on simulating one-dimensional quantum systems. At this point I also want to thank Johannes Hauschild and Ruben Verresen for valuable discussions in the context of Rydberg chains and corresponding simulations techniques. I want to express my deepest gratitude to my girlfriend Vera – not only for thoroughly proofreading this thesis and helping me to make it much more instructive, but also for steadily encouraging me and for giving me all the moral support I needed. Finally, I want to thank my parents, who enabled me to follow my way, which led me to Innsbruck and in further consequence to my PhD studies. I am thankful for their unconditional support, even though this implied living far away from one another.

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Contents

Zusammenfassung iii

Abstract v

Acknowledgements vii

1 Introduction 1

I Computational Methods 7

2 Tensor Networks 9 2.1 Notation ...... 9 2.2 Complexity of Contractions ...... 11 2.3 Finding the Optimal Contraction Order ...... 12 2.4 Python Implementation ...... 16 2.4.1 Memory Representation ...... 16 2.4.2 Fusing and Splitting Indices ...... 18 2.4.3 Contracting Tensor Networks ...... 18 2.5 Technical Details of Tensor Contractions ...... 20 2.6 Graphics Processing Units ...... 21 2.7 Tensor Decompositions ...... 24 2.8 Dominant Eigenvectors ...... 26 2.9 Geometric Series and Linear Equations ...... 29

3 Tensor Network States 33 3.1 Motivation ...... 33 3.2 The Tensor Network State Ansatz ...... 36 3.3 Entanglement Entropy ...... 37 3.4 Examples ...... 40

4 Matrix Product States 43 4.1 Historical Context ...... 43 4.2 The Ansatz ...... 43 4.3 Computing Observables ...... 45 4.4 Canonical Form ...... 46 4.5 Computing Observables Revisited ...... 50

ix Contents

4.6 Correlation Length ...... 51 4.7 Matrix Product Operators ...... 53 4.8 Finding MPS with Maximal Overlap ...... 54 4.9 Finding the Dominant MPS of an MPO ...... 58 4.10 Concluding Remarks ...... 63

5 Projected Entangled-Pair States 65 5.1 The Ansatz ...... 65 5.2 Computing Observables ...... 68 5.3 Boundary MPS Contractions ...... 70 5.4 Corner Transfer Matrix Contractions ...... 73 5.5 Correlation Length ...... 78 5.6 Channel Environments ...... 80 5.7 Energy Optimisation and Gradients ...... 84 5.8 Concluding Remarks ...... 89

II Research 91

6 Floating Phases in One-Dimensional Rydberg Ising Chains 93 6.1 Introduction ...... 93 6.2 Model and Expected Phases ...... 94 6.3 Method ...... 97 6.4 Phase Diagram with Floating Phases ...... 100 6.5 Conclusion ...... 101

7 Finite Correlation Length Scaling in Lorentz-Invariant Gapless iPEPS Wave Functions 103 7.1 Introduction ...... 103 7.2 Infinite PEPS ...... 104 7.2.1 Contraction ...... 105 7.2.2 Energy Optimisation ...... 107 7.2.3 Correlation Length ...... 107 7.3 Quantum Critical Behaviour in (2 + 1)D Conformal Field Theory ...... 108 7.3.1 Overview ...... 108 7.3.2 iPEPS Results for the Transverse-Field Ising Model ...... 111 7.4 Continuous Symmetry Breaking ...... 118 7.4.1 Overview ...... 118 7.4.2 푆 = 1/2 Antiferromagnetic Heisenberg Model ...... 120 7.4.3 푆 = 1/2 XY model ...... 122 7.5 Discussion and Interpretation ...... 124 7.6 Conclusions ...... 125 7.A Appendix: Further Results on the Transverse-Field Ising Model ...... 127 7.B Appendix: Variational iPEPS Energies ...... 128

x 1 Introduction

If someone is examining a piece of iron, they will notice several features: For example, it has a silvery shiny surface, it is conducting electrical currents if a voltage is applied, and it can be magnetised – at least if it is examined at room temperature and atmospheric pressure. For a physicist this raises the question, where these properties originate from and from which first principles this can be explained. Thanks to von Laue, Knipping, and Friedrich it is nowadays known that such a piece of iron is composed of a humongous number of atoms, which are arranged in a regular lattice [1]. The objective is therefore to deduce a material’s properties from the interplay of these constituents. The entirety of atoms composing a quantum many-body system is governed by the Schrödinger equation [2],

푖휕푡 |휓⟩ = ℋ |휓⟩ , (1.1) which can be seen as the axiom of solid state physics in the non-relativistic limit. Here, ℋ is the Hermitian Hamiltonian operator dictated by the physical system. It is rather simple to formulate the Schrödinger equation for a system consisting of an arbitrary number of interacting nuclei and electrons,

풩 2 풩 2 ℰ 2 ℰ 2 풩 ×ℰ 퐏푗 1 푄푗푄푘 퐩푗 1 푒2 1 푄푗푒 ℋ = ∑ + ∑ + ∑ + ∑ − ∑ , (1.2) 푗 2푀푗 4휋 푗<푘 ‖퐑푗 − 퐑푘‖ 푗 2푚 4휋 푗<푘 ‖퐫푗 − 퐫푘‖ 4휋 푗,푘 ‖퐑푗 − 퐫푘‖ where 푃푗, 푀푗, and 푄푗 (푝푗, 푚, and 푒) are the momentum operators, masses, and charges of the nuclei (electrons) and 풩 (ℰ) is the set of nuclei (electrons). However, solving the Schrödinger equation for this Hamiltonian is practically a futile endeavour.

Effective Models

For this reason, physicists derive effective models from realistic models like the one given in equation (1.2), which are approximations, but capture the essential physics and are of great help at understanding certain aspects of the real world [3]. A prominent example of an effective model in the field of (theoretical) solid state physics is the Hubbard model, which neglects vibrations of the nuclei, considers the dynamics of electrons only, and truncates the range of

1 Chapter 1 Introduction

Coulomb interactions between electrons. In the most basic Hubbard model, electrons can only hop between nearest-neighbour sites of the lattice and the interaction between them is limited to a contact interaction,

† † ℋ Hubbard = −푡 ∑ ∑ (푐푗,휎 푐푘,휎 + 푐푘,휎 푐푗,휎 ) + 푈 ∑ 푛푗,↑푛푗,↓. (1.3) ⟨푗,푘⟩ 휎 푗

Although this is an enormous simplification compared to the more realistic model, the simple half-filled Hubbard model on different lattices can still describe phenomena such asmetal- insulator transitions [4], Lifshitz transitions [5], as well as topological insulators [6, 7]. Fur- thermore, a comprehensive solution of the doped Hubbard model is believed to shed new light on the phenomena of high-temperature superconductivity [8, 9]. In many cases, including the case of the Hubbard model, these effective models are still extremely challenging, even though they are much simpler than the original model. From the Hubbard model at half filling and in the limit of strongly repulsive interactions, i. e. 푈 ≫ 푡, one can obtain another effective model, the so-called Heisenberg model,

ℋ HB = ∑ 퐒푗 ⋅ 퐒푘, (1.4) ⟨푗,푘⟩

1 with the spin-1/2 operators 퐒 = (휎푥 , 휎푧, 휎푧)푇 [3]. This seemingly trivial model is quite de- 푗 2 푗 푗 푗 lusive: Although this pure spin model has no fermionic mobile degrees of freedom left, it can be still very challenging to solve it, depending on the underlying lattice. Typically, one is interested in the ground state, which is the state the system takes on at absolute zero temperature. The ground state of the Heisenberg model on a square lattice is a Néel antiferromagnet, in which neighbouring spins are aligned antiparallel, but without perfect classical antiferromagnetic order due to quantum fluctuations [10]. However, the ground state of the Heisenberg model on the kagome lattice, which is composed of corner-sharing triangles, is still investigated [11–13]. The gapped ℤ2 spin liquid and the gapless U(1) Dirac spin liquid are two potential candidates for the ground state. These spin liquids are exotic quantum states with long-range entanglement, but without any magnetic order due to strong quantum fluctuations14 [ ].

Continuous Quantum Phase Transitions

Another effective spin model is the so-called transverse-field Ising model,

푧 푧 푥 ℋ TFI = − ∑ 휎푗 휎푘 − ℎ ∑ 휎푗 , (1.5) ⟨푗,푘⟩ 푗 where the first term represents a ferromagnetic nearest-neighbour Ising interaction andthe second term gives a transverse field with strength ℎ. In contrast to the Heisenberg model, the ferromagnetic transverse-field Ising model does not experience frustration effects, regardless of the lattice. For small ℎ, the transverse-field Ising model has a twofold degenerate ferromagnetic 푧 ground state with |⟨휎푗 ⟩| > 0, whereas for large ℎ, the ground state is unique and polarised in 푧 푥-direction with ⟨휎푗 ⟩ = 0, which is colloquially called a paramagnetic state.

2 An important observation is that the transverse-field Ising model has a built-in ℤ2 symmetry, i. e. the Hamiltonian is invariant under spin inversions for every value of ℎ, but the ordered ferromagnetic ground state spontaneously breaks this symmetry in the thermodynamic limit. However, the disordered paramagnetic ground state is invariant under spin inversions. This link between the formation of order in the system and spontaneous symmetry breaking is a very fundamental mechanism and is ubiquitous in nature [15]. For instance, when a liquid freezes and hence becomes a solid, the continuous translation symmetry is broken down to a discrete symmetry. The two different phases in the transverse-field Ising model are connected via a phase transition at a critical point with transverse field ℎ = ℎ푐. In contrast to phase transitions known from daily life, e. g. the melting of ice to water, or a ferromagnet becoming a paramagnet when heated above the Curie temperature, which happen at some finite temperature 푇 > 0, the considered phase transition in the transverse-field Ising model occurs also at 푇 = 0. Such a phase transition at absolute zero temperature is called a quantum phase transition and the corresponding critical point is called a quantum critical point [16]. In contrast to first order phase transitions, which display a discontinuity in the free energy known as latent heat, in condensed matter physics one is mainly concerned with so-called continuous or second order phase transitions. The quantum phase transition of the transverse- field Ising model is of this continuous type. Considering a connected correlation function ofa transverse-field Ising ground state (apart from the critical point), one observes that in thelimit of long distances, correlations decay exponentially in both phases with a characteristic length scale, called correlation length 휉,

|퐫푗 −퐫푘 |→∞ 푧 푧 푧 푧 −|퐫푗 −퐫푘 |/휉 ⟨휎푗 휎푘 ⟩ − ⟨휎푗 ⟩⟨휎푘 ⟩ ∝ 푒 . (1.6) However, as one approaches the critical point, the correlation length diverges, which is the characteristic feature of a continuous phase transition [15]. Directly at the critical point, the correlation length is infinite, which implies algebraic decay of correlations and a scale invariant system. This is quite astonishing, as these strong correlations between all sites of the system emerge from interactions between nearest neighbours only. When approaching the critical point from the ferromagnetic phase, the vanishing of the order parameter is described by 푧 훽 ⟨휎푗 ⟩ ∝ (ℎ푐 − ℎ) , (1.7) and when approaching the critical point from either the ferromagnetic phase or the disordered phase, the diverging of the correlation length is described by −휈 휉 ∝ |ℎ푐 − ℎ| , (1.8) where 훽 and 휈 are so-called critical exponents. A critical point is described by a set of (commonly six) critical exponents, which are related via the so-called (hyper)scaling relations, where only two of the critical exponents are independent. It is amazing that these critical exponents are equal for numerous different physical systems and that their corresponding critical behaviour does not depend on microscopic details of the model, but rather on the underlying mechanism of spontaneous symmetry breaking or the number of spatial dimensions. Systems with equal critical exponents are said to belong to the same universality class [15, 17].

3 Chapter 1 Introduction

Strongly Correlated Systems

Weakly correlated systems can commonly be described in terms of effective single-body theories or by applying perturbation theory. An example is the well-known mean-field theory, which reduces a many-body system to a single particle interacting with an effective field generated by the rest of the particles. Mean-field descriptions cannot account for correlations in a system, as the resulting quantum states are product states, and therefore cannot describe critical behaviour accurately. This manifests, for example, in the fact that critical properties obtained within a mean-field description are described by the same critical exponents, e.g. 훽 = 1 , regardless of the actual universality class [17]. 2 In contrast, systems that cannot be described in terms of such a simple theory, are called strongly correlated systems. They are dominated by the interactions between the individual particles and in order to determine their properties the entirety of particles has to be considered. They are of particular interest in condensed matter physics due to their rich features emerging even from very simple interactions. However, they are also among the most challenging systems in this field. Prominent examples of such systems, which are eagerly studied in today’s research are high-temperature superconductors [8], fractional quantum Hall systems [18–20], and quantum spin liquids [11–14]. A very successful way to tackle such strongly correlated systems are numerical methods due to the nonexistence of reliable analytical approaches for the general case. Unfortunately, there is no holy grail of a single method to solve each and every many-body system. However, there are several numerical methods around and each of them has its own sweet spot. For instance, the exact diagonalisation method can be applied to arbitrary physical systems, but is limited to very small systems [21]. In contrast, quantum Monte Carlo methods can deal with systems of much larger size, but cannot be used to address general fermionic or frustrated spin systems [22–24]. Therefore, new numerical methods are always welcome to overcome some of these limitations and face challenges from a different direction.

Tensor Network States

The main topic of this thesis are tensor network states, which are a particular family of numerical methods for addressing challenging strongly correlated systems. In 1992, White introduced a variational method – the so-called density matrix renormalisation group (DMRG) – which quickly became an extremely powerful tool for systems in one spatial dimension [25– 28]. However, the attempt to apply DMRG to two-dimensional systems revealed that small systems can be studied accurately, but large systems require an infeasible amount of computational resources. In parallel, it was discovered that the quantum state resulting from a DMRG simulations is a states that is nowadays known as matrix product state (MPS) – a specific instance of a tensor network state [29, 30]. This was the starting point for further developments, which were driven by quantum information theory [31–36]. It was realised that these MPSs systematically parametrise the low-entanglement corner of the Hilbert space by obeying an area law for the entanglement entropy for one-dimensional quantum systems. Since ground states of gapped Hamiltonians with local interactions are subject to such an area law for the

4 entanglement entropy [37], it becomes clear why DMRG does so well for one-dimensional systems. However, in the case of large two-dimensional systems DMRG fails, as an MPS cannot account for the amount of entanglement that is required by a general two-dimensional area law state. Another instance of a tensor network state that has to be mentioned at this point, as it is extensively studied within this thesis, is the projected entangled-pair state (PEPS), which is a straightforward generalisation of MPS for two-dimensional quantum systems [38]. A PEPS can account for an area law for the entanglement entropy for two-dimensional quantum systems, but is much more challenging from a computational point of view. In contrast to MPSs, they are still far from being a black-box tool, but they have already been used successfully to address a range of extremely challenging systems, e. g. various SU(푁 ) Heisenberg models [39–42], the 푡-퐽 model [43, 44], or the Shastry-Sutherland model [45, 46]. In conclusion, tensor network states with their corresponding algorithms are an exciting novel family of variational methods that can be used to systematically tune the amount of entanglement and correlations contained in a system and therefore trade accuracy for computation time. They give rise to promising numerical methods, that are well-suited to address open challenges in condensed matter physics. Within this thesis, state-of-the-art tensor network state methods are implemented and studied. They are improved both on a conceptual and a technical level and are used to make inroads in the understanding of strongly correlated physical systems.

Outline

This thesis is structured as follows. The first part introduces and discusses computational methods. Chapter 2 gives an introduction to tensor networks, which are the mathematical framework utilised throughout this thesis and provides a guide to implement numerical algorithms. In chapter 3, a general introduction to tensor network states is provided and compared to other numerical methods. MPSs and PEPSs are introduced as two specific instances of tensor network state in chapters 4 and 5. In the second part, the research projects performed within this thesis are presented. In chapter 6, MPSs are used to study chains of Rydberg atoms with van der Waals interactions, which feature an extended critical phase. A second research project, that is presented in chapter 7, gives an extensive study of correlation lengths obtained with PEPSs for gapless Hamiltonians and introduces a powerful finite correlation length scaling framework.

Computational Methods

2 Tensor Networks

Although this is a thesis in theoretical physics, this chapter is completely decoupled from physics. The aim of this chapter is to introduce the language used throughout this thesis – a language of tensors and tensor networks. First, the tensor network notation will be introduced, which is a convenient way to formulate tensor expressions and contractions of tensor indices. In a second step the costs of contracting tensors are assessed and how to minimise these costs when contracting three or more tensors. The last part of this chapter is a hands-on introduction on how to perform numerical computations with tensor networks using the Python programming language. After a demonstration of high-performance tensor network contractions using both CPUs and GPUs, ways to implement special operations such as tensor decompositions, computations of dominant eigenvectors, and computations of geometric series are illustrated. After going through this chapter the reader should be well prepared to develop their own tensor network algorithms.

2.1 Notation

Whenever one has to deal with mathematical expressions containing many tensors1 and indices, e. g.

∑ 푎푙푘푗푎푛푚푙푎푞푝푛푎푗푟푞, (2.1) 푗,푙,푛,푞 natural questions arising are: Which index is connected with which index of which tensor? How many open indices has the resulting object? How can one make sure to have written all the indices in all the right places? Fortunately, the notation introduced in this section will make these questions obsolete. In- stead of using longish expressions as shown above, the idea is to use a graphical notation that goes back to a graphical notation introduced by Penrose in the context of general relativ- ity [47], where each occurrence of a tensor is represented by a node and each index of a tensor is denoted by an edge connected to this node. This notation goes back to a tensor notation

1Depending on the scientific field, the understanding of what a tensor actually is, varies significantly. Withinthis thesis, tensors are simply considered as multi-index objects.

9 Chapter 2 Tensor Networks introduced The simplest example one could think of is a vector,

푣푗 = 푗 푣 . (2.2) Going one step further, a matrix can be expressed as

퐴푗푘 = 푗 퐴 푘 . (2.3) Note that in contrast to the example of a vector, for the matrix a node with non-symmetric shape is used, which prevents confusion of indices. However, one could use a symmetric shape to indicate a certain symmetry, e. g. 퐴푗푘 = 퐴푘푗 in the case of a matrix. By simply increasing the number of edges connected to a node, extending this notation to tensors of higher rank2 is straightforward. One thing that is not apparent from this graphical notation is the order of the indices. For many analytical calculations the actual order is irrelevant, but when it comes to storing a tensor on a computer the order needs to be fixed in some way. One of the most common operations in tensor expressions are contractions, i. e. summations over indices. Contractions of pairs of indices will be indicated by connecting the corresponding indices. A complex vector-vector product, i. e. a dot product, or a matrix-vector product then reads as

∗ ∗ ∑ 푣푗 푣푗 = 푣 푣 or ∑ 퐴푗푘푣푘 = 푗 퐴 푣 , (2.4) 푗 푘 respectively. Making use of this graphical notation, the example from equation (2.1) can be rewritten as

푘 푚 푝 푟 푎 푎 푎 푎 . (2.5)

This way it becomes obvious, which indices are contraction indices and which are open indices. Expressions of this kind, consisting of tensors and contractions connecting them, shall be called tensor networks here and henceforth. Note, that in the following, names of tensors will often be omitted in diagrams in order to simplify the notation. Unless needed, names of open indices will be omitted in diagrams as well and will be matched by their location and direction. Further, to distinguish different tensors, different shapes and colours will be used, e.g.

= . (2.6)

In addition, shapes will be used to identify certain properties of tensors. As already pointed out above, symmetries of index transpositions can be represented visually by choosing appropriate shapes. Triangular shaped tensors will be used in this work to identify isometric tensors:

2Unfortunately, the meaning of ‘rank’ is ambiguous as it can either refer to the rank of a matrix, which is the number of linearly independent rows or columns of a matrix, or to the number of indices of a tensor. Whenever the term rank is used in this thesis, it refers to the latter one.

10 2.2 Complexity of Contractions

Similar to an isometric matrix 푈 , for which 푈 †푈 = ퟙ, for an isometric tensor there exists an identity relation, e. g.

∗ = , (2.7) where the asterisk indicates the complex conjugation of the corresponding tensor. Indices not leaving the triangular tensor in the direction the triangle is pointing are the ones that have to be contracted to obtain an identity tensor. The notation as presented in this section is limited to contractions of pairs of indices only. However, this will be sufficient for the applications addressed in the following chapters.

2.2 Complexity of Contractions

At this point, the computational complexity of tensor (network) contractions shall be analysed. In practical applications this is an important quantity as it enables to estimate which problem sizes are feasible and for which a solution is out of reach. Considering a basic example of a contraction of two tensors,

휒2 휒4 , (2.8) 휒1 휒3 where the dimensions of the indices are indicated next to the edges, the computational complexity is 풪 (휒1휒2휒3휒4): The resulting tensor consists of 휒2휒3휒4 elements and for each of these elements a sum containing 휒1 terms has to be computed. In a more general fashion, the computational complexity of contracting two tensors is given by the product of all dimensions of the resulting tensor and of all dimensions of the contraction indices. Multiplying an 푚 × 푘 matrix with a 푘 × 푛 matrix is a simple case where this is well-known: For each entry of the resulting 푚×푛 matrix a sum over 푘 terms must be computed, so the computational complexity is 풪 (푚푛푘) (ignoring possible improvements via the Strassen algorithm [48]). Moving on from pairs of tensors to a first example of a tensor network consisting of three tensors,

휒 휒 푝 , (2.9) 휒 휒 one has to be more careful estimating the computational complexity. Simply computing the 휒 2 elements of the resulting tensor by iteratively computing the triple-sum, the computational complexity would be 풪 (푝휒 4). If this was the optimal way, it would be infeasible to perform contractions of larger networks, as the complexity would scale exponentially with the number of tensors. However, it is possible to compute intermediate tensors, which requires additional memory, but makes the overall computation cheaper. Contracting only the green and the orange tensor

11 Chapter 2 Tensor Networks of equation (2.9) in a first step and the intermediate tensor with the blue tensor in asecond step,

휒 휒 휒 휒 휒 푝 = 푝 = , (2.10) 휒 휒 휒 휒 reduces the computational complexity to 풪 (푝휒 3), which is the computational complexity of each of the two contraction steps. In reference [49] it is shown that any tensor network can be contracted asymptotically optimally by performing subsequent pairwise contractions. How- ever, the order of contractions is crucial: One could as well contract the orange and the blue tensor in a first step, followed by a contraction of the green and the intermediate tensor,

휒 휒 = = , (2.11) 휒 휒 but this results in a computational complexity of 풪 (푝휒 4), which is strictly larger than the 풪 (푝휒 3) of equation (2.10). Finding the optimal contraction order is a combinatorial problem becoming extremely difficult with increasing number of tensors that has been shown tobe NP-complete [50].

2.3 Finding the Optimal Contraction Order

Although the problem of finding an optimal contraction order for a tensor network isNP- complete, in this section a dynamic programming approach based on the work in reference [49], that is feasible for all tensor networks within the scope of this thesis, is presented. A novel feature of the approach presented here is its applicability not only to contractions connecting pairs of tensors, but an arbitrary number of tensors as well. An open-source implementation of this approach has been made publically available [51]. Starting with an arbitrary expression, consisting of a product of tensors and index contractions, all trace-like contractions, i. e. contractions for which the contraction index occurs in the index list of exactly one tensor, shall be executed first, e. g. the summations over 푗, 푘, and 푙 in

̃ ∑ 푎푗푚푏푘푘푚푛푐푙푙푙푛 = ∑ ̃푎푚푏푚푛 푛̃푐 . (2.12) 푗,푘,푙,푚,푛 푚,푛

In a second step, the set of tensors shall be divided into partitions, such that no tensors from different partitions are connected. Two tensors are connected if they have at least one common contraction index in their index list. For the example

∑ 푎푗푘푏푘푙푐푚푑푘푙푒푚, (2.13) 푘,푙,푚

12 2.3 Finding the Optimal Contraction Order

Input: sets of input tensor indices {푖푘 ⊂ 핀 ∶ 0 ≤ 푘 < 푛inputs}, set of output tensor indices 표 ⊂ 핀, index dimensions 푑 ∶ 핀 → ℕ Output: contraction tree 푡 ({0, 1, … , 푛inputs − 1}) 1 푐cap ← minimal possible cost 2 푖Σ = ⋃ 푖푘 ⧵ 표 푘 3 for 푛 ∈ {1, 2, … , 푛inputs} do 4 푆푛 ← ∅

5 for 푘 ∈ {0, 1, … , 푛inputs − 1} do 6 푐 ({푘}) ← 0 7 푡 ({푘}) ← 푘 8 푆1 ← 푆1 ∪ {{푘}} 9 while 푆푛inputs = ∅ do 10 for 푛 ∈ {2, 3, … , 푛inputs} do 푛 11 for 푚 ∈ {1, 2, … , ⌊ ⌋} do 2 12 for (푠1, 푠2) ∈ 푆푚 × 푆푛−푚 with 푠1 ∩ 푠2 = ∅ do 13 if indices(푠1) ∩ indices(푠2) ∩ 푖Σ ≠ ∅ then 14 푠1∪2 ← 푠1 ∪ 푠2 ′ 15 푐 ← 푐(푠1) + 푐(푠2) + cost2(푠1, 푠2) ′ ′ 16 if 푐 < 푐cap ∧ (푠1∪2 ∉ 푆푛 ∨ 푐 < 푐(푠1∪2)) then ′ 17 푐(푠1∪2) ← 푐 18 푆푛 ← 푆푛 ∪ {푠1∪2} 19 푡(푠1∪2) ← (푡(푠1), 푡(푠2))

20 푐cap ← 훾 × 푐cap Algorithm 2.1: Dynamic programming approach with cost capping for finding an optimal contraction order for a tensor network of connected tensors.

((0, 1), 2) ((0, 1), (2, 3))

(0, 1) (0, 1) (2, 3)

0 1 2 0 1 2 3

Figure 2.1: Illustration of two examples of contraction tree representations of contraction orders. Starting from the single tensors which are represented by the numbers in the lowermost line, the contraction tree is constructed by contracting pairs of tensors. The final contraction tree expression can be found in the tree’s root node.

13 Chapter 2 Tensor Networks the partitions are {푎, 푏, 푑} and {푐, 푒}. These partitions can be determined by traversing the graph using breadth-first-search [52]. For each partition the contraction order can be optimised independently and the optimal contraction for the whole tensor network can be obtained by forming an outer product of the optimal contractions of all partitions. Finally, the remaining optimisations of the contraction order for each of the tensor network partitions is performed individually by applying the procedure given in algorithm 2.1 to a single partition. This algorithm requires the following input: the set of indices for each of the 푛inputs input tensors of the partition, 푖푘 ⊂ 핀, the set of indices of the contraction result, 표 ⊂ 핀, and all index dimensions, 푑 ∶ 핀 → ℕ (with the set of all indices 핀). As a result the algorithm yields the optimal contraction order in the form of a contraction tree, which can be either a pair (푡1, 푡2) of two subtrees to be contracted or a single tensor 푘. For example, a contraction tree ((0, 1), 2) indicates that tensors 0 and 1 should be contracted in a first step, forming the intermediate tensor (0, 1), which is contracted with tensor 2 in a second step. An illustration of examples of contraction trees can be found in figure 2.1. Considering the example

휒 휒 2 푝 , (2.14) 0 휒 휒 1

one could choose the input indices to be 푖0 = {푚, 푛}, 푖1 = {푗, 푘, 푚}, 푖2 = {푗, 푙, 푛}, which implies the output indices 표 = {푘, 푙} and the dimensions 푑(푗) = 푝 and 푑(푘) = 푑(푙) = 푑(푚) = 푑(푛) = 휒. The optimal contraction given by the algorithm is either ((0, 1), 2) or ((0, 2), 1), as discussed in the previous section. The overall idea of algorithm 2.1 is to use a dynamic programming approach to construct and store an optimal contraction order for each subset 푠 ⊂ {0, 1, … , 푛inputs − 1} of 푛 tensors, where 푛 = |푠| ≤ 푛inputs, by combining already known optimal contractions for disjoint subsets of 푚 and 푛−푚 tensors, where 푚 < 푛, starting from the trivial case 푛 = 1. As soon as a (better) contraction order for a subset 푠 is found, the contraction tree is stored in 푡(푠) and the corresponding cost of the contraction in 푐(푠). Note that two subsets of tensors are only combined if they share contraction indices, i. e. intermediate outer products are ignored, which leads to asymptotically optimal contractions [49]. A major performance improvement is realised via the concept of cost capping [49]: Instead of constructing contractions for all possible subsets of tensors immediately, only contractions with a cost below a certain cost cap 푐cap are constructed. If it is not possible for a certain 푐cap to find a contraction of all tensors, 푐cap is increased by a factor of 훾 and the search is repeated. This increment factor is commonly chosen to be the smallest dimension among the occurring indices, 훾 = min {푑(ℓ) ∶ ℓ ∈ 핀}, but simply increasing the cost cap at a fixed rate, e. g. 훾 = 1.1, is a valid choice as well. The idea of cost capping is to avoid constructing intermediate contractions that are more expensive than the optimal overall contraction. It should be noted that even though this helps in a lot of practical cases, it does not reduce the complexity in general. For example, the worst case of a network with identical connections between each

14 2.3 Finding the Optimal Contraction Order pair of tensors,

, (2.15)

still leads to exponential scaling. The function indices(푠) used in algorithm 2.1 is defined as

indices(푠) = ⋃ 푖푘 ⧵ (푖Σ ⧵ ⋃ 푖푘) , (2.16) 푘∈푠 푘∉푠 and therefore yields the open indices of a contraction of a subset of tensors 푠, where 푖Σ is the set of all contraction indices of the considered partition. Note that this rather cumbersome form, including the subtraction of indices of tensors not contained in 푠, is necessary for the algorithm to work for cases with indices connecting three or more tensors as well – an improvement over the approach in reference [49]. The function cost2(푠1, 푠2) is used in algorithm 2.1 as a model to quantify the cost of contracting a pair of tensors. One possible choice is the product of all dimensions occurring in the contraction,

cost2(푠1, 푠2) = ∏ 푑(ℓ) with 푖 = indices(푠1) ∪ indices(푠2), (2.17) ℓ∈푖 which is proportional to the runtime complexity of the contraction. Although this choice works well in practice, more sophisticated models – ideally tailored for the target hardware – can be used as well. The ultimate version of cost2(푠1, 푠2) measures the runtime required for contracting 푠1 and 푠2 on the target hardware (with a timeout given by 푐cap), although this leads to larger runtimes for the optimisation process itself. In the end one has to find the right balance between the runtime for optimising a contraction (once) and the runtime of actually performing the optimised contraction (as often as needed). The initial choice of 푐cap naturally depends on the choice of cost2(푠1, 푠2). If possible, 푐cap should be initialised with a theoretical lower bound for the overall cost of the contraction, e. g. if the cost model from equation (2.17) is used,

푐cap = ∏ 푑(ℓ) (2.18) ℓ∈표 is appropriate. As a rule of thumb one should start with a value small enough not to explore too much of the unneeded search space containing far too expensive contraction orders. In summary, one should keep in mind that there is no ‘right’ cost model, initial cost cap, and increment strategy for the cost cap as there is no free lunch [53].

15 Chapter 2 Tensor Networks

푖 row-major order (C order) 1 푠0 = 4 푠 = 1 1 0 1 2 3 4 5 6 7 8 9 10 11 0 1 2 3

푖0 4 5 6 7 column-major order (Fortran order) 8 9 10 11 0 4 8 1 5 9 2 6 10 3 7 11 푠0 = 1 푠1 = 3

Figure 2.2: Illustration of row-major order (C order) and column-major order (Fortran order) for the example of a matrix.

2.4 Python Implementation

This section is intended as a hands-on introduction for people who are new to the field of tensor networks and want to implement numerical tensor network algorithms using the Py- thon programming language. Please note that this is not an introduction to Python itself and therefore the reader is assumed to have basic knowledge of Python and the corresponding scientific libraries such as NumPy and SciPy. In the scientific community other programming languages and technologies are used as well for implementing tensor network algorithms, e. g. C/C++ [54], Julia [55], or Matlab [56].

2.4.1 Memory Representation The first requirement for any tensor network algorithm is to have a representation of atensor in a computer’s memory. To accomplish this, n-dimensional arrays as provided by the NumPy library [57] are used. As computer memory is naturally linear, the concept of strides is used to store tensors of arbitrary rank. If a tensor with dimensions 푑0, 푑1, 푑2,… needs to be stored, so-called strides 푠0, 푠1, 푠2,… for all dimensions are used to map a tensor element 푎푗0푗1푗2… to its location ∑푘 푗푘푠푘 within a linear piece of memory containing ∏푘 푑푘 elements. Therefore, the stride 푠푘 is the distance between two elements in the linear computer memory when increasing the index 푗푘 of the corresponding dimension by one. 푘−1 Two common choices for strides of rank-푛 tensors are 푠푘 = ∏ℓ=0 푑ℓ (with 푠0 = 1) and 푛−1 푠푘 = ∏ℓ=푘+1 푑ℓ (with 푠푛−1 = 1). The first one is often called Fortran order or column-major order and the latter one C order or row-major order. They are illustrated for the example of a matrix in figure 2.2, where it becomes apparent that column-major (row-major) order implies that the stride for the column (row) index is the larger one. For arrays with more than two dimensions the concept of rows and columns becomes obsolete and hence column-major and row-major order are commonly called Fortran and C order originating from the strides used in these programming languages for multi-dimensional arrays. Both conventions lead to a contiguous alignment of the tensors in memory. A powerful feature of using strides is that arbitrary transpositions of tensors, i. e. permutations of indices, only require permuting the corresponding strides, but no expensive copying

16 2.4 Python Implementation

1 >>> a.shape 2 (2,3,4) 3 >>> b = a.transpose(0, 2, 1) 4 >>> b.shape 5 (2,4,3)

Listing 2.1: Transposing a tensor stored as NumPy array.

1 >>> a 2 array([[0.,1.,2.,3.], 3 [ 4.,5.,6.,7.], 4 [ 8.,9.,10.,11.], 5 [12.,13.,14.,15.]]) 6 >>> a.shape # a is a 4x4matrix 7 (4,4) 8 >>> a.itemsize # a requires8bytesperelementinmemory 9 8 10 >>> a.strides # a is C-ordered(stridesgiveninbytes) 11 (32,8) 12 >>> b = a[:, 1] # select2ndcolumnofa 13 >>> b 14 array([1.,5.,9.,13.]) 15 >>> b.strides # b is notcontiguousinmemory 16 (32,) 17 >>> c = a[1:3, 1::2] # select2ndto3rdrowandevery 18 >>> c # 2ndcolumnstartingfrom2ndcolumn 19 array([[5.,7.], 20 [ 9.,11.]]) 21 >>> c.strides 22 (32,16)

Listing 2.2: Slicing operations on NumPy arrays. of the actual memory storing the tensor entries is required. Tensors can be transposed using the NumPy function transpose as demonstrated in listing 2.1. Another characteristic of the strided storage is the capability of slicing operations as demonstrated in listing 2.2. They can be used to select parts of a tensor by specifying either a single index j or an index range j:k:s for each dimension. Whenever a single index is used for a dimension, the rank of the resulting tensor is reduced by one, e. g. if a single index is used to select a column from a matrix as shown in listing 2.2. In contrast to a single index, an index range j:k:s, defined by a start index j, a stop index k, and a step size s, is not modifying the rank of the tensor, as an arbitrary number of elements is selected along the corresponding dimension. Whenever the start index, the stop index, or the step size is not specified, they default to 0, the extent of the corresponding dimension, or 1, respectively. Omitting the second colon of an index range is equivalent to omitting the step size. Slicing operations can be implemented in a cheap fashion using strides, as only a view object to the same data in memory

17 Chapter 2 Tensor Networks

1 >>> a.shape 2 (2, 3, 4, 5, 6) 3 >>> b = a.reshape(6, 4, 30) # fuse 1st-2nd and 4th-5th index 4 >>> b.shape 5 (6, 4, 30) 6 >>> c = b.reshape(2, 3, 4, 5, 6) # split them again 7 >>> np.allclose(a, c) # ensure a and c are equal again 8 True

Listing 2.3: Fusing indices of a tensor. is created and hence no tensor entries have to be copied.

2.4.2 Fusing and Splitting Indices

Sometimes it is necessary to fuse or split indices, which can be achieved by using the NumPy function reshape. One particular example of this necessity are tensor decompositions, which will be discussed in section 2.7. An arbitrary number of indices can be fused to one super-index, which dimension is equal to the product of the dimensions of the fused indices. Listing 2.3 demonstrates how to fuse the first two and the last two indices of a rank-5 tensor. Note that only consecutive indices can be fused this way. Whenever non-consecutive indices shall be fused, the tensor has to be transposed beforehand. The inverse process of fusing indices is called splitting, which can be accomplished as well using the reshape function as shown in listing 2.3. However, splitting is only well-defined for an index that originates from an index fusion and if the dimensions after the splitting operation are the same as before the original fusion operation. One should be aware that there is some ambiguity, how the data is actually aligned within a fused super-index, e. g. if the two indices of a matrix are fused, the resulting vector contains the matrix either row-wise or column-wise. However, the only thing one usually has to care about is that super-indices, which are used as contraction indices later on, require the same order of the underlying indices among all the tensors involved in the contraction. Unlike transpose, which is – thanks to the strided storage – always cheap to apply, the application of reshape after applying transpose on an array requires the actual tensor memory to be reordered, which cannot be done in situ.

2.4.3 Contracting Tensor Networks

Undoubtably, the most important operation on tensor networks within this thesis are contractions. For contracting two tensors the NumPy function tensordot can be used. A much more powerful and versatile framework for performing tensor network contractions – also for three and more tensors – is provided by the Python package opt_einsum [58]. Let us con-

18 2.4 Python Implementation

1 import opt_einsum as oe 2 3 y = oe.contract( 4 x, [3, 4, 5], # e.g. 1st index of x (labelled as 3) ... 5 a, [6, 3, 0], # ... is contracted with the 2nd index of a 6 b, [1, 7, 4, 6], 7 c, [7, 5, 2], 8 [0, 1, 2], # order of output indices (optional) 9 optimize="dp" # use the dynamic programming optimiser 10 )

Listing 2.4: Contracting a tensor network using the opt_einsum package. sider an exemplary tensor network,

2 5 푐 2 7 4 푦 1 = 푥 푏 1 , (2.19) 6 0 3 푎 0 where the numbers enumerate the indices. If the tensors are present as NumPy arrays x, a, b, and c, listing 2.4 demonstrates a simple way to contract the tensor network given in equation 2.19. When calling contract, one has to supply all the input tensors together with enumera- tions of their indices and optionally the order of the output indices. If the output indices are omitted, a list containing all indices of input tensors occurring exactly once in increasing order is used implicitly. Input indices that are not at the same time output indices are summation indices. Note that listing 2.4 implicitly assumes an order of the indices from equation (2.19), e. g. the right index of 푎 (labelled as 0) is the third index of 푎 and becomes the first index of 푦. The function contract finds an optimised contraction order in a first step. Amongsev- eral optimisation strategies that are available in opt_einsum, one of them can be chosen using the parameter optimize. As a part of this thesis a dynamic programming optimiser as described in section 2.3 was implemented for opt_einsum [51] and can be selcted via optimize="dp". After an optimised contraction order has been found, the actual contraction according to this order is performed. In many cases a specific contraction is performed many times, e. g. in every iteration ofan algorithm, which makes it desirable to be able to optimise the contraction order independently of actually performing the contraction. For this reason opt_einsum provides the function contract_expression that only finds an optimised contraction order. Based onthe shapes of the input tensors it yields a function object that can be called to actually perform the contraction. In listing 2.5 exactly the same computation as in the previous listing 2.4 is performed, but the function object expr can be used to repeatedly perform the contraction without the need of optimising the contraction order first.

19 Chapter 2 Tensor Networks

1 import opt_einsum as oe 2 3 # prepare the contraction: 4 expr = oe.contract_expression( 5 x.shape, [3, 4, 5], 6 a.shape, [6, 3, 0], 7 b.shape, [1, 7, 4, 6], 8 c.shape, [7, 5, 2], 9 [0, 1, 2], 10 optimize="dp" 11 ) 12 13 # execute contraction: 14 y = expr(x, a, b, c)

Listing 2.5: Contracting a tensor network by separately optimising the contraction order using contract_expression from opt_einsum and executing the computation.

푚 푚 푙 푘 푗 푙

Figure 2.3: Illustration of how the tensor contraction ∑푙,푚 푎푗푙푚푏푙푘푚 can be mapped to a matrix- matrix multiplication.

2.5 Technical Details of Tensor Contractions

There are essentially three different kinds of tensor contractions: Either no tensor, one tensor or both tensors have open indices, e. g.

, , or . (2.20)

If indices are reordered and fused accordingly, these three kinds of tensor contractions can be reformulated as vector-vector, matrix-vector, or matrix-matrix multiplications. An example of mapping a tensor contraction to a matrix-matrix multiplication is illustrated in figure 2.3. For the mentioned linear algebra operations the various implementations of the well-known Ba- sic Linear Algebra Subprograms (BLAS) [59–64] provide the procedures DOT (vector-vector), GEMV, (matrix-vector), and GEMM (matrix-matrix) as high-performance solutions, which are accessible through NumPy [65]. This method for computing tensor contractions is commonly called Transpose-Transpose-GEMM-Transpose (TTGT) [66, 67]. As mentioned before, fusing

20 2.6 Graphics Processing Units indices after reordering them, i. e. applying NumPy’s reshape after tranpose to a tensor, might cause reordering of the tensor in memory. This reordering does not only increase the memory requirement, but also adds a significant overhead that especially affects bandwidth- bound contractions [67]. Recent proposals [66, 67] avoid the transposition overhead by only constructing blocks of the matrices, which are to be multiplied, on the fly. Note that high-performance GEMM- implementations perform a matrix-matrix multiplication in a block-wise fashion anyway, such that the blocks fit into the processor caches. This way the required copy operations can beover- lapped with the actual computations, which eliminates the transposition overhead in many cases, and therefore bandwidth-bound contractions can be transformed into compute-bound contractions. In the literature this novel approach for tensor contractions is called GEMM-like tensor-tensor multiplication (GETT) [67] and block-scatter-matrix tensor contraction (BSMTC) [66], for which open-source implementations can be found in the TBLIS library [68] and the tensor contraction code generator (TCCG) [69]. Another promising development is the incor- poration of the Strassen algorithm[48] into tensor contractions [70] on top of BSMTC, which reduces the asymptotic computational complexity. At this point it is worth mentioning a particular feature of the opt_einsum package presented in the previous section: When calling contract or executing a previously optimised contraction (using contract_expression), a backend to perform the computations can be selected. By default the NumPy-backend is used, but other (and even custom) backends can be provided and used as well. Due to the clean separation between optimisations and executions of tensor network contractions, opt_einsum can be used to follow the latest developments for tensor contractions. In the following section this feature will be used for offloading computations to graphics processing units.

2.6 Graphics Processing Units

Although the primary task of the graphics processing unit (GPU) in a computer is usually the rendering of pictures, which are to be displayed, they can be used to perform arbitrary computations as well. This is commonly known as general-purpose computing on GPUs (GPGPU). Compared to central processing units (CPUs), GPUs provide higher computation throughput at lower power consumption due to massive parallelism. However, GPUs have a very limited amount of dedicated memory and input and output data has to be moved between CPU and GPU memory, which introduces an additional overhead [71]. Therefore, as a general rule of thumb, the amount of memory transfers between CPU and GPU should be minimised and if possible, memory transfers should be overlapped with computations to increase the overall performance [72]. Listing 2.5 can be easily extended to perform the tensor network contraction on a GPU by selecting a proper backend as shown in listing 2.6. At this point only NVIDIA® hardware and the CUDA® framework are considered due to the lack of alternative GPUs with adequate support for IEEE 754 double precision. As a link between NumPy and the GPU the CuPy package [73] is used. Again it was assumed that the input tensors were given in the form of the NumPy arrays x, a,

21 Chapter 2 Tensor Networks

1 import cupy as cp 2 import opt_einsum as oe 3 4 # prepare the contraction: 5 expr = oe.contract_expression( 6 x.shape, [3, 4, 5], 7 a.shape, [6, 3, 0], 8 b.shape, [1, 7, 4, 6], 9 c.shape, [7, 5, 2], 10 [0, 1, 2], 11 optimize="dp" 12 ) 13 14 # copy numpy arrays (in cpu memory) to gpu: 15 x_gpu = cp.asarray(x) 16 a_gpu = cp.asarray(a) 17 b_gpu = cp.asarray(b) 18 c_gpu = cp.asarray(c) 19 20 # execute contraction on gpu: 21 y_gpu = expr(x_gpu, a_gpu, b_gpu, c_gpu, backend="cupy") 22 23 # copy results back to cpu memory: 24 y = y_gpu.get()

Listing 2.6: Contracting a tensor network on a GPU using CuPy. b, and c. In comparison to listing 2.5 only three changes are necessary to run the computations on the GPU: First, NumPy arrays, which are stored in CPU memory, are copied to GPU memory by calling cp.asarray. Second, the invocation of expr requires the additional argument backend="cupy" to perform the tensor contractions on the GPU. Third, the result is copied back from GPU to CPU memory by calling the method get on a CuPy array. In figure 2.4 the runtimes for executing expr from listings 2.5 (CPU) and 2.6 (GPU) for complex double-precision tensors are compared. The dimension of indices 1, 4, 6, and 7 is called 푑 ∈ {4, 9, 16, 25, 36} and for indices 0, 2, 3, and 5 it is called 휒 ∈ {32, 64, 128, 256, 512, 1024} and hence the naive computational complexity is 풪 (푑4휒 2 + 푑2휒 3). The CPU timings are obtained by running the contraction on a dual-socket Intel® Xeon® Gold 6130 system with NumPy linked to Intel® MKL 2019 and the GPU timings are obtained on an NVIDIA® TITAN V with CuPy and CUDA® 10.0. The presented runtimes correspond to a single contraction of the network in equation (2.19). On average, the GPU runtimes show a speedup of 10.3 compared to the CPU runtimes. One should note that this remarkable speedup is obtained even though all four input tensors and the resulting tensor are copied between CPU and GPU memory. In many cases some or all tensors do not have to be copied, which leads to even larger speedups. The missing GPU datapoints correspond to contractions that require more memory than the 12 GiB available on an NVIDIA® TITAN V. This illustrates a weak spot of GPUs: the limited amount of memory. A possibility to circumvent this limitation is to use out-of-core strategies

22 2.6 Graphics Processing Units

102 Intel® Xeon® Gold 6130 (x2) NVIDIA® TITAN V

100 ] s [

10−2 runtime

10−4

106 107 108 109 1010 1011 1012 푑4휒 2 + 푑2휒 3 (computational complexity)

Figure 2.4: Runtimes for contracting the tensor network from equation (2.19) by executing expr from listing 2.5 (green data points) and from listing 2.6 (orange data points) on the platforms indicated in the legend. The values for the dimensions are chosen as 푑 = 4 (circles), 푑 = 9 (squares), 푑 = 16 (triangles up), 푑 = 25 (triangles down), 푑 = 36 (pentagons) and 휒 ∈ {32, 64, 128, 256, 512, 1024}. The average speedup between GPU and CPU timings is 10.3

23 Chapter 2 Tensor Networks for tensor contractions, i. e. storing the entire tensor in CPU memory and copy only parts of the tensor to the GPU memory as they are needed for computations. If this is done in a clever way, such that the amount of memory transfers is minimised and transfers are overlapped with computations, the GPU peak performance can be reached and therefore the overhead due to memory transfers is amortised [72].

2.7 Tensor Decompositions

Another class of operations frequently used in tensor network algorithms are tensor decompositions, which are based on well-known matrix decompositions, e. g. QR decompositions, eigenvalue decompositions (EVDs), or singular value decompositions (SVDs) [44, 74]. Note that at this point ‘true’ tensor decompositions such as the Tucker decomposition [75, 76] are not considered. For demonstration purposes let us examine the QR decomposition: A matrix 퐴 ∈ ℂ푚×푛 can be decomposed as 퐴 = 푄푅 with 푄 ∈ ℂ푚×푝, 푅 ∈ ℂ푝×푛, and 푝 = min{푚, 푛}. 푄 is an isometric † matrix, i. e. 푄 푄 = ퟙ, and 푅 is upper triangular, i. e. 푅푗푘 = 0 for 푗 > 푘 [77]. If the indices of a tensor are divided into two sets and fused within each set, the QR decomposition can be applied to the emerging matrix. In the end, the indices of the matrices resulting from the decomposition are split again. The decomposition of a rank-4 tensor with two indices in both sets of indices, can be written as

= , (2.21) where the orange tensor is an isometric tensor, i. e.

∗ = , (2.22) and the blue tensor corresponds to an upper triangular matrix. Listing 2.7 shows a Python function, tqr, implementing a tensor QR decomposition as explained above. The parameters j and k are used to pass the corresponding matrix indices, e. g. j=[1, 2] and k=[0, 3] for the green tensor on the left-hand side in equation (2.21) if the indices of the green tensor are enumerated counter-clockwise starting from the right index. A special feature of this implementation is the possibility of unique QR decompositions: If the corresponding flag is set, the diagonal entries of the upper triangular matrix 푅 are enforced to lie on the positive real axis of the complex plane. This can be achieved by inserting a unitary matrix, 푊 = diag(푒푖 arg(푅00), 푒푖 arg(푅11), …), and its Hermitian conjugate between 푄 and 푅 and then absorbing it into 푄 and 푅:

퐴 = 푄푅 = 푄푊 푊 †푅 = 푄̃푅̃. (2.23)

Note that 푄̃ = 푄푊 is still an isometric matrix and 푅̃ = 푊 †푅 an upper triangular matrix with all its diagonal elements on the positive real axis of the complex plane.

24 2.7 Tensor Decompositions

1 import cupy as cp 2 import numpy as np 3 import scipy.linalg as la 4 5 def tqr(a, j, k, unique=False, backend="numpy"): 6 """ 7 Computes the QR decomposition of tensor a with corresponding 8 matrix indices j and k using the chosen backend "numpy" or 9 "cupy". 10 11 If unique is set, the corresponding diagonal elements of r are 12 real and non-negative and therefore the decomposition is unique. 13 """ 14 15 if sorted(j + k) != list(range(a.ndim)): 16 msg = "invalid indices j={} and k={} for rank-{} tensor" 17 raise ValueError(msg.format(j, k, a.ndim)) 18 19 dj = [int(a.shape[i]) for i in j] 20 dk = [int(a.shape[i]) for i in k] 21 a_mat = a.transpose(j + k).reshape(np.prod(dj), np.prod(dk)) 22 23 if backend == "numpy": 24 q, r = la.qr(a_mat, mode="economic") 25 xp = np 26 elif backend == "cupy": 27 q, r = cp.linalg.qr(a_mat, mode="reduced") 28 xp = cp 29 else: 30 raise ValueError("uknown backend '{}'".format(backend)) 31 32 if unique: 33 if xp.iscomplexobj(r): 34 w = xp.exp(1j * xp.angle(xp.diag(r))) 35 else: 36 w = xp.sign(xp.diag(r)) 37 q *= w # q <-- matmul(q, diag(w)) 38 r *= w[:,None].conj() # r <-- matmul(diag(w*), r) 39 40 p = q.shape[1] 41 return q.reshape(dj + [p]), r.reshape([p] + dk)

Listing 2.7: Implementation of the tensor QR decomposition in Python.

25 Chapter 2 Tensor Networks

1 >>> a.shape 2 (2, 3, 4, 5) 3 >>> q, r = tqr(a, [2, 1], [0, 3], unique=True) 4 >>> q.shape 5 (4, 3, 10) 6 >>> r.shape 7 (10, 2, 5) 8 >>> np.allclose(a, oe.contract(q, [2, 1, 4], r, [4, 0, 3])) 9 True

Listing 2.8: Computing the QR decomposition for the example in equation (2.21) using the function tqr from listing 2.7.

With the parameter backend one can choose whether the QR decomposition shall be performed on the CPU or on the GPU. Depending on the choice, the tensor has to be supplied as a NumPy or a CuPy array. Listing 2.8 shows how to use the tqr function to perform the QR decomposition for the example from equation (2.21) assuming that the indices of the green tensor are enumerated counter-clockwise starting from the right index. Other decompositions, such as EVDs or SVDs, can be implemented in a very similar way as the QR decomposition in listing 2.7: First one has to form the respective matrix, then the actual decomposition is performed, and the indices of the resulting matrices have to be split again to obtain the original shape of the tensor.

2.8 Dominant Eigenvectors

In some tensor network algorithms it is required to find the dominant eigenvector, i. e. the eigenvector corresponding to the largest eigenvalue in magnitude, of some tensor network, e. g.

= 휇dom × with |휇dom| maximal, (2.24) where the green tensor is the dominant eigenvector, corresponding to the eigenvalue 휇dom, of the tensor network consisting of the orange and the blue tensor. Note that whenever the subscript ‘dom’ is used throughout this thesis, the dominant eigenvalue is indicated. In principle it is possible to achieve this by contracting the corresponding tensor network and applying an eigendecomposition as shown in section 2.7,

휒 휒 휒 휒 휒 휒 2 2 푝 = = 휒 휒 . (2.25) 휒 휒 휒 휒 휒 휒

The circular matrix in the eigendecomposition is diagonal containing all the eigenvalues on its

26 2.8 Dominant Eigenvectors diagonal and the green and the blue tensor are reciprocally inverse to each other, i. e.

= and = . (2.26)

If the 푗-th eigenvalue is the dominant one, 휇dom and the dominant eigenvector from equation (2.24) can be selected as

휇dom = 푗 푗 and = 푗 . (2.27)

At this point, it should be noted that storing the contracted tensor (the red one in equation (2.25)) requires memory of size 풪 (휒 4) and computing the eigendecomposition has a runtime complexity of 풪 (휒 6) [77]. An alternative approach for computing the dominant eigenvector would be to use a Krylov subspace method, e. g. the implicitly restarted Arnoldi method [78] or the Krylov-Schur method [79]. The overall idea is to find a Krylov decomposition of order 푘 ≪ 푛,

† 퐴푉푘 = 푉푘퐵푘 + 푣푘푏푘 , (2.28)

푛×푛 푘×푘 푛×푘 푛 푘 where 퐴 ∈ ℂ , 퐵푘 ∈ ℂ , 푉푘 ∈ ℂ , 푣푘 ∈ ℂ , and 푏푘 ∈ ℂ . 퐴 is the matrix of which the dominant eigenvector shall be computed. 푉푘 is an isometric matrix consisting of the columns 푣0, …, 푣푘−1. The vectors 푣0, …, 푣푘 form an orthonormal basis of the Krylov subspace, span{푣0, 퐴푣0,…, 푘 퐴 푣0}. Note that a Krylov decomposition is fully determined by a single vector, 푣0, as an order 푘 decomposition can be extended to order 푘 + 1 by applying 퐴 to 푣푘 and reorthonormalising the Krylov subspace basis. 퐵푘 is the projection of the matrix 퐴 onto the Krylov subspace with † the residual 푣푘푏푘 . † The target is to find a Krylov decomposition with 푣푘푏푘 ≈ 0 and for which the dominant eigenvalue 휇dom of 퐴 is an eigenvalue of 퐵푘. In this case the corresponding eigenvector 푥 of 퐵푘 gives the dominant eigenvector 푉푘푥 of 퐴 can be obtained. For both, the implicitly restarted Arnoldi method and the Krylov-Schur method, this is achieved by expanding a Krylov decomposition of order 푘 to order 푘 + 푚 and then truncating it back to order 푘 again with a simultaneous transformation of the Krylov subspace basis. The purpose of this basis transformation is to shift the spectrum of 퐵푘 towards 휇dom. The main difference between these two Krylov subspace methods is the way how the transformations are constructed. The required number of applications of 퐴 strongly depends on the eigenvalue structure of 퐴. If there is a gap between the dominant and the other eigenvalues, the required number of applications of 퐴 onto a vector, 푁app, is in practice much smaller than the dimension of 퐴. 2 The runtime complexity for the example from above is therefore 풪 (푝휒 푁app). At this point, the dimensions 푝 and 휒 are in principle independent, but for the applications in the following chapters 푝 will always be much smaller than 휒. Listing 2.9 shows a function domeigs that can be used to compute the dominant eigenvector of a tensor network. It uses the SciPy binding to the ARPACK [80] implementation of the implicitly restarted Arnoldi method. As a first parameter the function requires a function f that

27 Chapter 2 Tensor Networks

1 import cupy as cp 2 import numpy as np 3 import scipy.sparse.linalg as spla 4 5 def domeigs(f, args, x0, ncv=20, backend="numpy"): 6 """ 7 Computes the dominant eigenpair (mu, x), such that 8 f(x, *args, backend=backend) = mu * x. 9 10 The numpy vector x0 is used as the first Krylov subspace basis 11 vector and no more than ncv Krylov basis vectors are used. 12 13 Supported backends are "numpy" and "cupy". 14 """ 15 16 shape = x0.shape 17 18 if backend == "numpy": 19 def matvec(x): 20 y = f(x.reshape(shape), *args, backend="numpy").ravel() 21 return y 22 elif backend == "cupy": 23 def matvec(x): 24 x = cp.asarray(x) 25 y = f(x.reshape(shape), *args, backend="cupy").ravel() 26 return y.get() 27 28 op = spla.LinearOperator( 29 matvec=matvec, 30 dtype=x0.dtype, 31 shape=[np.prod(shape)]*2 32 ) 33 mu, x = spla.eigs(op, 1, v0=x0.ravel(), which="LM", ncv=ncv) 34 return mu[0], x[:, 0].reshape(shape)

Listing 2.9: Finding dominant eigenvalue and eigenvector.

1 >>> f = oe.contract_expression( 2 >>> x.shape, (2, 3), 3 >>> a.shape, (4, 2, 0), 4 >>> b.shape, (4, 3, 1) 5 >>> ) 6 >>> mu, x = domeigs(f, (a, b), x0) 7 >>> np.allclose(f(x, a, b), mu * x) 8 True

Listing 2.10: Computing the dominant eigenpair from equation (2.24) using the function domeigs from listing 2.9.

28 2.9 Geometric Series and Linear Equations contracts the tensor network corresponding to the matrix-vector product, e. g. the network on the left-hand side of equation (2.24). This function f is called from domeigs with a tensor of the same shape as the resulting eigenvector as the first parameter. If backend="numpy" (backend="cupy") is set, this tensor is a NumPy (CuPy) array. The second parameter, args, of domeigs is used to pass additional parameters to f – usually the other tensors of the network. The paramter x0 is used as the first Krylov basis vector. The better this initial guess for x0 is, the fewer iterations are required to obtain a converged result. In practical applications, the tensors of the tensor network, of which the dominant eigenvector shall be computed, often only change slightly between iterations of an algorithm. It is then advisable to use the previous eigenvector as an initial guess for x0 in the following call of domeigs. With the parameter ncv the maximum number of Krylov basis vectors can be set. Note that no matter which backend is used, the input vector x0 and the resulting eigenvector are always passed as NumPy arrays (and are therefore stored in CPU memory) due to the usage of ARPACK. In listing 2.10 it is demonstrated, how the domeigs function can be used to find the dominant eigenvector of equation (2.24).

2.9 Geometric Series and Linear Equations

Another class of operations used in some of the tensor network algorithms in this thesis involve geometric series [74, 81], where a tensor network acts on a tensor very much like a matrix on a vector, e. g.

푘 ∞ = + + + ⋯ = ∑ , (2.29) 푘=0 ( ) where the exponent 푘 in the rightmost tensor diagram indicates 푘 applications of the tensor network within the parentheses. For a matrix 퐴 ∈ ℂ푛×푛 an equivalent task is to compute

∞ 푥 = ∑ 퐴푘푏, (2.30) 푘=0 where 푥, 푏 ∈ ℂ푛. If ‖퐴‖ < 1, i. e. all eigenvalues of 퐴 are smaller than one in magnitude, the geometric series converges and 푥 = (1 − 퐴)−1푏. To determine 푥 numerically, instead of inverting (1 − 퐴) one would solve the linear system

(1 − 퐴)푥 = 푏, (2.31) which is especially beneficial when 퐴 is sparse and procedures such as the biconjugate gradient stabilized method (BiCGSTAB) [82] or the loose generalized minimal residual method (LGMRES) [83] are used. In this sense, if the magnitude of the dominant eigenvalue of the repeatedly applied tensor network is smaller than one, the red tensor from equation (2.29) can be computed by solving

29 Chapter 2 Tensor Networks the linear system

( − ) = . (2.32)

Unfortunately, a common scenario in the following chapters will be that the repeatedly applied tensor network has a unique dominant eigenvalue equal to one, which causes the geometric series to diverge. Going back to the case of equation (2.30), one can define a matrix 퐴̃ = 퐴 − 푟ℓ푇 , where 푟 is the right dominant eigenvector, i. e. 퐴푟 = 푟, and ℓ is the left dominant eigenvector, i. e. ℓ푇 퐴 = ℓ푇 , with ℓ푇 푟 = 1. As ‖퐴‖̃ < 1, the sum over powers of 퐴̃ converges and one has ∞ ∞ ∞ −1 푥 = ∑ 퐴̃푘푏 + ∑ 푟ℓ푇 푏 = (1 − 퐴̃) 푏 + ∑ 푟ℓ푇 푏. (2.33) 푘=0 푘=0 푘=0

The potentially diverging part is now isolated in the remaining sum and for all cases considered in this thesis some trick can be played to make all terms of this sum vanish, e. g. by having the freedom to choose 푏 such that ℓ푇 푏 = 0. This idea of subtracting the projector onto the dominant eigenspace can be used to evaluate the geometric series of equation (2.29) if the dominant eigenvalue is unique and equal to one and if the diverging part can be enforced to vanish. With the dominant left and right eigenvectors

= and = , (2.34) obeying the normalisation condition

= 1, (2.35) the limit of the geometric series can be computed by solving the linear equation

[ − ( − )] = . (2.36)

Listing 2.11 provides a Python function geosum that can be used to solve these types of linear equations to compute limits of geometric series with tensor networks. For solving the linear equation the LGMRES implementation of SciPy is used, but can be easily replaced by other linear solvers, e. g. BiCGSTAB. The first paramter of geosum is a function f that applies the tensor network from the geometric series to a tensor, which is passed as a first paramter to f. If backend="numpy" (backend="cupy") is set, this tensor is passed as a NumPy (CuPy) array. The second parameter of geosum, args, is a tuple of additional parameters passed to f – usually the tensors of the tensor network to be applied. Using the parameter b, the tensor, on which the tensor network is repeatedly applied, is specified, e. g. the grey tensor

30 2.9 Geometric Series and Linear Equations

1 import cupy as cp 2 import logging 3 import numpy as np 4 import scipy.sparse.linalg as spla 5 6 def geosum(f, args, b, p0=None, x0=None, backend="numpy", 7 atol=1e-15, rtol=1e-15): 8 """ 9 Computes x from linear equation: 10 (1 - g)(x) = b if p0 is None 11 (1 - (g - r l^T))(x) = b if p0 = (r, l) 12 13 g(x) = f(x, *args, backend=backend) 14 """ 15 16 xp = cp if backend == "cupy" else np 17 shape = b.shape 18 n = np.prod(shape) 19 x0 = None if x0 is None else x0.ravel() 20 21 def g0(x): 22 y = f(x.reshape(shape), *args, backend=backend).ravel() 23 return x - y 24 25 if p0 is None: 26 g1 = g0 27 else: 28 r, l = p0 29 g1 = lambda x: g0(x) + xp.dot(x, l) * r 30 31 if backend == "cupy": 32 g = lambda x: g1(cp.asarray(x)).get() 33 else: 34 g = g1 35 36 op = spla.LinearOperator(matvec=g, dtype=b.dtype, shape=(n, n)) 37 x, info = spla.lgmres(op, b.ravel(), x0, atol=atol, tol=rtol) 38 39 if info != 0: 40 logging.warning("lgmres failed with info={}".format(info)) 41 42 return x.reshape(shape)

Listing 2.11: Computing the limit of a tensor network geometric series by solving a linear equation using LGMRES.

31 Chapter 2 Tensor Networks

1 # create tensors: 2 p, chi = 2, 16 3 a = np.random.rand(p, chi, chi) 4 b = np.random.rand(chi, chi) 5 6 # prepare contraction: 7 f = oe.contract_expression( 8 b.shape, (2, 3), a.shape, (4, 2, 0), a.shape, (4, 3, 1), 9 optimize="dp" 10 ) 11 12 # scale dominant eigenvalue to 0.9: 13 mu, x = domeigs(f, (a, a.conj()), np.random.rand(chi, chi)) 14 a /= abs(mu / 0.9)**0.5 15 16 # compute geometric series: 17 x = geosum(f, (a, a.conj()), b) 18 19 # validate result: 20 assert np.allclose(f(x, a, a.conj()) + b, x)

Listing 2.12: Computing the limit of a geometric series with a graphical representation as in equation (2.29) using the geosum function from listing 2.11. Note that the randomly generated tensor a is rescaled, such that the dominant eigenvalue of the tensor network is smaller than one in magnitude. in equation (2.29). The parameter x0 can be used to supply a starting guess for the solver of the linear equation. With the optional parameter p0=(r, l) the projector onto the dominant eigenspace can be defined, if f applies a tensor network with a unique dominant eigenvalue equal to one. Both, r and l must be a NumPy (CuPy) array if backend="numpy" (backend="cupy"). The parameters atol and rtol specify the absolute and relative tolerance, which are passed to the LGMRES solver. Listing 2.12 demonstrates how the function geosum can be used to compute the limit of a geometric series as the one in equation (2.29).

32 3 Tensor Network States

This chapter starts with formulating the challenge of ground state computations for quantum many-body systems and gives an overview of various methods that are used to address this challenge. Variational methods are illustrated as one family of such methods and tensor network states (TNSs) will be introduced as a particular example of a variational ansatz. After discussing the entanglement properties of TNSs and the potential benefit of having quantum states obeying an area law for the entanglement entropy, this chapter will conclude with a list of examples of specific instances of TNSs.

3.1 Motivation

A common, albeit often nontrivial task in condensed matter physics is to find the ground stateof a system, i. e. the state the system adopts at absolute zero temperature. The ground state is the eigenstate corresponding to the smallest eigenvalue of the stationary Schrödinger equation [2],

ℋ |휓⟩ = 퐸 |휓⟩ , (3.1) in which the Hermitian Hamiltonian matrix ℋ is dictated by the considered physical system. If the system consists of 푁 sites and the state of each isolated site can be described by an element of the 푝-dimensional Hilbert space ℂ푝, the state of the whole system is an element of 푁 the 푝푁 -dimensional Hilbert space ℍ = ℂ푝 . The most straightforward way to find the ground state of such a system is to use the so-called exact diagonalisation (ED) method [21], where one directly computes the ground state by applying the Lanczos algorithm [84]. The biggest strength of ED is its applicability to practically any lattice model one can think of in combination with truly exact results. If a model with a certain system size can be addressed with ED, the results represent the gold standard among all numerical results. However, the weak spot of ED is its limitation to small system sizes due to the inevitable exponential scaling of the Hilbert space dimension. By exploiting symmetries, e. g. lattice symmetries or the conservation of magnetisation, the Hamiltonian can be brought into block-diagonal form with a block for each symmetry sector. This drastically reduces the dimension of the matrices to which the Lanczos algorithm has to be applied and therefore helps to push system sizes as far as possible. However, the ‘exponential hard wall’ for the example 푁 −1 푧 of a spin 푆 = 1/2 system with a Hamiltonian commuting with ∑푗=0 푆푗 has been reported to be

33 Chapter 3 Tensor Network States

(a) (b)

ℍvar |휓0⟩ ℍvar |휓0⟩

ℍ ℍ

Figure 3.1: Illustrations of variational manifolds ℍvar within a Hilbert space ℍ. In the ideal case, the ground state |휓0⟩ is contained in the variational manifold, cf. subfigure (a), whereas a badly chosen variational manifold is far away from the ground state, cf. subfigure (b). at 푁 = 42 sites in 2011. Algorithmical improvements for discrete spatial symmetries enabled simulations of up to 푁 = 50 spin 푆 = 1/2-sites in 2018 [85]. Moore’s law [86, 87] may push this limit a few sites further every few years, but simply doubling the number of sites appears to be a hopeless endeavour. The family of quantum Monte Carlo (QMC) methods [22–24] is an alternative approach to the challenge of finding ground states of quantum many-body systems. The overall ideaof QMC methods such as stochastic series expansion QMC, worldline QMC, or determinantal QMC is to compute observables of interest by stochastic importance sampling over configura- tions of the system instead of directly computing and storing the ground state as a vector. This allows for the numerical effort to scale only polynomially with the number of sites, rather than exponentially. If QMC methods are applicable, they can be used to study much larger systems compared to ED. Schuler et al. [88] for example were able to study the critical transverse field Ising model in two spatial dimensions with ED up to 푁 = 40 and with QMC up to 푁 = 900. The accessibility of large systems is crucial when it comes to extrapolating finite-size results to the thermodynamic limit, 푁 → ∞. However, for frustrated and fermionic systems, statistical weights can become negative unless an adequate basis is found, which is known as the infam- ous negative sign problem [89], that causes an exponential slowing down of QMC simulations and keeps these systems out of reach. Another family of methods are the so-called variational methods. The underlying key concept is the Rayleigh-Ritz principle [90–92],

⟨휓|ℋ |휓⟩ |휓0⟩ = argmin , (3.2) |휓⟩∈ℍ ⟨휓|휓⟩ which states that the ground state |휓0⟩ is the element of the Hilbert space that minimises the energy functional on the right-hand side of equation (3.2). Instead of using the entire Hilbert space with its exponentially many dimensions as search space, it is replaced by a significantly smaller variational manifold ℍvar ⊂ ℍ, that is typically spanned by an ansatz wave function

34 3.1 Motivation

|휓0⟩ ℍvar(푟)

푟

ℍ

Figure 3.2: Illustration of how a refinement parameter 푟 can be used to tune the size of a variational manifold ℍvar(푟) to tune computational expensiveness for accuracy. Ideally the distance of the true ground state |휓0⟩ to the variational manifold decreases with increasing size of the variational manifold. depending on a number of parameters scaling at most polynomially in 푁 . The smaller the variational manifold is, the simpler it is to find the ground state candidate, i. e. the element from the manifold minimising the energy functional. One should note that in contrast to ED and QMC, variational methods are not truly exact methods. A variational method can only be exact, if the ground state of a system is contained in the variational manifold. However, it is a nontrivial task to construct a manifold that both contains the ground state and is small at the same time, which is illustrated in figure 3.1. In practice, good variational manifolds yield ground state candidates having the same symmetries and physical properties as the true ground state and are quantitatively as close as possible. Often it is desirable to have a refinement parameter that can be used to tune the size of the variational manifold and therefore trade computational expensiveness for accuracy as illustrated in figure 3.2. Examples for variational manifolds are Gutzwiller wave functions [93], Laughlin wave functions [19], correlator product states [94–97], or neural network states [98], just to name a few. Another family of variational ansatz wave functions are the so-called tensor network states (TNSs) [99–101], which are the main focus of this thesis. The remaining part of this chapter gives a general introduction to TNSs and the two following chapters focus on two particular instances of TNSs. In addition to a variational manifold, a fully fledged variational method requires an optimisation procedure to find the ground state candidate within the manifold. An optimisation procedure can be a purely mathematical recipe, e. g. a conjugate gradient or a quasi-Newton method [102], can be inspired by concepts from physics, e. g. imaginary time evolutions [32, 34, 103], or can be a stochastic procedure, e. g. differential evolution104 [ ] or basin-hopping [105]. Note, that this list highlights a few examples, but is far from begin exhaustive. In gen-

35 Chapter 3 Tensor Network States eral, optimising the energy is nontrivial due to the sheer amount of parameters a variational manifold often brings along.

3.2 The Tensor Network State Ansatz

Considering a quantum many-body system with 푁 sites and a local 푝-dimensional Hilbert space, any state of the system can be written as

휎0 휎1 ⋯ 휎푁 −1 |휓⟩ = ∑ |휎0, 휎1, …⟩ , (3.3) 휎0,휎1,… where for each site 푗 ∈ {0, … , 푁 − 1} the local basis states are enumerated with 휎푗 ∈ {0, … 푝 − 1}. The wave function coefficient in its most general form can be seen as ahugerank-푁 tensor as illustrated in equation (3.3), consisting of 푝푁 numbers. A TNS [99–101] is a variational ansatz for such a wave function, where a tensor network is used as an ansatz for the wave function coefficient, e. g.

휎1

휎0 |휓⟩ = ∑ ⋯ |휎0, 휎1, …⟩ . (3.4) 휎0,휎1,…

The open indices pointing up are called physical indices and all other indices are called virtual indices or sometimes just bonds. Their corresponding dimensions are commonly called physical dimensions and bond dimensions. If the local Hilbert spaces are all 푝-dimensional as stated above, all physical dimensions are equal to 푝, but in principle one could have a different physical dimension for each site. Note that the red tensor in equation (3.4) does not have a physical index intentionally, as it is perfectly valid to have tensors only connecting other tensors without carrying physical indices. The number of parameters of a TNS is given by the number of tensors used in the ansatz and the chosen bond dimensions. Each of these bond dimensions can be chosen as an arbitrary nonnegative integer and therefore they act as refinement parameters for a TNS as it should become apparent in the following. If all bond dimensions are chosen to be one, a TNS boils down to a product state independent of the connectivity of the tensors. A TNS provides direct access to the wave function itself and therefore allows computations of e. g. observable expectation values, correlation functions, or entanglement entropies. For the TNS from equation (3.4) the expectation value of a single-site observable for example can

36 3.3 Entanglement Entropy

be computed by contracting the tensor network

∗

∗ ∗

⟨휓|퐴푗|휓⟩ = , (3.5)

where the observable given by the lime-coloured matrix at the vertical centre is evaluated at the site to which the physical index of the blue tensor belongs. The tensors in the lower half are associated to |휓⟩, whereas the complex conjugate tensors in the upper half originate from ⟨휓|. Note that the expression in equation (3.5) only gives the correct expectation value, if the state is normalised, i. e. ⟨휓|휓⟩ = 1. Otherwise one has to divide the right-hand side by ⟨휓|휓⟩, which can be obtained by contracting the same tensor network but leaving out the lime-coloured matrix and connecting the blue tensors directly. The TNS ansatz is not only used as an ansatz for finite systems as shown by the example in this section, but can be used as an ansatz for systems in the thermodynamic limit as well [34, 103]. For infinite-size systems only a unit cell of tensors is defined and the actual tensor network consists of infinitely many repetitions of this unit cell, connected by virtual indices.

3.3 Entanglement Entropy

In this section it will be shown that for a TNS the entanglement entropy 푆퐴∶퐵 for any bipartition of the system into subsystems 퐴 and 퐵 is limited,

푆퐴∶퐵 ≤ ∑ log(휒푗), (3.6) 푗∈풞

where 풞 is the set of bonds – with corresponding dimensions 휒푗 – connecting subsystems 퐴 and 퐵 and simultaneously minimising the right-hand side of equation (3.6). From this it follows that a TNS can be interpreted as parametrisation of the low-entanglement corner of the Hilbert space.

Considering a general many-body state |휓⟩ and a bipartition into the first 푁퐴 sites and the

37 Chapter 3 Tensor Network States remaining 푁퐵 sites, applying a tensor-SVD to the wave function coefficient gives

휎0 휎1 ⋯ 휎푁 −1 |휓⟩ = ∑ |휎1, 휎2, …⟩ (3.7) 휎1,휎2,…

휎1 ⋯ 휎푁퐴 휎푁퐴+1 ⋯ 휎푁

= ∑ ∗ |휎1, 휎2, …⟩ . (3.8) 휎 ,휎 ,… 1 2 휁

From the isometric tensors two sets of basis states for the subsystems can be constructed,

휎1 ⋯ 휎푁퐴 (퐴) |휙푗 ⟩ = ∑ |휎1, … , 휎푁퐴 ⟩ (3.9) 휎 ,…,휎 1 푁퐴 푗

휎푁퐴+1 ⋯ 휎푁 |휙(퐵)⟩ = ∑ |휎 , … , 휎 ⟩ , (3.10) 푗 ∗ 푁퐴+1 푁 휎 ,…,휎 푁퐴+1 푁 푗

(퐴) (퐴) (퐵) (퐵) which are orthonormal, i. e. ⟨휙푗 |휙푘 ⟩ = ⟨휙푗 |휙푘 ⟩ = 훿푗푘. Inserting them into equation (3.8) gives the Schmidt decomposition [106]

(퐴) (퐵) |휓⟩ = ∑ 휁푗 |휙푗 ⟩ |휙푗 ⟩ , (3.11) 푗 from which the entanglement entropy for the chosen bipartition can be computed as

2 2 푆퐴∶퐵 = − ∑ 휁푗 log (휁푗 ) . (3.12) 푗

2 The normalisation of a wave function implies ∑푗 휁푗 = 1 and the entanglement entropy becomes maximal if all 휁푗 are equal. Therefore, if the number of nonzero singular values is 푟, the entanglement entropy is limited by

푆퐴∶퐵 ≤ log(푟). (3.13)

Going back to the example of the TNS from equation (3.4) and assuming a bipartition into the three sites on the left and the four sites on the right, the corresponding tensors canbe

38 3.3 Entanglement Entropy contracted,

휎2

휎1 |휓⟩ = ∑ ⋯ |휎1, 휎2, …⟩ (3.14) 휎1,휎2,…

휎1 휎2 ⋯

= ∑ |휎1, 휎2, …⟩ , (3.15) 휎1,휎2,… ∏ 휒푗 푗∈풞 where 풞 is the set of virtual indices, each of them having a bond dimension of 휒푗, connecting the subsystems such that ∏푗∈풞 휒푗 is minimal. At this point it is implicitly assumed that the dimension of the left index of the red tensor is smaller than the product of the dimensions ofthe other two indices. From this contraction it becomes apparent that the number of nonzero singu- 푁퐴 푁퐵 lar values in a tensor-SVD as illustrated in equation (3.8) is limited by min{푝 , 푝 , ∏푗∈풞 휒푗}, i. e. by each of the effective matrix dimensions, as the number of nonzero singular values ofa product of an 푚 × 푘 matrix with a 푘 × 푛 matrix is limited by min{푚, 푘, 푛} [107]. These limits for the number of nonzero singular values in combination with the limit from equation (3.13) yield 푆 ≤ 푁퐴 log(푝), 푆 ≤ 푁퐵 log(푝), and the most important limit 푆 ≤ log(∏푗∈풞 휒푗) = ∑푗∈풞 log(휒푗). The bond dimensions of a TNS are therefore systematically limiting the amount of entanglement the state can contain. In the edge case of each bond dimension equal to one, a TNS is nothing but a product state, i. e. a state without any entanglement at all. Increasing the bond dimensions of a TNS yields a larger variational manifold, which allows the TNS to account for more entanglement. A TNS can therefore be seen as an efficient parametrisation of the low-entanglement corner of the Hilbert space with the bond dimensions as refinement parameters. A state is said to obey an area law for the entanglement entropy if for any bipartition into subsystems 퐴 and 퐵, the entanglement entropy is limited by

푆퐴∶퐵 ∈ 풪 (|휕퐴|) , (3.16) where 휕퐴 is a real space surface enclosing the sites of 퐴, but none of 퐵 [37]. For example, if 퐴 is a disk in a two-dimensional system, the entanglement entropy between this disk and the remaining of the system never grows faster than the circumference of the disk. A TNS can be constructed to obey such an area law if the virtual indices are chosen to connect the tensors accordingly. This will be illustrated in the following section on the basis of prominent examples of TNSs. But why are these area law states interesting at all? For a Hamiltonian with local interactions and an energy gap between the ground state and excited states, the entanglement entropy of the ground state is commonly obeying an area law. For systems in one spatial dimension this connection can be proven rigorously [108] and for higher-dimensional systems there is strong

39 Chapter 3 Tensor Network States evidence for it to hold [37]. However, almost all quantum states from a Hilbert space are so- called volume law states violating this area law. Therefore, if one wants to find the ground state of a gapped Hamiltonian with local interactions, parametrising the low-entanglement corner of a Hilbert space using TNSs obeying an area law seems to be a favourable approach.

3.4 Examples

Undoubtedly the most prominent example of a TNS is the so-called matrix product state (MPSs) [109], also called tensor train decomposition in the mathematical literature [110],

휎1 휎2 ⋯ |휓⟩ = ∑ 푝 |휎 , 휎 , …⟩ , (3.17) 휒 1 2 휎1,휎2,… intended for studying one-dimensional systems. The tensors in the MPS ansatz are connected by virtual indices with a bond dimension of 휒 and hence the MPS consists of 풪 (푁 푝휒 2) parameters. For a bipartition into a block of length 퐿 and the rest of the system, one or two bonds need to be cut, depending on whether the first or last site of the system is part of the block, which gives an upper bound for the entanglement entropy of 푆(퐿) ≤ log(휒) or 푆(퐿) ≤ 2 log(휒) respectively. Therefore, an MPS is always obeying an area law for the entanglement entropy. Chapter 4 is entirely dedicated to MPSs, their properties and computational techniques. Another TNS ansatz that can be used for simulating one-dimensional systems is the tree tensor network (TTN) [111],

휎1 휎2 ⋯ 푝

|휓⟩ = ∑ 휒 |휎1, 휎2, …⟩ . (3.18) 휎1,휎2,… 휒

This TTN consists of 풪 (푁 푝2휒 + 푁 휒 3) parameters if the bond dimension 휒 is chosen to be the same for all vertical levels. The entanglement entropy between a consecutive block of 퐿 sites and the rest of the system is limited by 푆(퐿) ≤ 2 log(휒) and hence a TTN obeys an area law for the entanglement entropy. TTNs can in principle also be used for higher-dimensional systems [112]. However, if a TTN should be an area law state for a higher-dimensional system, the bond dimensions need to grow rapidly in the vertical direction of the tree, e. g. for a two-dimensional 퐿 × 퐿 system the bond dimensions are required to scale exponentially in 퐿. A particular field of applications for TTNs are quantum chemistry problems, as trees are a natural way to describe the geometry of many molecules [113–116]. The multi-scale entanglement renormalisation ansatz (MERA) [117–120] is a further instance

40 3.4 Examples of a TNS ansatz,

휎1 휎2 ⋯ 푝 푝

|휓⟩ = ∑ 푝 |휎1, 휎2, …⟩ . (3.19) 휎1,휎2,… 푝

푝

It aims at simulating critical one-dimensional systems. The green unitaries are called disentanglers, which should remove local entanglement between two neighbours, and in a next step the orange isometries perform a renormalisation step that halves the number of sites. Two layers of this kind are applied successively until only two sites are left, which are connected by a terminal tensor. If the disentanglers and renormalisation tensors are the same in each layer, the MERA is commonly called a scale-invariant MERA. A binary MERA as shown in equation (3.19) is defined by 풪 (푁 푝4) parameters. The distinguishing feature of MERA is their ability to account for an amount of entanglement beyond that of area law states. To separate a block of 퐿 consecutive sites from the rest of the tensor network, a number of bonds scaling logarithmically in 퐿 needs to be cut and therefore 푆(퐿) ∈ 풪 (log(퐿)) is possible, which is the entanglement scaling of one-dimensional systems at quantum critical points. The MERA ansatz can in principle be generalised for higher dimensions, but it turns out that from a practical and numerical point of view this is infeasible. Projected entangled-pair states (PEPSs) [38] are a straightforward generalisation of MPSs for studying two-dimensional systems, for which the TNS ansatz can be written as

⋯

휎2 |휓⟩ = ∑ |휎 , 휎 , …⟩ . (3.20) 휎1 1 2 휎1,휎2,… 퐷

Note that although the round tensors seem to indicate a spatial symmetry here, this is not necessarily the case. As the visual representation of tensor symmetries in three-dimensional tensor network diagrams is hard to grasp, this concept is discarded at this point. A PEPS can be parametrised by 풪 (푁 푝퐷4) parameters and to divide the system into two subsystems 퐴 and 퐵, one has to cut a number of bonds proportional to the circumference of 퐴. Hence, a PEPS obeys an area law for the entanglement entropy for two-dimensional systems. Chapter 5 provides a more profound treatment of PEPSs and associated algorithms. The last example highlighted in this section does not introduce a new TNS, but aims at illustrating that the tensor network connectivity is not rigidly coupled to the spatial geometry.

41 Chapter 3 Tensor Network States

Although MPSs are a natural choice for studying one-dimensional systems, they can be used – at least in principle – for studying arbitrary systems if an appropriate spatial alignment for the tensors is chosen. In a common application, an MPS is used to address systems defined on a two-dimensional lattice [109], as indicated by

⋯

휎2 |휓⟩ = ∑ 휎1 |휎1, 휎2, …⟩ . (3.21) 휎1,휎2,…

휒

Note that equations (3.17) and (3.21) are identical as the connectivity of the tensor network is exactly the same. However, the aim of equation (3.21) is to illustrate the spatial alignment of the sites on a two-dimensional square lattice. In contrast to PEPSs, MPSs are not able to account for area law states if the bond dimension 휒 is independent of the system size. For a system consisting of 퐿 × 퐿 sites, bond dimensions up to 휒 = 퐷퐿 are required to account for the same amount of entanglement as a PEPS with bond dimension 퐷 can account for.

42 4 Matrix Product States

It is now time to discuss the first instance of a tensor network state in more detail: the matrix product state (MPS). This chapter follows the great lecture notes of Vanderstraeten, Haegeman, and Verstraete [74], but in contrast presents algorithms specifically for the case of multi-site unit cells. Beside a general introduction of MPSs including the canonical form and basic techniques such as the computation of local observables and correlation lengths, the highlight of this chapter are algorithms to find an MPS with maximal overlap to another reference MPSand to find a fixed point MPS ofa matrix product operators (MPO). The basic idea of this chapter is to focus on techniques that are required in the following chapter as well for addressing two- dimensional quantum systems.

4.1 Historical Context

The history of MPSs is quite diffuse. The ansatz that is nowadays called MPS has been used for decades with neither the name nor the attention it has today and has its early roots going back to Baxter [121]. The work of Affleck, Kennedy, Lieb, and Tasaki [122, 123] led to the introduction of MPSs under the name of finitely correlated states [124, 125]. In the following, they were used for analytical variational studies [126–131] and in the course of these studies the name MPS apparently emerged. Decoupled from this early history of MPSs, White developed the famous density matrix renormalisation group (DMRG) [25, 26], which quickly became an extremely powerful numerical method for addressing one-dimensional quantum systems [27, 28]. It was then realised that the results from both infinite-system and finite-system DMRG are actually MPSs [29, 30], which can be seen as the starting point of the modern MPS research. In the subsequent years, the development of MPSs was mainly driven by insights from the field of quantum information [31–36], which finally led to MPSs, as they are known and used today. For more details about the history of MPSs, the interested reader is referred to reference [27]

4.2 The Ansatz

For a system consisting of 푁 sites, where the state of each isolated site can be described by a state from a 푝-dimensional Hilbert space, the MPS ansatz for open boundary conditions is

43 Chapter 4 Matrix Product States

휎0 휎1 ⋯ 푝 |휓⟩ = ∑ 휒0 휒1 |휎0, 휎1, …⟩ , (4.1) 휎 ,휎 ,… 0 1 0 1 where 휒푗 is the bond dimension connecting the tensors of sites 푗 and 푗 + 1. Note that although all tensors are are of the same colour, there can be a different tensor for each site. The site of a tensor is indicated by the index underneath the tensor. Remembering the discussion of entanglement for general TNS in section 3.3, there are natural upper bounds for the bond 2 2 dimensions: 휒0 ≤ 푝, 휒1 ≤ 푝 , …, 휒푁 −3 ≤ 푝 , 휒푁 −2 ≤ 푝. Therefore, any state from the Hilbert space can be written as an MPS if the bond dimensions are sufficiently large. Usually, these upper bounds only play a role at the borders of the MPS. Due to the possible exponential growth of the bonds with distance from the boundaries, the inner bonds are truncated by 휒푗. A slightly different ansatz incorporating periodic boundary conditions for an MPS readsas

휎0 휎1 ⋯ 푝 |휓⟩ = ∑ 휒0 휒1 |휎0, 휎1, …⟩ . (4.2) 휎 ,휎 ,… 0 1 0 1 In contrast to the MPS for open boundary conditions, the MPS for periodic boundary conditions contains a loop in its tensor network. Although this seems to be very unimpressive at the moment, it is of great conceptual importance as discussed later in the context of the canonical form. Another implication of this observation is that there is only one theoretical upper bound 푁 /2 for all the bond dimensions in this case: 휒푗 ≤ 푝 . For both options of boundary conditions, 2 the MPS can be described by 풪 (푁 푝휒 ) parameters, where 휒 = max푗 휒푗. Another variant of an MPS ansatz is the infinite MPS (iMPS),

휎0 휎1 ⋯ |휓⟩ = ∑ 푝 |휎0, 휎1, …⟩ , (4.3) 휒0 휒1 휎0,휎1,… which is constructed by infinitely many repetitions of a unit cell consisting of 푁uc tensors. As indicated by the colours of the tensors, 푁uc = 3 was chosen for the example in equation (4.3). Although indicating the unit cell size via colours is very illustrative, in most cases the iMPS representation

휎0 휎1 푝 |휓⟩ = ∑ 휒0 휒1 |휎0, 휎1, …⟩ (4.4) 휎0,휎1,… 0 1 푁uc − 1 0 1 2 will be used. In the case of an iMPS 풪 (푁uc푝휒 ) parameters are necessary to describe the state. In all three cases of an MPS, to separate a block of 퐿 consecutive tensors from the rest, at most two bonds need to be cut, which limits the entanglement entropy for this bipartition by 푆(퐿) ≤ 2 log(휒). (4.5) Therefore, if an MPS is used as an ansatz for a one-dimensional system, the state is obeying an area law for the entanglement entropy by construction. Note that the remaining of this chapter solely focuses on the case of iMPSs.

44 4.3 Computing Observables

4.3 Computing Observables

As a first warm-up exercise one of the most basic operations for an iMPS – the evaluationof a single-site observable – shall be considered. Having the general case of an arbitrary TNS in

equation (3.5) in mind, a single-site expectation value for an iMPS can be computed with

∗ ∗ ∗ ∗ ∗ ∗ ∗

⟨휓|퐴푗|휓⟩ = , (4.6)

where the 3-site unit cell from equation (4.3) is assumed and the norm of the state reads as

∗ ∗ ∗ ∗ ∗ ∗ ∗

⟨휓|휓⟩ = . (4.7)

To evaluate these expressions, it is necessary to compute the fixed points of the transfer

matrices first,

∗ ∗ ∗ ∗ ∗ ∗

= 휇 × and = 휇 × , (4.8) where 휇 is the dominant eigenvalue. Only the case of a unique dominant eigenvalue 휇 is considered, in which the iMPS is called injective [35] and boundary effects have no physical meaning. Note that almost all states in an iMPS manifold are injective. In the case of an injective iMPS 휇 is a real positive number. Therefore, the state can be normalised by rescaling an iMPS tensor,

1 ← × , (4.9) √휇 such that the dominant eigenvalue of the transfer matrix becomes one, and by normalising the overlap of the dominant left and right eigenvector,

← . (4.10) /

Using the normalised tensors, the expectation value can finally be written as ∗

⟨휓|퐴푗|휓⟩ ⟨퐴 ⟩ = = . (4.11) 푗 ⟨휓|휓⟩

45 Chapter 4 Matrix Product States

4.4 Canonical Form

A natural question arising when dealing with MPSs is how one can determine whether two MPSs, defined via their tensors, are equal. Unfortunately, simply comparing the tensor elements does not answer this question, as the same MPS can be defined via infinitely many different sets of tensors. One can apply a gauge transformation, defined by 퐺 ∈ GL (휒), on two neighbouring tensors,

퐺 퐺−1 ⟶ , (4.12) 푗 푗 + 1 푗 푗 + 1 where 퐺 and 퐺−1 are absorbed into the two MPS tensors. This gauge transformation alters the two MPS tensors, but the state itself is unchanged. Luckily, there are ways to fix the gauge of an MPS. The most straightforward and atthesame time numerically most stable way of achieving this is to iteratively perform unique tensor-QR decompositions1 until a fixed point is reached:

QR+ ⟶

QR+ ⟶ ⋯

QR+ ⟶ . (4.13)

For this example a single-site translationally invariant MPS defined by the grey rank-3 tensor is assumed as initial state. Somewhere an identity matrix is inserted, which is indicated by the grey matrix in the first line. This matrix is contracted with the MPS tensor to its right andthen a unique tensor-QR decomposition is computed. As a consequence, the matrix moves one step to the right. This operation of contracting the matrix with the MPS tensor to the right and then computing a unique tensor-QR decomposition is repeated until the matrix is converged, which

1As explained in section 2.7, a QR decomposition can be enforced to be unique, if the diagonal elements of the upper triangular matrix are chosen to be positive real numbers.

46 4.4 Canonical Form is indicated by the transition of the colour from grey to purple. It should be noted that the initial choice of the grey matrix does not affect the result as only injective MPSs are considered. As soon as the matrix is converged, each tensor-QR decomposition yields the same orange isometric tensor, which is therefore converged as well. From this observation the fixed point equation

= (4.14) can be derived. Note that the grey and the orange MPS tensor both define exactly the same physical state, having exactly the same bond dimension 휒, and the two MPSs are related via a gauge transformation. The MPS given by the orange isometric tensors is called an MPS in left canonical form due to the property

∗ = . (4.15)

The presented approach to bring an MPS into (left-)canonical form is applicable to iMPSs and MPSs with open boundary conditions. However, it cannot be applied to MPSs with periodic boundary conditions, due to the loop in the tensor network. Starting from the MPS in left canonical form, the same procedure can be applied a second time, but now pulling the matrix from right to left, which leads to another fixed point equation,

= . (4.16)

Again, the blue MPS tensor defines the same physical state as the orange tensor and in analogy to above, the MPS given by the blue isometric tensor is called an MPS in right canonical form. Using the fixed point equation (4.16), the green matrix can be moved freely through the MPS consisting of both orange and blue isometric tensors,

, (4.17) which is called an MPS in mixed canonical form. Note that the norm of the state only depends

on the green bond matrix, ∗ ⟨휓|휓⟩ = , (4.18) which is why in the following it will always be assumed that the matrix has been rescaled properly such that ⟨휓|휓⟩ = 1. In some cases it is convenient to have a central MPS tensor,

= , (4.19)

47 Chapter 4 Matrix Product States

Input: MPS ∀푗 ∈ {0, … , 푁uc − 1}, tolerance 휏 > 0 푗 Output: MPS in mixed-canonical form

, , , ∀푗 ∈ {0, … , 푁uc − 1} 푗 푗 푗 푗 1 for 푗 ∈ {0, … , 푁uc − 1} do

2 ⟵

3 repeat 4 for 푗 ∈ {0, … , 푁uc − 1} do QR+ 5 ⟶ 푗 푗 + 1 푗 + 1

1 ∗ − 2 6 ⟵ ( ) ×

7 휀푗 ⟵ ‖ − ‖ 푗 + 1 max

8 ⟵

9 until max푗 휀푗 < 휏; 10 repeat 11 for 푗 ∈ {0, … , 푁uc − 1} do QR+ 12 ⟶ 푗 + 1푗 + 1 푗 + 1 − 1 2 13 ⟵ ( ) ×

14 휀푗 ⟵ ‖ − ‖ max

15 ⟵

16 until max푗 휀푗 < 휏; 17 for 푗 ∈ {0, … , 푁uc − 1} do

18 ⟵ 푗 푗

Algorithm 4.1: Bringing an MPS with an 푁uc-site unit cell into canonical form.

48 4.4 Canonical Form

Input: MPS in mixed-canonical form

, , , ∀푗 ∈ {0, … , 푁uc − 1} 푗 푗 푗 푗 Output: MPS in mixed-canonical form with diagonal bond matrices

, , , ∀푗 ∈ {0, … , 푁uc − 1} 푗

1 for 푗 ∈ {0, … , 푁uc − 1} do SVD 2 ⟶ ∗ 푗

3 ⟵ , ⟵ ∗ 푗 푗 푗 + 1 푗 + 1

4 ⟵ , ⟵ ∗ 푗 푗 + 1 푗 + 1

5 ⟵ 푗 푗 Algorithm 4.2: Fixing the unitary gauge freedom of an MPS in mixed canonical form by enforcing the bond matrices to be diagonal. instead of having the green bond matrix, e. g. when it comes to computing observables. So far only the case of a single-site unit cell has been considered. However, the procedure of bringing an MPS into canonical form works for an MPS with nontrivial unit cell as well, as shown in algorithm 4.1. The resulting tensors are then related via

= = . (4.20) 푗 푗 푗 푗 − 1 푗 Note that there is still remaining gauge freedom of applying unitary transformations on the bonds, e. g. two neighbouring left-canonical tensors can be transformed as

∗ ⟶ (4.21) 푗 푗 + 1 푗 푗 + 1 without altering the state or loosing the left canonical gauge, as the resulting tensors are still isometric. However, this freedom can be fixed by enforcing the bond matrices to be diagonal. This can be achieved by computing SVDs of the bond matrices as shown in algorithm 4.2. Comparing equations (4.17) and (3.8) reveals that the singular values 휁푘 of the bond matrix between sites 푗 and 푗 + 1 determine the entanglement entropy,

휒푗 −1 2 2 푆푗 = − ∑ 휁푘 log (휁푘 ) , (4.22) 푘=0

49 Chapter 4 Matrix Product States for a bipartition into two half-infinite chains separated between sites 푗 and 푗 + 1.

4.5 Computing Observables Revisited

Thanks to the mixed canonical form of an MPS, computing expectation values of observables becomes especially simple, e. g. the expectation value of a single-site observable can be computed as

∗ ∗

⟨퐴푗⟩ = = (4.23)

⋯ 푗 − 1 푗 푗 + 1 ⋯ 푗 due to the vanishing isometric MPS tensors to the left and the right. In the same way,the expectation value of a nearest-neighbour observable can be computed as

∗ ∗ ∗ ∗

⟨퐴푗⟩ = = . (4.24)

푗 푗 + 1 푗 푗 + 1

Here it should become apparent that the location of the green central tensor can be chosen freely by moving it using the relations from equation (4.20). A bit more involved is the computation of infinite sums of two-site observables decaying exponentially over distance,

∗ ∗ ∗ ∗ ∞ ∞ 푘 푘 ⟨∑ 훽 퐴푗퐵푗+푘+1⟩ = ∑ 훽 , (4.25) 푘=0 푘=0 푗 푗 + 1 푗 + 푘 푗 + 푘 + 1 with |훽| < 1. Note that the site of the left observable is fixed, whereas the site of the rightob- servable moves through the unit cell. In a first step, the infinite sum containing all observables on a half-infinite line to the left,

푚 ∗ ∗ ∗ ∞ = ∑ 훽푚푁uc ( ) , (4.26) 푚=0

푗 푗 + 1 푗 + 퐿uc

50 4.6 Correlation Length shall be computed. This can be achieved by solving the linear equation

⎛ ∗ ∗ ⎞ ∗ ⎜ 푁 ⎟ ⎜ − 훽 uc × ⎟ = , (4.27) ⎜ ⎟

⎝ 푗 + 1 푗 + 퐿uc ⎠ 푗 as described in section 2.9. From this intermediate result, the expectation value can be computed as

∗ ∗ ∗ ∞ 푁uc−1 푘 ⟨∑ 훽 퐴푗퐵푗+푘+1⟩ = ∑ . (4.28) 푘=0 푘=0 푗 + 1 푗 + 푘 푗 + 푘 + 1 Note that after evaluating equation4.25 ( ) for some 푗, it can be evaluated for 푗 + 1 without the need of solving the linear equation again. One simply has to update the red tensor containing observables for a half-infinite line,

∗ ∗

+ 훽 × → , (4.29)

푗 + 1 푗 + 1 and then proceed with reevaluating equation (4.28). Unlike exponentially decaying observables, long-range observables cannot be evaluated directly. However, a common technique is to approximate a long-range function, e. g. 푓 (푟) = 1 , 푟3 by a sum of 푛 exponentials, 푛−1 푟 푓 (푟) = ∑ 훼푗훽푗 + 휀(푟), (4.30) 푗=0 such that the numerical error |휀(푟)| is minimised on an interval 푟 ∈ [1, 푟max] [132, 133]. Using this approximation, the previously presented technique for evaluating exponentially decaying observables can be used to evaluate each term of equation (4.30) individually.

4.6 Correlation Length

Considering an MPS with a single-site unit cell, an arbitrary two-point correlation function can be computed as 푟 − 1 ∗ ∗ ∗

⟨퐴푗퐵푗+푟 ⟩ = ( ) . (4.31)

51 Chapter 4 Matrix Product States

If the right column of tensors is considered as a vector 푣 on which the transfer matrix 푇 is applied 푟 − 1 times and the left column as a vector 푢, the correlation function can be written as

푇 푟−1 ⟨퐴푗퐵푗+푟 ⟩ = 푢 푇 푣. (4.32)

Denoting the projection of 푢 (푣) onto the eigenspace of 푇 corresponding to the eigenvalue 휇푗 with 푢푗 (푣푗) enables to rewrite the correlation function as

푟→∞ 푇 푟−1 푇 푟 ⟨퐴푗퐵푗+푟 ⟩ ⟶ 푢0 푣0 + 휇1 푢1 푣1 + 풪 (휇2) (4.33) in the limit of large 푟 with |휇0| > |휇1| > |휇2| > … Note that the dominant eigenvalue 휇0 = 1 as the MPS is properly normalised in the canonical form. As the first term is independent of 푟 and the second term is decaying exponentially with increasing 푟, the correlation function can be identified with

푟→∞ −푟/휉 ⟨퐴푗퐵푗+푟 ⟩ ⟶ ⟨퐴푗⟩⟨퐵푗+푟 ⟩ + const. × 푒 , (4.34) where the first part is the constant disconnected part of the correlation function and thecor- relation length can be determined to be 1 휉 = − . (4.35) log |휇1|

In the case of a nontrivial unit cell, i. e. 푁uc > 1, a transfer matrix consisting of 푁uc columns of tensors has to be considered for computing the subdominant eigenvalue 휇1 and consequently the correlation length is computed as 푁 휉 = − uc . (4.36) log |휇1| Therefore, no matter how large the bond dimension 휒 is chosen, correlation functions always decay exponentially and the correlation length of an MPS is always finite. When studying critical systems, for which 휉 = ∞, MPS ground state candidates introduce an effective length scale 휉(휒). For conformally invariant critical points with a central charge 푐 it has been shown within a framework called finite-correlation length scaling or finite entanglement scaling [134, 135] that this finite-휒 correlation length scales as 6 휉(휒) ∼ 휒 휅 with 휅 = . (4.37) 푐 ( 12 + 1) √ 푐

Further, for a conformally invariant critical system, this finite correlation length 휉(휒) can be used to extrapolate observables to the limit 휉 → ∞, e. g. the ground state energy density of a one-dimensional system

퐴 푒(휉(휒)) = 푒(∞) + + … (4.38) 휉 2(휒)

52 4.7 Matrix Product Operators

Even in the case of a non-critical system, much larger bond dimensions are required to converge the correlation length in contrast to other observables. Especially, close to a critical point it is practically impossible to converge the correlation length as a function of the bond dimension. However, the second gap of the transfer matrix spectrum can be used to obtain the correct correlation length in the limit 휒 → ∞ [136],

1 1 휇 (휒) = + const. × | 1 | . (4.39) 휉(휒) 휉(∞) 휇2(휒)

4.7 Matrix Product Operators

The concept of MPSs for states of a Hilbert space can be generalised to operators on this Hilbert space – so-called matrix product operators (MPOs),

휎0 휎1 ⋯ 푝 휂0 휂1 ′ ′ 푊 = ∑ ∑ |휎0, 휎1, …⟩ ⟨휎0, 휎1, …| . (4.40) ′ ′ 푝 휎0,휎1,… 휎0,휎1,… ′ ′ 휎0 휎1 ⋯ Initially, these MPOs have been introduced to represent density operators for computing quantum states at finite temperature137 [ ]. In analogy to MPSs, the dimensions 휂0, 휂1,… of the indices connecting the MPO tensors are called bond dimensions. In general, applying an MPO with bond dimension 휂 to an MPS with bond dimension 휒 yields a new MPS with bond dimension 휒휂,

휂 = 휒휂 . (4.41) 휒

An operator commonly represented by an MPO is the Hamiltonian. For example, considering a Hamiltonian with nearest-neighbour and on-site interactions for a one-dimensional quantum system,

ℋ = ∑ 퐴푗퐵푗+1 + ∑ 퐶푗, (4.42) 푗 푗 a corresponding MPO can be found with a respective MPO tensor for site 푗,

ퟙ푗 0 0 푙 푘 = (퐴푗 0 0 ) . (4.43) 퐶푗 퐵푗 ퟙ푗 푘푙 If the Hamiltonian is modified, such that the two-site interaction is no longer restricted to nearest neighbours, but instead is exponentially decaying over distance,

푘−푗−1 ℋ = ∑ 훽 퐴푗퐵푘 + ∑ 퐶푗, (4.44) 푗<푘 푗

53 Chapter 4 Matrix Product States with |훽| < 1, the corresponding MPO tensor only needs to be slightly altered,

ퟙ푗 0 0 푙 푘 = (퐴푗 훽 0 ) . (4.45) 퐶푗 퐵푗 ퟙ푗 푘푙 Further details on the construction of MPOs for Hamiltonians as illustrated above can be found in references [109, 138]. For even more complex Hamiltonians a more sophisticated approach based on finite automata can be used to construct the corresponding139 MPOs[ , 140].

4.8 Finding MPS with Maximal Overlap

A common task in the context of MPS algorithms is the truncation of bond dimensions, i. e. maximising the overlap between a reference MPS with bond dimension 휒 and an MPS with bond dimension 휒 ′ < 휒,

휒 휒 ′ ≈ . (4.46)

This can be achieved by using an algorithm based on tangent space projections as introduced in reference [74], which is summarised in algorithm 4.3 for MPSs with a single-site unit cell. Starting from a uniform MPS with bond dimension 휒, the MPS maximising the overlap with bond dimension 휒 ′ is computed iteratively. The resulting MPS is obtained in mixed-canonical form. The dominant eigenvectors computed in line 3 are used together with the tensor of the reference MPS to compute updated versions of the central MPS tensor and the MPS bond matrix in line 4. These two tensors are normalised in line 5 by dividing the tensors by the square root of the respective norm expressions. In lines 6–8, updated versions of the left- and right-canonical tensors of the resulting MPS are computed based on the new central tensor and the bond matrix using unique tensor QR decompositions. Finally, the error of the canonical fixed point equation (4.16) is computed in line 9. This scheme is iterated until the error is below the tolerance 휏. Note that the MPS tensors are independently initialised with random numbers in line 1, i. e. the two tensors neither give rise to the same MPS nor are they isometric tensors. However, as the algorithm converges, the resulting tensors consistently represent the MPS in mixed-canonical form. Once the algorithm is converged, the two dominant eigenvalues 휇dom in line 3 are equal for both equations and are equal to the overlap per site between the reference MPS and the resulting MPS. Algorithm 4.3 can be generalised to nontrivial unit cells with 푁uc sites. There are two approaches, called the sequential and the parallel approach [141], which are demonstrated in algorithms 4.4 and 4.5, respectively. In the sequential approach, only the tensors occurring in the fixed point equation (4.20) for exactly one 푗 are updated at once. This requires computing four of the 2푁uc environment tensors (two for each direction), where only two of them (one for each direction) require computing a dominant eigenvector. After updating MPS tensors for site 푗 and updating the corresponding environment tensors, the MPS tensors for site 푗 + 1 are updated. In one iteration of the sequential approach, the MPS tensors for all sites are updated – one after another.

54 4.8 Finding MPS with Maximal Overlap

Input: MPS with bond dimension 휒 , tolerance 휏

Output: MPS in mixed-canonical form with bond dimension 휒 ′ < 휒

, , ,

1 initialise , randomly

2 repeat ∗ ∗ 3 = 휇dom × , = 휇dom ×

4 ⟵ , ⟵ ∗

∗ ! ! 5 = = 1

QR+ QR+ 6 ⟶ , ⟶

QR+ QR+ 7 ⟶ , ⟶

8 ← ∗ , ← ∗

9 휀퐿 = ‖ − ‖ , 휀푅 = ‖ − ‖ max max 10 until max {휀퐿, 휀푅} < 휏; Algorithm 4.3: Maximising the overlap between a reference MPS (with bond dimension 휒 and a single-site unit cell) and the resulting MPS (with bond dimension 휒 ′ in mixed- canonical form).

55 Chapter 4 Matrix Product States

Input: MPS with bond dimension 휒 ∀푗 ∈ {0, … , 푁uc − 1}, tolerance 휏 푗 Output: MPS in mixed-canonical form with bond dimension 휒 ′ < 휒

, , , ∀푗 ∈ {0, … , 푁uc − 1} 푗 푗 푗 푗

1 initialise , ∀푗 ∈ {0, … , 푁uc − 1} randomly

2 repeat 3 for 푗 ∈ {0, … , 푁uc − 1} do

∗ ∗ ∗ ∗ 4 = 휇 × , = 휇 ×

푗 − 1 푗 푗 + 푁uc − 1 푗 − 1 푗 + 1 푗 + 푁uc 푗 + 1 푗 + 1

∗ ∗ 5 ← , ←

푗 푗 − 1 푗 푗 푗 푗 + 1

6 ⟵ , ⟵ 푗 − 1

푗 푗 + 1 푗 − 1 푗

∗ ∗

∗ ! ! ! 7 ⟵ , = = = 1

푗 − 1 푗 푗 + 1 푗 푗 푗 − 1

QR+ QR+ 8 ⟶ , ⟶

QR+ QR+ 9 ⟶ , ⟶

10 ← ∗ , ← ∗

(푗) (푗) 11 휀퐿 = ‖ − ‖ , 휀푅 = ‖ − ‖ 푗 푗 max 푗 − 1 푗 max (푗) (푗) 12 until max {max {휀퐿 , 휀푅 } | 푗 ∈ {0, … , 푁uc − 1}} < 휏; Algorithm 4.4: Maximising the overlap between a reference MPS (with bond dimension ′ 휒 and a unit cell of 푁uc sites) and the resulting MPS (with bond dimension 휒 in mixed- canonical form) using the sequential approach.

56 4.8 Finding MPS with Maximal Overlap

Input: MPS with bond dimension 휒 ∀푗 ∈ {0, … , 푁uc − 1}, tolerance 휏 푗 Output: MPS in mixed-canonical form with bond dimension 휒 ′ < 휒

, , , ∀푗 ∈ {0, … , 푁uc − 1} 푗 푗 푗 푗

1 initialise , ∀푗 ∈ {0, … , 푁uc − 1} randomly

2 repeat

∗ ∗ ∗ ∗ 3 = 휇 × , = 휇 ×

0 1 푁uc 0 0 푁uc − 1 0 0 4 for 푗 ∈ {0, … , 푁uc − 2} do

∗ ∗ 5 ← , ←

푗 + 1 푗 푗 + 1 푗 − 1 푗 − 1 푗

6 for 푗 ∈ {0, … , 푁uc − 1} do

7 ⟵ , ⟵ ,

푗 − 1 푗 푗 + 1 푗 푗 + 1 ∗

∗ ! ! = = 1

푗 푗

8 for 푗 ∈ {0, … , 푁uc − 1} do QR+ QR+ 9 ⟶ , ⟶

QR+ QR+ 10 ⟶ , ⟶ 푗 − 1

11 ← ∗ , ← ∗

(푗) (푗) 12 휀퐿 = ‖ − ‖ , 휀푅 = ‖ − ‖ 푗 푗 max 푗 − 1 푗 max (푗) (푗) 13 until max {max {휀퐿 , 휀푅 } | 푗 ∈ {0, … , 푁uc − 1}} < 휏; Algorithm 4.5: Maximising the overlap between a reference MPS (with bond dimension ′ 휒 and a unit cell of 푁uc sites) and the resulting MPS (with bond dimension 휒 in mixed- canonical form) using the parallel approach.

57 Chapter 4 Matrix Product States

In the parallel approach, the idea is to update the MPS tensors for all sites at once. For this, all of the 2푁uc environment tensors are computed, but only two of them require computing a dominant eigenvector. Therefore, per iteration of the parallel approach two dominant eigenvectors are computed, whereas in the sequential approach 2푁uc dominant eigenvectors are computed. Note that computing such an eigenvector is the most costly operation in algorithms 4.4 and 4.5. However, the sequential approach commonly requires fewer iterations to converge to the same level of accuracy. Within the scope of this thesis, it has never been observed that the parallel approach outperformed the sequential approach. Nevertheless, it is hard to proof that the sequential approach runs faster in general. Instead of directly truncating the bond dimension of a reference MPS, a common use case of this overlap optimisation scheme is to truncate the bond dimension of an MPS resulting from an application of an MPO to an MPS,

휂 휒 ′ ≈ , (4.47) 휒 where 휒휂 < 휒 ′. To achieve this, one could simply apply the MPO to the MPS, which results in an intermediate MPS with bond dimension 휒휂, and then apply one of the algorithms 4.3, 4.4, or 4.5. However, constructing this intermediate MPS is more expensive than necessary. By simply replacing the lime tensors in algorithms 4.3, 4.4, and 4.5 by the corresponding stacks of lime and turquoise tensors from the left-hand side of equation (4.47) and promoting the environment tensors to rank-3 tensors, the computational complexity of a matrix-vector product from an ei- 2 2 ′ ′ ′ 2 genvector computation can be reduced from 풪 (푁uc푝휒 휂 휒 ) to 풪 (푁uc푝휒휂휒 (휒 + 푝휂 + 휒 )).

4.9 Finding the Dominant MPS of an MPO

The probably most versatile algorithm in the context of MPSs is presented in this section, which addresses the challenge of finding the fixed point of an MPO by approximating it withanMPS of bond dimension 휒,

휒 휒 ≈ . (4.48)

Depending on the correlation length of the exact fixed point, a certain value of 휒 is required to obtain an MPS approximating the exact fixed point with a certain quality. One should keepin mind that in principle, if the exact fixed point has an infinite correlation length, no finite value of 휒 is sufficient to represent this state. In the literature, this algorithm for finding fixed point MPS is often simply called VUMPS, as it was originally introduced as variational uniform MPS algorithm [141]. The procedure for

2Strictly speaking, this modification only results in a speedup in the case of ‘small’ physical dimensions 푝. How- ever, in real world applications this is always the case.

58 4.9 Finding the Dominant MPS of an MPO

Input: MPO , tolerance 휏

Output: Fixed point MPS in mixed-canonical form , , ,

1 initialise , randomly

2 repeat ∗ ∗

3 = 휇dom × , = 휇dom ×

4 = 휇dom × , = 휇dom × ∗

∗ ! ! 5 = = 1

QR+ QR+ 6 ⟶ , ⟶

QR+ QR+ 7 ⟶ , ⟶

8 ← ∗ , ← ∗

9 휀퐿 = ‖ − ‖ , 휀푅 = ‖ − ‖ max max 10 until max {휀퐿, 휀푅} < 휏; Algorithm 4.6: Finding the dominant MPS for an MPO with a single-site unit cell.

59 Chapter 4 Matrix Product States single-site unit cells is summarised in algorithm 4.6. As it is based on tangent space projections in the same way as the algorithm presented in the previous section, the overall structure is very similar to algorithm 4.3. The major difference is the way of updating the central MPS tensor and the bond matrix in line 4, which are normalised in line 5. Updating the left- and right- canonical tensors (lines 6–8), based on the new central MPS tensor and bond matrix, as well as evaluating the numerical error (line 9) works exactly as in algorithm 4.3. In addition to finding dominant eigenvectors of one-dimensional quantum mechanical operators in the form of MPOs, algorithm 4.6 can be used to compute partition functions (per site) of two-dimensional classical statistical mechanics systems, if the partition function can be written as an infinite two-dimensional network121 [ , 142, 143], e. g.

풵 = . (4.49)

Depending on the model and its couplings, this network does not necessarily have a trivial single-site unit cell, but could for example look like

풵 = , (4.50)

0 1 2 where the colours indicate a unit cell consisting of three sites (0, 1, 2). In the case of a nontrivial unit cell consisting of 푁uc sites, one has to find MPS tensors for each site of the lattice. Sucha two-dimensional lattice can be formally defined by introducing nearest-neighbour lists, which are two permutations giving the nearest neighbour in both 푥- and 푦-direction for each site. For the lattice from equation (4.50) the corresponding nearest-neighbour lists are

0 1 2 0 1 2 nn = ( ) and nn = ( ) , (4.51) 푥 1 2 0 푦 2 0 1 or simply nn푥 = (1, 2, 0) and nn푦 = (2, 0, 1) if a short-hand notation for permutations is used. To identify relative locations in the two-dimensional lattice, instead of this enumeration, a coordinate-based scheme will be used. For the example from equation (4.50), if (푥, 푦) denotes site 0, then (푥 + 2, 푦 + 1) denotes site 1. A generalised version of algorithm 4.6 for nontrivial unit cells is depicted in algorithm 4.7, where coordinates are used to identify the location of tensors within the unit cell. At the

60 4.9 Finding the Dominant MPS of an MPO

Input: MPO 푦 ∀(푥, 푦) ∈ 핃, tolerance 휏

푥 Output: Fixed point MPS in mixed-canonical form

푦 , 푦 , 푦 , 푦 ∀(푥, 푦) ∈ 핃 푥 푥 푥 푥

1 initialise , ∀(푥, 푦) ∈ 핃 randomly

2 repeat 3 compute dominant eigenvectors for (푥, 푦) ∈ 핃 from

푦 + 1 ∗ ∗ 푦 + 1 푦 + 1 ∝ and ∝ 푦 + 1 푦 푦 푥 푥 + 1 푥 + 1 푥 − 1 푥 푥 − 1 4 compute dominant eigenvectors for (푥, 푦) ∈ 핃 from

푦 + 1 푦 + 1

푦 + 1 ∝ 푦 + 1 and 푦 + 1 ∝ 푦 + 1 푥 푥 푦 푦 푥 − 1 푥 푥 + 1 푥 푥 푥 + 1

5 for (푥, 푦) ∈ 핃 do

∗ 푦 ∗ 푦 ! ! 6 = = 1 푦 푦 푥 푥 7 for (푥, 푦) ∈ 핃 do QR+ QR+ 8 ⟶ , ⟶

QR+ QR+ 9 ⟶ , 푦 ⟶ 푥 − 1

10 ← ∗ , ← ∗

(푥,푦) 푦 (푥,푦) 푦 11 휀퐿 = ‖ − ‖ , 휀푅 = ‖ − ‖ 푥 푥 max 푥 − 1 푥 max (푥,푦) (푥,푦) 12 until max {max {휀퐿 , 휀푅 } | (푥, 푦) ∈ 핃} < 휏; Algorithm 4.7: Finding the dominant MPS for MPOs with nontrivial unit cell using the parallel approach.

61 Chapter 4 Matrix Product States bottom (left or right) of a tensor network diagram, the 푥-coordinate (푦-coordinate) for the corresponding column (row) of tensors can be found. Although it can sometimes be tricky to see, which site of the unit cell corresponds to a certain tensor, there is a simple trick to avoid mistakes: Imagine, each index in equation (4.50) and each virtual index in the resulting MPS has a different dimension. Then by simply matching the dimensions, the correct unitcell site can be derived for each tensor. When implementing these algorithms, it is also strongly recommended to test the implementation with different dimensions for all indices, as errors can be easily spotted. The set 핃 indicates the 푁uc sites of the unit cell. When it is used to iterate over the sites of the unit cell with (푥, 푦) ∈ 핃, the actual values of 푥 and 푦 do not matter, but they are used to indicate relative positions of the tensors within the unit cell. A particularity of algorithm 4.7 are the dominant eigenvalue problems in lines 3 and 4. The given expressions only relate two tensors, but do not directly give an eigenvalue problem. However, such expressions can be iteratively applied, e. g. in line 3, mapping a tensor for site 푥 to site 푥 + 1, then mapping from site 푥 + 1 to site 푥 + 2, and so on, until site 푥 is reached again, which then results in an eigenvalue problem

푦 + 1 ∗ ∗ 푦 + 1

푦 + 1 = 휇dom × 푦 + 1 , (4.52) 푦 푦 푥 푥 + 1 푥 + ℓ 푥 where ℓ is the length of the cycles of nn푥 . Note that for each cycle there is one eigenvalue equation to solve and the remaining ℓ − 1 eigenvectors are obtained by applying the corresponding transfer operators. The same works for the eigenvalue equations given in line 4 and the cycles of nn푦 [144, 145]. Algorithm 4.7 is using the idea of the parallel approach from algorithm 4.5, as all central MPS tensors and bond matrices are updated at once. One could as well think of a sequential approach, where the tensors for all sites along a vertical cycle are updated at once. Therefore, if nn푦 consists of only one cycle, the sequential and the parallel approach are identical. An example for a lattice with more than one cycle per direction, where the sequential and the parallel approach differ, is

. (4.53)

Another important application of the algorithm presented in this section is the contraction of the norm of projected entangled-pair states as shown in the following chapter. This contraction

62 4.10 Concluding Remarks is necessary for computing observables and correlation lengths, or to compute energy gradients to optimise projected entangled-pair states. Furthermore, a modified version of algorithm 4.7 can be used to efficiently find ground states of one-dimensional quantum141 systems[ ].

4.10 Concluding Remarks

To sum up, this chapter introduced MPSs and discussed them mainly in the context of infinite- size systems. Not only basic operations such as computations of observables and correlation lengths were illustrated, but also algorithms to bring MPSs into canonical form, to find MPSs with maximal overlap to reference MPSs, and to find MPSs approximating the fixed points of MPOs were presented. The latter two algorithms are instances of novel family of tangent space methods. The aim of this chapter was to focus on the aspects of MPSs that are required as a basis for the topics in the following chapter about PEPSs. Therefore, many aspects from the wide field of MPSs have not been discussed in this chapter. From a conceptual point ofview,the fundamental theorem of MPS [146] is of great importance. It is a vital technical tool for analytical considerations of MPSs. Further, it can be seen as a basis for implementing global symmetries for MPSs. From a numerical point of view, constructing MPSs obeying a certain symmetry yields block-sparse MPS tensors, which allows for simulations with much larger effective bond dimensions [147–150]. Another entire branch of MPS algorithms that has not been discussed at all are energy optimisations, which are naturally an essential tool for a variational ansatz. Historically, DMRG was the first MPS optimisation method, which was initially developed independently fromthe MPS ansatz [25, 26, 151, 152]. Despite its age, DMRG is, without any doubt, still the most widely used method for finding MPS ground state candidates. The popularity of DMRG is certainly increased by the existence of software packages such as iTensor [54] and TenPy [153], which are providing reliable and powerful (i)DMRG implementations. A novel MPS energy optimisation method based on tangent space projections with a very clean mathematical foundation has been introduced in reference [141] and has been shown to outperform DMRG in several physical scenarios. Another way to optimise MPS tensors is by simply employing a gradient minimisation technique, e. g. a conjugate gradient (CG) method [154] or a quasi-Newton method such as the Broyden-Fletcher-Goldfarb-Shanno (BFGS) algorithm [155–158]. To apply one of these methods, one needs a recipe to compute the gradient, such as the one given in reference [74]. Further, one could perform time evolutions of MPSs. To accomplish this, the most common method is the time-evolving block decimation (TEBD) algorithm [31, 32, 34, 159], which first requires a Trotter expansion of the evolution operator and then applies a series of local evolution gates to the MPS, keeping the bond dimension fixed. Another technique, that is more sophisticated, but at the same time more complex to implement, is the so-called time-dependent variational principle (TDVP) [160–162], which is another tangent space method. When performing real time evolutions of MPSs, one should keep in mind that the entanglement entropy can grow linearly in time, i. e. 푆(푡) − 푆(푡 = 0) ∝ 푡, due to the Lieb-Robinson bounds [163–168], but an MPS can only account for 푆 ≤ log(휒).

63 Chapter 4 Matrix Product States

In addition to real time evolutions, TEBD and TDVP can as well be used to perform imaginary time evolutions to find ground state candidates. However, imaginary time evolutions are clearly outperformed by methods as DMRG. Not only ground states can be found using MPSs, but also excitations can be computed in the form of Bijl-Feynman excitation spectra [169, 170] on top of ground states using a quasiparticle ansatz [161, 171]. In this chapter, only systems on discrete chains were considered. However, with continuous MPSs (cMPSs) continuous field theories can be simulated as well[172–175].

64 5 Projected Entangled-Pair States

After discussing MPSs and some related algorithms in the previous chapter, it is nowtimeto move forward to face two-dimensional systems. In a first step, projected entangled-pair states (PEPSs) are introduced for several two-dimensional lattice geometries. Further, two ways to contract PEPSs are discussed, which are the basis of almost every PEPS computation or algorithm. Such a contraction is required, for example, to compute observables or correlation lengths. Finally, so-called channel environments for PEPSs are introduced, which pave the way to state-of-the-art energy minimisation techniques.

5.1 The Ansatz

PEPSs are a straightforward generalisation of MPSs for two-dimensional systems [38],

휎0 휎1 ⋯ 푝

∑ |휎0, 휎1, …⟩ , (5.1) 휎0,휎1,… 퐷 where again, a system consisting of 푁 sites, for which the state of each isolated site can be described by a state from a 푝-dimensional Hilbert space, is considered. The bond dimension 퐷 is the dimension of the virtual indices connecting the PEPS tensors and the number of parameters of this variational ansatz is 풪 (푁 푝퐷4). For these three-dimensional tensor network diagrams, the convention of indicating tensor symmetries via their shapes is discarded, as the shapes are hard to grasp visually. However, all tensor network diagrams that are required to formulate algorithms in the context of PEPSs, are in the end two-dimensional diagrams, in which the symmetry indications via tensor shapes will be restored. In equation (5.1), a PEPS with open boundary conditions is shown. By connecting the tensors in the top row with the tensors in the bottom row, a PEPS with cylinder boundary conditions is obtained. If additionally the left column of tensors is connected with the right columnof tensors, the resulting PEPS has torus boundary conditions.

65 Chapter 5 Projected Entangled-Pair States

The generalisation of PEPSs to describe systems directly in the thermodynamic limit is called infinite PEPS (iPEPS) [103],

휎0 휎1 ⋯ 푝

∑ |휎0, 휎1, …⟩ , (5.2) 휎 ,휎 ,… 0 1 퐷

for which an arbitrary unit cell of 푁uc tensors can be chosen. In the example of equation (5.2), a unit cell of two chequerboard-like aligned sites is used. The choice of the unit cell is crucial in variational studies. For example, the unit cell as illustrated in equation (5.2) would be a good choice to describe a state with Néel order, but commonly fails to grasp a state with an incompatible unit cell. In the case of multiple physical phases with energy minima close to the true ground state energy, different unit cells can be used to find ground state candidates from each of the competing phases and extrapolate to 퐷 → ∞ for each ground state candidate in order to identify the best candidate [42, 44, 45]. The number of variational parameters for an 4 iPEPS is 풪 (푁uc푝퐷 ). Note that this chapter mainly focuses on the case of iPEPSs, but many concepts discussed in this chapter are helpful to gain insights for finite PEPSs as well. An advantage of iPEPSs is that results only have to be converged in the bond dimension 퐷. In contrast, with finite PEPSs, where results need to be converged in the bond dimension 퐷 for each system size 푁 , before the final result can be obtained from finite-size extrapolations. So far, the PEPS shown in equations (5.1) and 5.2 suggest to use these ansatz wave functions for systems on square lattices. Indeed, the square lattice is the preferred lattice in the PEPS universe, as most algorithms are tailored for this lattice. Nevertheless, PEPSs can be used in the context of various other lattice geometries as well. Systems on a honeycomb lattice [41], for example, can be treated using the ansatz

, (5.3)

where in addition to the bonds resembling the honeycomb lattice, dashed indices with dimension 퐷 = 1 are added. These do not increase the number of variational parameters or any properties of the state, but restore a square lattice, such that existing algorithms can be directly applied. This trick of adding artificial bonds of dimension 퐷 = 1 works in the case of the honeycomb lattice, as it has a smaller coordination number 푧 = 3 compared to the square lattice with 푧 = 4. However, the kagome lattice already has a coordination number of 푧 = 4. Therefore, adding

66 5.1 The Ansatz bonds to the PEPS ansatz for the kagome lattice,

, (5.4)

would cause the coordination number to exceed 푧 = 4. A possible solution to evade this obstacle is to group sites into super-sites [176]. In the specific kagome case, three sites can be grouped into one super-site,

= , (5.5) where two virtual indices of the kagome PEPS are grouped into a single bond of the square lattice PEPS. This approach increases the physical dimension from 푝 to 푝3, but restores the square lattice structure. At this point it should be noted that the computational complexity of PEPS contractions, which are discussed later in this chapter, and which are in almost all cases the most time-consuming part of PEPS algorithms, are independent of the physical dimension (or can at least be formulated in such a way) and therefore do not suffer from increasing 푝 to 푝3. Further, using super-sites can give a more fine-grained control over the amount of entanglement contained in the system: Directly using the kagome PEPS from equation (5.4) with uniform bond dimensions limits the effective bond dimensions between super-sites to 퐷 ∈ {1, 4, 9, 16, …}, whereas using super-sites enables 퐷 to be any nonnegative integer. Another common lattice is the triangular lattice, which has a coordination number of 푧 = 6. As blocking sites to super-sites does not reduce the coordination number, this approach cannot be used to transform a PEPS on a triangular lattice into a PEPS on a square lattice. Therefore, a common technique to simulate systems on a triangular lattice using PEPSs, is to treat the system as a system on a square lattice with additional next-nearest-neighbour interactions [40], as illustrated by

, (5.6)

where the dotted lines do not indicate virtual indices of the PEPS, but only interactions between sites. So far, this section illustrated that a PEPS is a versatile variational ansatz that is applicable to a wide range of two-dimensional lattice geometries. Based on this generalisation of the MPS ansatz for two-dimensional systems, it would be simple to generalise PEPSs even further to

67 Chapter 5 Projected Entangled-Pair States three-dimensional systems, although there is a lack of efficient algorithms to simulate three- dimensional systems with tensor network states. Similar to MPSs, gauge transformations can be applied to PEPSs, leaving the state invariant. For example, for a PEPS on a square lattice with a single-site unit cell, an invertible transformation 퐺 ∈ GL (휒) applied to the bond in 푥-direction,

퐺−1 퐺 → , (5.7) does not change the state. Consequently, there is an infinite number of different tensors describing the same physical state. It can also be shown that if two PEPSs define the same state, they can be related via a gauge transformation [177]. However, due to the bond loops in the PEPS ansatz, there is no unique canonical form for PEPSs, as it is known for MPSs. As already discussed in chapter 3, the entanglement properties are the distinctive feature of tensor network states. Dividing a system, described by a PEPS, into two partitions, 퐴 and the remainder 퐵,

, (5.8)

where the shaded area corresponds to the partition 퐴, requires to cut a number of bonds equal to the circumference |휕퐴| of 퐴. As these bonds have a dimension of 퐷, the entanglement entropy of such a bipartition is limited by

푆퐴∶퐵 ≤ |휕퐴| × log(퐷). (5.9)

Therefore, a PEPS is by construction obeying an area law for the entanglement entropy. In contrast to MPSs, where the entanglement entropy simply saturates with increasing subsystem size, one has a nontrivial area law for PEPSs. Due to the strong evidence for two-dimensional systems described by a gapped Hamiltonian with local interactions to have a ground state obeying an area law, PEPSs are promising ground state candidates for such systems [37].

5.2 Computing Observables

The expression for the expectation value of an observable in equation (3.5) can be generalised to PEPS tensors with the help of the double layer tensors

∗ ∗ ∶= and ∶= , (5.10)

68 5.2 Computing Observables where the green tensors are the PEPS tensors and the orange matrix is an arbitrary observable. The corresponding expectation value can be written as

⋮

푦 + 1

⟨휓|퐴푥,푦 |휓⟩ = 푦 . (5.11) 푦 − 1

⋮

⋯ 푥 − 1 푥 푥 + 1 ⋯ Note that here the same coordinate-based indexing of unit cell sites as introduced in section 4.9 is used to identify relative positions of tensors. At the bottom (left) of diagrams 푥-coordinates (푦-coordinates) are indicated and the nearest-neighbour lists nn푥 and nn푦 define the lattice. The norm ⟨휓|휓⟩ can be obtained by contracting the same nework with the orange observable tensor substituted by a green norm tensor. For a finite-size system, this two-dimensional tensor network is of finite size as well. Never- theless, the contraction of the entire network has been proven to be #P-complete [178]. From a more practical point of view: The numerically least expensive way of contracting a network of 푁푥 × 푁푦 sites is column-wise,

= = =

= = = ⋯ , (5.12)

or row-wise if 푁푥 < 푁푦 . Contracting the network this way for a PEPS with physical dimension 푝 and bond dimension 퐷 has a runtime complexity of at least 풪 (푝퐷2×min{푁푥 ,푁푦 }+4) and a memory complexity of at least 풪 (퐷2×min{푁푥 ,푁푦 }+2). Due to this exponential scaling in the linear system size, exact contractions of finite systems are in principle possible, but are infeasible for larger system sizes. Luckily, there are controllable approximate methods to perform these contractions. After this short intermezzo discussing finite systems, the focus will be again on infinitesys- tems in the following. To contract infinite-size systems, there are three possible routes to

69 Chapter 5 Projected Entangled-Pair States follow: boundary MPS contractions [74, 103], corner transfer matrix (CTM) contractions [44, 179], and direct renormalisation group (RG) methods [180–184]. Boundary MPS and CTM contractions are very similar, as both of them focus on finding an approximate environment. For example, the evaluation of a single-site observable as in equation (5.11) then reads as

⟨휓|퐴푥,푦 |휓⟩ = , (5.13) where the surrounding green tensor is the environment obtained by the contraction method. Note that in both methods this environment tensor is not a single tensor but consists of a network of several tensors. The two methods are introduced and discussed in more detail in sections 5.3 and 5.4 In contrast to boundary MPS and CTM contractions, direct RG methods, such as tensor RG (TRG) [180], tensor-entanglement RG (TERG) [181], second RG (SRG) [182], higher-order TRG (HOTRG) [183] or tensor network renormalisation (TNR) [184], aim to directly renormalise the PEPS tensors. In TRG, for example, tensor-SVDs are applied to the PEPS double-layer tensors,

휒 SVD ⟶ , (5.14)

where the number of singular values is truncated to 휒. In a second step, the tensors enclosed by the dashed lines are contracted. The tensors resulting from this contraction are the renormalised versions of the PEPS double-layer tensors. This is repeated until the tensors converge to a fixed point. While RG contraction methods could allow for insight into the entanglement structure of PEPSs they the lack versatility of the other implementations.

5.3 Boundary MPS Contractions

The starting point of boundary MPS contractions, which were the first iPEPS contraction method introduced [103], is the observation that the tensor network from equation (5.11) can be considered as infinitely many applications of an MPO, which has to result in a fixed point, corresponding to the dominant eigenvector of the MPO. This fixed point shall be obtained in

70 5.3 Boundary MPS Contractions the form of an MPS with bond dimension 휒,

푦 + 1 휒 휒 ≈ 푦 + 1 , (5.15) 푦 ⋯ 푥 − 1 푥 푥 + 1 ⋯ ⋯ 푥 − 1 푥 푥 + 1 ⋯ where it is assumed that the dominant eigenvalue of the MPO is unique. The contraction dimension 휒 can be used to control the quality of this approximation. Unless the MPO tensors are invariant under rotations, the fixed point for the opposite direction has to be obtained as well,

⋯ 푥 − 1 푥 푥 + 1 ⋯ ⋯ 푥 − 1 푥 푥 + 1 ⋯ 푦 휒 ≈ 푦 − 1 휒 . (5.16) 푦 − 1

If the MPO tensors have a two-fold spatial rotation symmetry, the boundary MPS tensors in equations (5.15) and (5.16) are identical. However, in the following it is assumed that the boundary MPS tensors might be different, depending on their orientation in tensor network diagrams. In section 4.9 it was discussed, how these boundary MPS fixed points can be obtained. Al- gorithm 4.6 can be used to determine the boundary MPS tensors for PEPSs with a single-site unit cell and algorithm 4.7 for PEPSs with a nontrivial unit cell. With these boundary MPS tensors the single-site expectation value from equation (5.11) can be written as

⋮

푦 + 1 푦 + 1 푦 ≈ 푦 . (5.17)

푦 − 1 푦 − 1 ⋯ 푥 − 1 푥 푥 + 1 ⋯ ⋮

⋯ 푥 − 1 푥 푥 + 1 ⋯ In addition to the boundary MPS tensors one needs to determine horizontal fixed point tensors based on the boundary MPS tensors,

푦 + 1 푦 + 1 푦 + 1 푦 + 1 푦 ∝ 푦 and 푦 ∝ 푦 . (5.18)

푦 − 1 푦 − 1 푦 − 1 푦 − 1 푥 푥 + 1 푥 + 1 푥 − 1 푥 푥 − 1

71 Chapter 5 Projected Entangled-Pair States

As already discussed in section 4.9, by iteratively inserting these relations into themselves, eigenvalue equations for each of the horizontal fixed point tensors can be obtained. Note once more that if two boundary MPSs are not equal, also the two green fixed point tensors might be different. The environment is normalised by enforcing 휇dom = 1 for the dominant eigenvalue. This can be achieved by dividing one of the PEPS tensors by √|휇dom|, which causes the corresponding PEPS double-layer tensor to be divided by |휇dom|, and absorbing the complex phase of 휇dom into the boundary MPS tensors. In addition to normalising the dominant eigenvalue, also the overlap between the left and right fixed point tensors has to be normalised,

푦 + 1 ! 푦 = 1, (5.19)

푦 − 1 푥 − 1 푥 푥 + 1 by rescaling the central boundary MPS tensors accordingly. With these properly normalised fixed point tensors the single-site expectation value can finally be simplified to

푦 + 1 푦 + 1

⟨퐴푥,푦 ⟩ = 푦 = 푦 . (5.20) 푦 − 1 푦 − 1 ⋯ 푥 − 1 푥 푥 + 1 ⋯ 푥 − 1 푥 푥 + 1

Another example is the evaluation of a two-site expectation value for nearest neighbours,

푦 + 1

⟨퐴푥,푦 퐵푥+1,푦 ⟩ = 푦 , (5.21) 푦 − 1 푥 − 1 푥 푥 + 1푥 + 2 where one should recognise that the centre boundary MPS tensor can be moved in the same way as it can be moved for any other MPS in mixed-canonical form using equation (4.20). Note that observables computed this way are obtained only approximately due to the finite contraction dimension 휒. This is especially important when it comes to computing energies for a state, as these energies can in principle be smaller than the true ground state energy. Therefore, expectation values of observables must be converged in 휒. So far, only vertical fixed points, i. e. top and bottom boundary MPSs, were discussed inthis section. Of course, it is possible to compute horizontal fixed points, i. e. left and right boundary MPSs, as well, which can be used, for example, to compute two-site observables along a vertical bond. Another use case for both horizontal and vertical fixed points are the so-called channel environments, which will be discussed in section 5.6

72 5.4 Corner Transfer Matrix Contractions

5.4 Corner Transfer Matrix Contractions

The idea of using CTMs for computing partition functions of classical statistical mechanics systems dates back to Baxter’s work in the 1960s [121, 142, 143]. At this time it was an analytical tool, but became also a numerical tool with the advent of the CTM renormalisation group (CT- MRG) algorithm [185, 186], which aimed at computing CTMs for classical statistical mechanics systems. Shortly after the introduction of iPEPSs in reference103 [ ], which uses a very simple boundary MPS technique for contractions, also CTMRG was started being used for iPEPS contractions [179]. CTMRG quickly became a widely used contraction method due to its natural treatment of nontrivial unit cells. However, it lacks a clean mathematical foundation until today. The most important objects in CTM contractions are the CTMs, e. g. for the upper left and upper right infinite quarter-planes,

휒 푦 + 2 휒 푦 + 2 푦 푦 ≈ 푦 + 1 and ≈ 푦 + 1 , (5.22) 휒 휒 푦 푦 푥 푥 푥 − 2푥 − 1 푥 푥 푥 + 1푥 + 2 where the infinitely many bonds on the edges are combined into single effective bondswith contraction dimension 휒, which controls the quality of the approximation. Note that there are in principle four different CTMs, which can be identified by their orientation in tensor network diagrams. In addition to the CTMs, one further needs to introduce half-line transfer tensors (HLTTs), e. g. for infinite half-lines to the left and to the right,

휒 휒 푦 ≈ 푦 and 푦 ≈ 푦 . (5.23) 휒 휒 푥 − 2푥 − 1 푥 푥 푥 + 1푥 + 2 푥 푥

If these CTMs and HLTTs are known, the single-site expectation value from equation (5.11) can be computed as

푦 + 1

⟨휓|퐴푥,푦 |휓⟩ = 푦 , (5.24)

푦 − 1

푥 − 1 푥 푥 + 1

73 Chapter 5 Projected Entangled-Pair States and a two-site expectation value for nearest neighbours as

푦 + 1

⟨휓|퐴푥,푦 퐵푥+1,푦 |휓⟩ = 푦 . (5.25)

푦 − 1

푥 − 1 푥 푥 + 1 푥 + 2

Before one can evaluate expectation values using a CTM contraction, the CTMs and HLTTs have to be determined. This can be done by performing a renormalisation approach, the so- called CTMRG. The key idea of CTMRG is to insert a line into the system and absorb one site of this line into a HLTT and the two infinite half-lines into two CTMs. This process of absorbing the additional sites can be performed by applying RG projectors as shown in algorithm 5.1. For a full CTMRG iteration, this procedure has to be applied once for each of the four directions and these iterations are repeated until the CTM environment is converged. To test for convergence, a common method is to evaluate PEPS observables and test whether they are converged. Unfortunately, this test is by far not as clean as the one used in algorithm 4.7 for computing boundary MPSs. It is not clear whether converged local observables imply a converged CTM environment. Therefore, one must be careful when evaluating, for example, correlation lengths or energy gradients, for which convergence was never tested. A technical detail that is much more important than one might think at first sight, is indicated by the comment in line 5 of algorithm 5.1. If the CTMs and HLTTs are not properly normalised with each RG transformation, the tensor entries will either diverge or vanish. Therefore the entries of these tensors should be rescaled. A common strategy is to enforce that the largest entry of each tensor is of magnitude one. The true magic of CTMRG lies in the construction of the RG projectors. Among the many different proposals for constructing them [43, 179, 187, 188], the specific construction from reference [44] is presented in algorithm 5.2, which results in a stable CTMRG implementation. However, it should be noted that the inversion of the singular values is unfavourable for the numerical precision. The truncated SVD, where at most 휒 singular values are kept, is the step responsible for limiting the contraction dimension to values no larger than 휒. Due to the following inversion, singular values smaller than 푠0 × 휏 should be discarded, where 푠0 is the largest singular value. The numerical tolerance 휏 is commonly chosen as 휏 ∈ [10−14, 10−12], as a smaller values of 휏 might cause CTMRG to become unstable, whereas larger values of 휏 introduce an artificial limit to 휒 allowing for less correlations in the CTM environment. It should be noted that the construction of the RG projectors is the part of CTMRG with the 6 3 6 2 largest computational complexity, 풪 (푁uc퐷 휒 + 푁uc푝퐷 휒 ), where the first term is usually the dominating one. Therefore, with increasing contraction dimension 휒, CTMRG iterations are asymptotically much more expensive than single iterations of algorithm 4.7. Traditional SVD implementations cannot be parallelised as good as tensor contractions and therefore do not profit a lot from running them on modern multi-core architectures and quickly become

74 5.4 Corner Transfer Matrix Contractions

Input: PEPS double-layer tensors 푦 ∀(푥, 푦) ∈ 핃,

푥

CTMs and HLTTs 푦 , 푦 , 푦 ∀(푥, 푦) ∈ 핃,

푥 푥 푥

푦 + 1 푦 + 1

RG projectors 푦 , 푦 ∀(푥, 푦) ∈ 핃

푥 푥 Output: Renormalised CTMs and HLTTs

, , ∀(푥, 푦) ∈ 핃

1 for (푥, 푦) ∈ 핃 do

푦 + 1

2 푦 + 1 ← 푦 푥 푥 푥

푦 + 1

3 푦 + 1 ← 푦 푥 푥 − 1 푥 푥

푦 + 1

4 푦 + 1 ← 푦 푥 푥 − 1 푥

5 Rescale CTM and HLTT entries, such that no entry is larger than one in magnitude. Algorithm 5.1: Renormalisation of a CTM environment for adding a row of tensors to the 4 3 6 2 lower half-plane, which has a runtime complexity of 풪 (푁uc퐷 휒 + 푁uc푝퐷 휒 ).

75 Chapter 5 Projected Entangled-Pair States

Input: PEPS double-layer tensors 푦 ∀(푥, 푦) ∈ 핃,

푥

푦 푦 CTM environment , ∀(푥, 푦) ∈ 핃 (for all directions)

푥 푥

푦 + 1 푦 + 1

Output: RG projectors 푦 , 푦 ∀(푥, 푦) ∈ 핃

푥 푥 1 for (푥, 푦) ∈ 핃 do

푦 + 3 푦 + 3

푦 + 2 QR 푦 + 2 QR 2 ⟶ , ⟶ 푦 + 1 푦 + 1

푦 푦

푥 푥 + 1 푥 + 2 푥 + 3

SVD휒 푠 3 ⟶ ∗

푦 + 1 푠−1/2 4 푦 ←

푥 + 1

푦 + 1 푠−1/2 5 푦 ← ∗

푥 + 1

Algorithm 5.2: Construction of CTMRG projectors for adding a row of tensors to the lower half-plane, according to reference [44], which has a runtime complexity of 6 3 6 2 풪 (푁uc퐷 휒 + 푁uc푝퐷 휒 ).

76 5.4 Corner Transfer Matrix Contractions the bottleneck in CTMRG. Therefore, the technique of randomised SVDs [189–191] should be considered for implementing CTMRG, as they are specifically designed to perform well on massively parallel computer architectures. If one takes a closer look and compares CTM environments with boundary MPS contractions it should become apparent that they are very closely related. For example, an infinite half-plane could be approximated using HLTTs as well as a boundary MPS, which directly leads to the equality

휒 휒 = . (5.26)

From this observation it follows that the HLTTs define an MPS that is equivalent to the boundary MPS – but in a different gauge. Therefore, HLTTs and CTMs are not uniquely defined and one can apply gauge transformations onto the bonds between them in exactly the same way as it has been discussed for MPSs, cf. equation (4.12). This relation has been used in reference [188] to establish a new method, called fixed point corner method, to compute CTM environments, which however was not able to outperform tangent space methods computing boundary MPS contractions. Comparing the evaluations of single-site observables in equations (5.20) and (5.24), also the CTMs can be related to the orthogonal fixed point tensors for boundary MPSs,

푦 + 1 푦 + 1 푦 = 푦 . (5.27)

푦 − 1 푦 − 1 푥

푥

Over several years, CTM contractions have provided valuable services for research and applications in the context of PEPSs, mainly because of their maturity with respect to large unit cells. However, it has to be taken into account that the new generation of tangent space methods for computing boundary MPS contractions has several advantages compared to CTMRG. Most importantly, it has a much cleaner mathematical foundation, which provides trustworthy convergence testing, and has a lower computational complexity in the limit of large contraction dimension 휒. In the following, boundary MPSs will be used exclusively as PEPS contractions. However, due to the close relation between CTM environments and boundary MPSs, it is always possible to replace the boundary MPS by CTMs and HLTTs if equations are adjusted accordingly.

77 Chapter 5 Projected Entangled-Pair States

5.5 Correlation Length

Following from the evaluation of nearest-neighbour observables in equation (5.21), any two- point correlation function along a line in 푥-direction can be written as

푟 − 1

⟨퐴푥,푦 퐵푥+푟,푦 ⟩ = ( ) . (5.28)

In analogy to the evaluation of the correlation length for an MPS as discussed in section 4.6, the eigenvalues of the transfer matrix, which can be computed from

푗 = 휇푗 × 푗 , (5.29)

where 휇0 = 1 > |휇1| > |휇2| > …, can be used to compute the correlation length, 1 휉 = − . (5.30) log |휇1| Note that for the sake of simplicity the case of a single-site unit cell was considered in equations (5.28) and (5.29). For PEPSs with a nontrivial unit cell and a nearest-neighbour list nn푥 , which has cycles of length ℓ, the transfer matrix in equation (5.29) consists of ℓ columns of tensors and the correlation length can be determined as ℓ 휉 = − . (5.31) log |휇1| Note that general PEPSs can be anisotropic states and therefore the correlation length can have different values 휉푥 and 휉푦 for the corresponding directions. The correlation length for the 푦-direction can be determined from the eigenvalues of the vertical transfer matrix,

= 휇푗 × 푗 . (5.32) 푗

Unlike for MPSs, for which the correlation length can grow with increasing bond dimension, but is necessarily finite (cf. section 4.6), PEPSs can have infinite correlation length, i. e. 휉 = ∞, already for the simplest nontrivial case of 퐷 = 2. This can be shown with the example of the so-called Ising-PEPS [192], a Rokhsar-Kivelson type wave function,

−훽퐻(휎 ,휎 ,…)/2 |휓⟩ = ∑ 푒 0 1 |휎0, 휎1, …⟩ , (5.33) 휎0,휎1,…

78 5.5 Correlation Length

0.12

0.10

| | | 0.08 1 1 휇 | | |

log 0.06 =

1 휉 0.04 critical Ising-PEPS 휉 = ∞ 0.02 TFI, ℎ = 3.044 38, 퐷 = 3 휉 = 10.83(1) 0.00 0.00 0.02 0.04 0.06 0.08 |휇1 | log | | |휇2 |

Figure 5.1: Two examples of correlation length estimations according to the model in equation (5.34). The green points (circles) show data obtained for the critical Ising- PEPS, which has infinite correlation length by construction, with contraction dimensions 휒 ∈ [8, 32]. The corresponding fit is in agreement with 휉 = ∞. The orange points (squares) show data obtained from a PEPS, that has been optimised for the transverse-field Ising model (for details on the model cf. chapter 7) at ℎ = 3.044 38 with 퐷 = 3, evaluated with contraction dimensions 휒 ∈ [32, 160]. From the corresponding fit, the correlation length for this PEPS can be determined as 휉 = 10.83(1). where 퐻 is the Hamilton function of a classical statistical mechanics system. If 퐻 is the Hamilton function of the classical Ising model, the state is called Ising-PEPS and can be em- 1 푧 푧 bedded into a 퐷 = 2-PEPS. The expectation value ⟨휎푗 휎푘 ⟩ of the Ising-PEPS is equal to the correlation function of the classical Ising model at inverse temperature 훽. Therefore, if 훽 is chosen to be equal to the critical inverse temperature 훽푐 of the Ising model, the Ising-PEPS displays algebraically decaying correlations and therefore has 휉 = ∞. It is much more difficult to obtain the correlation length of a PEPS converged in the contraction dimension 휒 compared to local observables. The possibility of having a PEPS with infinite correlation length makes this even more challenging, as a PEPS contraction with finite contraction dimension 휒 necessarily yields a finite correlation length 휉.2 This means, one has to distinguish between the difficulty of converging large correlation lengths and the impossibil-

1For an arbitrary statistical mechanics system consisting of classical 푝-level particles, such a Rokhsar-Kivelson PEPS can be constructed with 퐷 = 푝. 2This follows from exactly the same chain of arguments as given in section 4.6 for the finiteness of correlation lengths of MPSs.

79 Chapter 5 Projected Entangled-Pair States ity of converging infinite correlation lenghts in 휒. Luckily, the technique from reference [136], that was already used to perform bond dimension extrapolations to accurately determine the correlation length of MPSs in section 4.6, can be used as well to perform contraction dimension extrapolations for PEPSs. More specifically, if the correlation length determined fora PEPS with contraction dimension 휒 according to equation (5.31) is denoted with 휉(휒), the true correlation length 휉(∞) can be extracted from the relation

1 1 휇 (휒) = + const. × log | 1 | , (5.34) 휉(휒) 휉(∞) 휇2(휒) where 휇1(휒) and 휇2(휒) are the subdominant eigenvalues of the transfer matrix, as illustrated in equation (5.28). In figure 5.1 this extrapolation is demonstrated for the critical Ising-PEPS, which has 휉 = ∞ by construction, and a PEPS optimised for the transverse-field Ising model (for details on the model cf. chapter 7), which has finite correlation length. For the critial Ising-PEPS the model from equation (5.34) indeed suggests 휉 = ∞. In the transverse-field Ising case, it is remarkable that already with moderate contraction dimensions up to 휒 = 160, the correlation length can be determined with a relative error of less than 0.1 %, although 휉(휒 = 160) is still 8 % smaller than 휉(∞). Despite the fact that PEPSs can in principle have infinite correlation length, it has been observed that when they are optimised for gapless Hamiltonians, for which the true ground state has infinite correlation length, the energetically best PEPS has only finite correlation length [193, 194]. This led to the conjecture that the set of critical PEPSs only constitutes a fine- tuned an nongeneric submanifold of the entire PEPS manifold. However, the finite correlation lengths obtained for gapless models are not only a curse, but are a blessing as well: They can be understood as induced finite length scales that can be used to extrapolate observables to the 휉 → ∞ limit for gapless models. This manifests in the finite correlation length scaling framework, which consists of field theoretically inspired Casimir scaling formulae, e.g.

퐴 푒(휉) = 푒(∞) + + … , (5.35) 휉 3 for the energy density 푒. For a more detailed discussion of this matter, the interested reader is referred to chapter 7, which is dedicated to this finite correlation length scaling.

5.6 Channel Environments

So far, it was discussed how one can evaluate two-point observables along a horizontal or a vertical line of the lattice by using either boundary MPS or CTM contractions. But what to do,

80 5.6 Channel Environments if an expectation value such as

푦 + 2

⟨퐴푥,푦 퐵푥+3,푦+2⟩ = (5.36) 푦

푥 푥 + 3 shall be computed? One possible solution would be to find fixed points based on boundary MPS tensors,

푦 + 4 푦 + 4

푦 + 3 푦 + 3

푦 + 2 = 푦 + 2 , (5.37) 푦 + 1 푦 + 1

푦 푦 푥 푥 + 1 푥 + 1 which can then be used to compute the expectation value as

푦 + 3

푦 + 2

⟨퐴푥,푦 퐵푥+3,푦+2⟩ = 푦 + 1 . (5.38) 푦

푦 − 1 푥 − 1 푥 푥 + 1푥 + 2푥 + 3푥 + 4

However, the major problem with this approach for evaluating an observable ⟨퐴푥,푦 퐵푥+훿푥 ,푦+훿푦 ⟩ is the undesirable exponential scaling in min{훿푥 , 훿푦 }. Luckily, Vanderstraeten et al. [195] introduced the concept of channel environments, which allows to find environments for arbitrarily placed sites circumventing the exponential scaling. The cornerstone of this idea is an object called corner matrix, which allows to ‘bend’ boundary MPSs, such that they can be used to approximate infinite quarter-planes. Formally, this

81 Chapter 5 Projected Entangled-Pair States approximation of an infinite quarter-plane can be written as

푦 + 2 푦 + 2 푦 + 1 푦 + 1 ≈ , (5.39) 푦 푦 푦 푥 푥 + 1푥 + 2 푥 푥 푥 + 1푥 + 2 where the grey matrix connecting the two boundary MPSs is the corner matrix. Note that the orange left- and blue right-canonical boundary MPS tensors belong to boundary MPSs for two different directions, as indicated by the orientation of the tensors. Although CTMs approximate the same infinite quarter-planes, it will become clear in the following that channel environments are much more versatile than CTMs. The starting point for computing the corner matrices is the fixed point equation

푦 + 1 푦 + 1

푦 푦 ≈ , (5.40) 푦 − 1 푦 푦 − 1 푦 − 1 푥 − 1푥 − 1 푥 푥 + 1 푥 − 1 푥 푥 푥 + 1 which relates the corner matrices for sites (푥, 푦) and (푥 − 1, 푦 − 1). Note that there are 4푁uc different corner matrices: one per direction and site in the unit cell. In the following, thefixed point tensors

푦 푦 푦 푦 − 1 푦 − 1 ∝ 푦 − 1 and 푦 − 1 ∗ ∝ (5.41) 푥 푥 푥 + 1 푦 − 1 ∗ 푦 − 1 푥 푥 푥 + 1 푥 − 1 푥 푥 − 1 are required to construct the necessary tangent space projections. For the complex conjugate tensors in equation (5.41) the rule of identifying tensors by their orientation is slightly adap- ted: A tensor marked with an asterisk is the complex conjugate of the tensor with opposite orientation. It is important to note that therefore each of the two fixed point relations contains tensors of a single boundary MPS only. To obtain eigenvalue equations for the corner matrices, the relations from equation (5.40)

82 5.6 Channel Environments can be projected onto adequate tangent spaces [195, 196], which results in

푦

푦 ∝ 푦 − 1 . (5.42) 푦 − 1 ∗ 푥 − 1 푦 − 1 ∗ 푥 − 1푥 − 1 푥 푥

As already discussed in section 4.9, by iteratively inserting these relations into themselves, eigenvalue equations for each of the corner matrices can be obtained. After the corner matrices are computed from these eigenvlaue equations, they havetobe properly normalised,

푦 + 1

푦 + 1 ! 푦 = 1. (5.43)

푦 − 1

푦 − 1 푥 − 1푥 − 1 푥 푥 + 1푥 + 1

Note that the green terminal tensors are the ones defined in equation (5.18) and are therefore different from the orange and blue terminals introduced in5.41 equation( ). Once these corner matrices are at hand, arbitrary two-point correlation functions can be written as

푦 + 훿푦 + 1

푦 + 훿푦

푦 + 훿푦 − 1

푦 + 1

푦 + 1 ⟨퐴푥,푦 퐵푥+훿푥 ,푦+훿푦 ⟩ = . (5.44) 푦

푦 − 1

푦 − 1 푥 − 1 − 1 + 1 푥 푥 푥 푥 푥 + 1 푥 + 훿 푥훿 푥 + 훿 푥 + 훿

83 Chapter 5 Projected Entangled-Pair States

As promised in the beginning, with channel environments two-point observables of this kind can be evaluated without the undesirable exponential complexity scaling in min{훿푥 , 훿푦 }, but with a mild linear scaling in both 훿푥 and 훿푦 . Although the motivation for the introduction of channel environments in this section was the evaluation of arbitrary two-point observables, they are basis for applications of much larger importance: computing energy gradients [81], static structure factors [81], or excitation spectra [195] for PEPSs.

5.7 Energy Optimisation and Gradients

With the advent of iPEPSs in reference [103] also the imaginary time evolution technique was implemented for the first time for PEPSs, which aims to find a ground state candidate |휓0⟩ by minimising the ground state energy as

푒−휏ℋ |휙⟩ |휓0⟩ = lim , (5.45) 휏→∞ ‖푒−휏ℋ |휙⟩‖ starting from some initial PEPS |휙⟩. As in most cases the evolution operator 푒−휏ℋ is not directly applicable to a PEPS, it has to be decomposed into a sequence of local gates using a Suzuki- Trotter expansion, which introduces a numerical error depending on the chosen imaginary step size. Without any doubt, imaginary time evolutions played an important role for PEPSs to become as popular as they are nowadays. They enabled PEPSs to be used to study e. g. fermionic models [197], various SU(푁 ) Heisenberg models [39–42], the 푡-퐽 model [43, 44], or the Shastry- Sutherland model [45, 46], just to name a few of the most remarkable achievements. However, with the introduction of a new generation of direct variational approaches [81, 198], it became clear that the imaginary time evolution technique introduces uncontrollable errors that are beyond the errors due to the Suzuki-Trotter expansion and cannot be exclusively attributed to the finiteness of 휒. This direct variational optimisation approach manifests either in the form of a sequence of generalised eigenvalue problems [198] or as a gradient minimisation problem [81]. The two approaches are closely related, as both of them rely on exactly the same contraction of all interaction terms in the system. Such an interaction contraction can be either performed on the basis of CTMRG [198] or on the basis of boundary MPS contractions [81]. Due to the observation that gradient minimisations are more stable than optimisations based on generalised eigenvalue problems, especially with respect to the initial state, this section will focus on how to compute energy gradients for PEPSs. In the following, energy gradients for a nearest-neighbour Hamiltonian, i. e.

ℋ = ∑ ℎ푗,푘, (5.46) ⟨푗,푘⟩ are computed. Note that the method presented here can be generalised to more complicated interaction connectivities, e. g. for interaction plaquettes over four sites as shown in reference [196]. The starting point is the energy gradient of a quantum state in its most general

84 5.7 Energy Optimisation and Gradients form, ⟨휓|ℋ |휓⟩ ∇⟨휓|ℋ |휓⟩ ∇⟨휓|휓⟩ ∇⟨ℋ ⟩ = ∇ = − ⟨ℋ ⟩ . (5.47) ⟨휓|휓⟩ ⟨휓|휓⟩ ⟨휓|휓⟩ Subtracting the expectation value from each term of the Hamiltonian,

ℎ푗,푘 → ℎ푗,푘 − ⟨ℎ푗,푘⟩ (5.48) which results in nothing but an energy shift,

ℋ → ℋ − ⟨ℋ ⟩, (5.49) such that ⟨ℋ ⟩ = 0, leaving the structure of the energy levels invariant, simplifies the energy gradient to ∇⟨휓|ℋ |휓⟩ ∇⟨ℋ ⟩ = . (5.50) ⟨휓|휓⟩ The energy functional can be seen as a function depending on the independent PEPS tensors and their complex conjugates. Therefore, the energy gradient is computed as derivative with respect to the complex conjugate PEPS tensors. This requires the use of a double-layer tensor, where the complex conjugate tensor is left out,

∇ ∶= . (5.51)

As the energy is an extensive quantity and infinite systems are considered, the gradient of the energy density 푒 = ⟨ℋ ⟩/푁 is computed. In the case of a nontrivial unit cell, the gradient consists of derivatives with respect to each of the PEPS tensors,

푇 ∇푒 = (∇0푒 ⋯ ∇푁uc−1푒) , (5.52) where ∇푗 is the derivative with respect to the 푗-th PEPS tensor within the unit cell. Note that in the following again coordinate-based indexing for sites in the unit cell is used. The gradient of the energy density can be determined as

⎛ ⎞ ⎜ ⎟ ⎜ 훿푦 ⎟ ⎜ 훿푦 ⎟ ∇ ∇ ∇푒푥,푦 = ∑ ⎜ + ⎟ , (5.53) 훿푥 ,훿푦 ∈ℤ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎝ 훿푥 훿푥 ⎠

85 Chapter 5 Projected Entangled-Pair States which contains an infinite number of interaction terms. In the following, channel environments will be used to obtain contractions containing all interaction terms. In a first step, interactions along a single half-line are considered, which can formally be computed with an infinite geometric series,

푦 + 2 푦 + 2 ∞ 푦 + 1 = ∑ 푦 + 1 . (5.54) 푘=0 푦 푦 푥 푥 푥 − 푘 푥 − 푘 − 2 푥 − 푘 − 1 푥 − 푘 + 1

Note that the transfer matrix of this series has a unique dominant eigenvalue 휇dom = 1. There- fore, to compute this infinite sum by solving a linear equation, the trick of subtracting the dominant projector has to be used, as discussed in section 2.9. For this, in addition to the green terminal tensor, the other-hand side eigenvectors are required as well, which can be computed by solving

푦 + 2 푦 + 2 푦 + 2 ! 푦 + 1 ∝ 푦 + 1 with 푦 + 1 = 1 (5.55) 푦 푦 푦 푥 − 1 푥 푥 − 1 푥 푥 + 1 using the techniques introduced previously. The tensors containing all interactions along a line can be obtained by solving the linear equation

⎛ 푦 + 2 푦 + 2 ⎞ ⎜ ⎟ ⎜ − 푦 + 1 + 푦 + 1 ⎟ ⎜ ⎟ ⎜ 푦 푦 ⎟

⎝ 푥 + 1 푥 + 푁uc 푥 + 1 푥 ⎠

푁uc−1 = ∑ . (5.56) 푘=0

Note that due to enforcing ⟨ℋ ⟩ = 0, the term corresponding to the potentially diverging term of the rightmost-hand side of equation (2.33) vanishes. Further note that this linear equation

86 5.7 Energy Optimisation and Gradients has to be solved only for one tensor for each cycle of nn푥 (nn푦 for contracting interactions along a vertical line). The other tensors can be obtained from the relation

푦 + 2 푦 + 2 푦 + 2

푦 + 1 = 푦 + 1 + 푦 + 1 (5.57) 푦 푦 푦 푥 + 1 푥 푥 + 1 푥 − 1 푥 푥 + 1

By making use of channel environments, all interaction terms on a line orthogonal to the line from the previous term can be computed as

푦 + 2 푦 + 1 푦 + 1 푦 + 1 푦 + 1 푦 + 1 푦 푦 = + 푦 푦 − 1 푦 − 1 푦 − 1 푥 푦 − 1 푦 − 1 푥 − 1푥 − 1 푥 푥 + 1 푥 − 1푥 − 1 푥 푥 + 1 푦 + 1 푦 + 1 푦 + 1 푦 + 1 푦 푦 + + 푦 − 1 . (5.58) 푦 − 1 푦 − 1 푦 − 1 푦 − 2 푥 − 1푥 − 1 푥 푥 + 1 푥 − 1푥 − 1 푥 푥 + 1

In the next step, all interaction terms along parallel lines on an infinite half-plane are combined into a single tensor, which can be represented by another geometric series,

푦 + 2 푦 + 2 ∞ 푦 + 1 = ∑ 푦 + 1 . (5.59) 푘=0 푦 푦 푥 푥 − 푘 푥 − 푘 + 1 푥

87 Chapter 5 Projected Entangled-Pair States

Once more, this geometric series can be cast into a linear equation,

푦 + 2 ⎛ 푦 + 2 푦 + 2 ⎞ ⎜ ⎟ 푦 + 1 ⎜ − 푦 + 1 + 푦 + 1 ⎟ ⎜ ⎟ 푦 ⎜ 푦 푦 ⎟

푥 ⎝ 푥 + 1 푥 + 푁uc 푥 + 1 푥 ⎠

푦 + 2 푁uc−1 = ∑ 푦 + 1 , (5.60) 푘=0 푦 푥 − 푘 푥 − 푘 + 1 푥 which has to be solved for only one tensor per cycle in nn푥 (nn푦 for the vertical versions of this tensor). The remaining tensors can be obtained from the relation

푦 + 2 푦 + 2 푦 + 2

푦 + 1 = 푦 + 1 + 푦 + 1 . (5.61) 푦 푦 푦 푥 + 1 푥 푥 + 1 푥 + 1 Combining these terms containing the contracted interactions, the derivative of the energy density with respect to one of the PEPS tensors is computed as

푦 + 1 푦 + 1

∇푥,푦 푒 = 푦 ∇ + 푦 ∇ 푦 − 1 푦 − 1

푦 − 1 푦 − 1 푥 − 2푥 − 1푥 − 1 푥 푥 + 1푥 + 1 푥 − 1푥 − 1 푥 푥 + 1푥 + 1

푦 + 1

+ 푦 ∇ + rotations. (5.62)

푦 − 1 푥 − 1 푥 푥 + 1 Note that for the sake of brevity, only three of the total twelve terms are given explicitly in equation (5.62). The other nine terms are simply the rotated versions of these three terms. If only rotationally invariant PEPSs are considered, the gradient can be computed by just evaluating the three terms in equation (5.62) and manually symmetrising the gradient.

88 5.8 Concluding Remarks

Once the gradient is available, the energy of a PEPS can be optimised by using an arbitrary gradient minimisation technique, e. g. a conjugate gradient (CG) method [154] or a quasi- Newton method such as the Broyden-Fletcher-Goldfarb-Shanno (BFGS) algorithm [155–158]. Commonly, BFGS is observed to be more robust than CG. However, one should keep in mind 2 2 8 that BFGS is storing an approximate Hessian matrix, which consists of 풪 (푁uc 푝 퐷 ) elements. This can become a limitation when going to large values of 퐷. The contraction dimension 휒 controls the accuracy of the approximate contraction and therefore also the accuracy of the gradient. Optimisations commonly start from states close to product states, which require only small values of 휒 to obtain a proper gradient, and move to (strongly) correlated states, which require much larger values of 휒 to obtain gradients guiding the optimisation into the right direction. It is therefore important to adapt 휒 over the course of an optimisation. To increase the bond dimension of a boundary MPS, algorithm 4.3, 4.4, or 4.5 can be used. The numerical error of the gradient can be related to the smallest singular value of the boundary MPS bond matrices [144]. Although energy optimisations using gradient minimisations are among the most precise techniques for finding PEPS ground state candidates, one should be aware that the energy landscape of the PEPS manifold might be highly nontrivial. Therefore, one never has the guar- antee that a gradient method really ends up in the global energy minimum. A simple strategy, one can always give a try, is to start from different random states and compare the minimisation results. Another strategy would be to precondition a gradient minimisation by starting from the result of a (rough) imaginary time evolution. A different approach to compute gradients is the so-called automatic differentiation (AD) technique, which was used in reference [199] for the first time to compute PEPS energy gradients. In a nutshell, the idea behind AD is to formulate a computation graph, which is a directed acyclic graph containing all the computation steps, to obtain the energy of a PEPS. Subsequently, the gradient is computed by automatically evaluating all terms emerging from applying the chain rule to the computation graph. Two popular frameworks providing computations of AD gradients are TensorFlow [200] and PyTorch [201]. The AD technique looks very appealing due to the little effort required for implementing state-of-the-art PEPS optimisations compared with the channel environment based gradient computations as illustrated previously in this section. However, up to now, AD gradients were only determined based on energy computations driven by CTMRG, which have a much less clean mathematical foundation than tangent space boundary MPS computations. Therefore, AD gradients are lacking a proper numerical error analysis, which would be a desirable feature to obtain reliable results.

5.8 Concluding Remarks

This chapter discussed the PEPS ansatz for various lattice geometries and its entanglement properties. Due to the importance of PEPS contractions for basically any PEPS algorithm, the two most common contraction techniques – boundary MPS and CTM contractions – were illustrated. Further, it was shown how to compute correlation lengths for PEPSs, how to cope with finite contraction dimensions, and how correlation lengths of PEPSs behave when they are optimised for gapless Hamiltonians. Contractions were extended by channel environments,

89 Chapter 5 Projected Entangled-Pair States which allowed for efficient and elegant computations of energy gradients for state-of-the-art PEPS energy optimisations. However there are many more topics in the rapidly developing field of PEPSs that were not discussed in this chapter. For example, in reference [202] it was shown how one can construct PEPSs which are invariant under global Abelian symmetry transformations, such as ℤ푞 or U(1) spin symmetries. The benefit of symmetric PEPSs is a drastically reduced number of parameters, as the PEPS tensors turn into block-sparse tensors. This allows for larger effective bond dimensions in simulations. Further, also SU(2)-symmetric PEPSs have been constructed for the square lattice [203]. In reference [204], PEPS symmetries are introduced in a very general way, allowing not only for on-site symmetries, but also for lattice and time-reversal symmetries. The time-evolving block decimation (TEBD) algorithm, which has been ported from MPSs to PEPSs [103, 179, 197, 205–207], can be used to perform imaginary-time evolutions to find ground state candidates in the PEPS manifold as discussed in section 5.7. In addition to this, the TEBD algorithm can be used to perform real time evolutions as well. Nevertheless, as in the case of MPSs, the amount of entanglement can grow quickly as a function of time and PEPSs might become a bad approximation of the exact time-evolved state. Another whole branch of the PEPS literature is dealing with so-called fermionic PEPSs, which are promising candidates for describing interacting fermionic systems [197, 208–210]. So far, it was only discussed, how pure quantum states can be described by PEPSs. How- ever, if an additional physical index is added to PEPS tensors, the result is a so-called projected entangled-pair operator (PEPO), which relates to a PEPS as an MPO relates to an MPS. These PEPOs have been used to describe mixed quantum states to study systems at finite temperature [211, 212]. Although PEPSs are mainly used for finding ground states of two-dimensional systems, one can use PEPSs to formulate a Bijl-Feynman ansatz [169, 170] to compute excitation spectra on top of these ground states [195, 196, 213].

90 II

Research

6 Floating Phases in One-Dimensional Rydberg Ising Chains

This chapter is a reprint with permission of the authors of the preprint by Michael Rader and Andreas M. Läuchli published as arXiv:1908.02068v1 on 6th August, 2019.

We report on the quantitative ground state phase diagram of a van der Waals interacting chain of Rydberg atoms. These systems are known to host crystalline phases locked to the underlying lattice as well as a gapped, disordered phase. We locate and characterise a third type of phase, the so-called floating phase, which can be seen as a one-dimensional ‘crystalline’ phase which is not locked to the lattice. These phases have been theoretically predicted to exist in the phase diagram, but were not reported so far. Our results have been obtained using state-of-the-art numerical tensor network techniques and pave the way for the experimental exploration of floating phases with existing Rydberg quantum simulators.

6.1 Introduction

The theoretical and experimental investigation of strongly interacting quantum matter is one of the central current topics in condensed matter, quantum optics and even high-energy or nuclear physics. For this endeavour, quantum simulators play an important role, as their controlled conditions, high tunability and flexibility offer to make powerful inroads towards adeeper understanding of quantum matter in and out of equilibrium. With the recent advent of quantum simulators based on Rydberg atoms trapped with optical tweezers or in optical lattices [214–222] there is a natural interest in systems which can be natively studied using these existing platforms. In particular, in the recent Harvard experiments [217, 222, 223], long one-dimensional chains of Rubidium atoms have been trapped and several aspects of quantum many-body physics have been studied by exciting atoms to interacting Rydberg states. An exemplary study is the experimental investigation of quantum Kibble-Zurek dynamics when quenching the system from a gapped, disordered phase into a gapped, crystalline ordered phase [222]. This experiment tests the nature of the zero temperature quantum phase transition(s) from the disordered into the various crystalline phases, as

93 Chapter 6 Floating Phases in One-Dimensional Rydberg Ising Chains the quantum Kibble-Zurek scaling form depends on critical exponents, such as the correlation length exponent 휈 or the dynamical critical exponent 푧 of the transitions. In this context it might come as a surprise that an accurate quantitative theoretical study of the ground state phase diagram of the one-dimensional Rydberg model studied in reference [222] is currently lacking, despite a long history of the subject in various contexts [224– 234]. It is our goal to close this gap by performing state-of-the-art numerical simulations based on an infinite-system tensor network approach. We report accurate boundaries of themain crystalline phases and unveil the quantitative extent of a so-called floating, incommensurate solid phase.

6.2 Model and Expected Phases

We study an infinite chain of Rydberg atoms described by the Hamiltonian

Ω 푛푗푛푙 ℋ = − ∑ 휎푥 − Δ ∑ 푛 + 푉 ∑ . (6.1) 푗 푗 6 2 푗 푗 푗<푙 (푙 − 푗)

The local Hilbert space for an atom at site 푗 is spanned by the ground state |푔푗⟩ and the excited 푥 Rydberg state |푟푗⟩. The single-site operators are 휎푗 = |푔푗⟩ ⟨푟푗| + |푟푗⟩ ⟨푔푗| and 푛푗 = |푟푗⟩ ⟨푟푗|. The parameters Ω and Δ are called Rabi frequency and laser detuning. Throughout this work we use a unit lattice spacing and 푉 = 1, such that the corresponding Rydberg-Rydberg interaction gives an energy penalty for excited Rydberg atoms, which decreases with distance 푟 between these atoms with a characteristic van der Waals 1/푟6 power law. This Rydberg chain can be mapped onto a spin chain model 1, and it is therefore sufficient to explore the physics for Δ ≤ 휁 (6), as the remaining region can be obtained by virtue of the mapping. For Ω = 0, i. e. in the absence of quantum fluctuations, the model is purely classical and the ground states form a complete Devil’s staircase of crystalline phases, as shown by Bak and Bruinsma [225]. These translation symmetry breaking crystals are characterised by coprime integers 푝 and 푞, where 푞 denotes the size of the unit cell and 푝 the number of atoms in the Rydberg state within the unit cell. The atoms in the Rydberg state are located as far apart as possible, given 푝 and 푞. The extent in detuning of a crystal only depends on 푞 [235], so that e. g. a crystal with 푝/푞 = 1/5 has the same width as the one with 푝/푞 = 2/5 in the limit Ω = 0. Note that due to the rapid 1/푟6 decay of the interactions, the widths of the plateaux shrink rapidly with increasing 푞. For large detuning and/or large Rabi frequency the interaction can be neglected and the ground state corresponds in essence to a product state of the single-site ground state dictated by the local terms. This ground state breaks no symmetry, has a large excitation gap and rapidly decaying correlations. In between the two limits considered there are only a few quantitative results so far for the van der Waals interacting case [222, 228, 229] (see references [236–238] for results for the dipolar 1/푟3 case). One generally expects the crystalline phases to be stable upon the addition

1This interacting Rydberg chain is equivalent to a spin one-half chain with power-law antiferromagnetic Ising interactions, a transverse field Ω , and a longitudinal field Δ − 휁 (6). This mapping is useful to understand that 2 the phase diagram features a symmetry, (Δ, 푛) ↦ (2휁 (6) − Δ, 1 − 푛), where 휁 (푛) is the Riemann Zeta function.

94 6.2 Model and Expected Phases

1/5 4 1/4 −1/6 3

푦 = Ω 1/3 2 1/2 1 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 푥 = Δ/Ω

100

10−1 Δ 10−2

10−3

10−4 10−3 10−2 10−1 Ω Figure 6.1: Phase diagram of the model defined in equation (6.1). Both panels show the same data, but the upper one uses the coordinates 푥 = Δ/Ω and 푦 = Ω−1/6, introduced in reference [222]. The coloured regions indicate 푝/푞 crystalline phases with broken translational invariance and the white region corresponds to the gapped, disordered phase. Grey regions indicate the floating phase, which fully coats the 1/5-lobe. Note that the dark parts of the lobes in the lower panel are equivalent to the lobes in the upper panel and the lighter parts do not contain actual simulation data, but are illustrations of the fact that these crystals are expected to extend down to Ω = 0. Only the dots correspond to actual simulation results. Each of them is the phase transition point obtained from a series of simulations along a horizontal or vertical line. The phase transition lines connecting these dots are guides to the eye. The black lines in the upper panel indicate the cuts presented in figures 6.3 and 6.4, the diamond symbols denote the two points shown in figure 6.2, and the circle in the lower panel indicates the location of the cut presented in figure 6.5.

95 Chapter 6 Floating Phases in One-Dimensional Rydberg Ising Chains

푥 = 4, 푦 = 4 128 1.75 96 푐 ≈ 0.9923 64 푥 = 2, 푦 = 1.86202 48 1.50 32 푐 ≈ 0.5023 24 16 1.25 12 8

푆 1.00 128 64 0.75 32 16 0.50 8 4 0.25 2

1 4 16 64 256 1024 4096 휉

Figure 6.2: Scaling of the entanglement entropy 푆 as a function of the correlation length 휉 for a system in the floating phase at 푥 = 4, 푦 = 4 (green circles) and for a system close the Ising phase transition between the 1/2-lobe and the disordered phase at 푥 = 2, 푦 = 1.86202 (orange squares). Labels next to the points indicate the used bond dimensions 휒. The dashed lines show fits according to the model 푆 = 푐 log(휉) + 6 const., from which estimates for the central charge 푐 are extracted.

96 6.3 Method of the Rabi drive Ω, up to a strength which is in line with its width in detuning. The plethora of proposed scenarios of the quantum fluctuation induced melting of the crystalline phases is however quite diverse [227, 229, 231, 232], and requires an accurate numerical investigation. In reference [228] it was advocated in the Rydberg context, that in general the transition from a commensurate 푝/푞 crystal to the gapped, disordered phase proceeds via an extended, incommensurate and gapless phase, the so-called floating phase. Such phases have a long history [226, 239] and have been introduced in the context of commensurate-incommensurate transitions in solids. In modern parlance these phases are Luttinger liquids [240, 241], whose characteristics, such as the Luttinger parameter 퐾 and the ‘Fermi’ momentum 푘퐹 can depend on microscopic parameters, here Ω and Δ. Other aspects however, such as the central charge 푐 = 1 and the qualitative power-law decay of correlations are universal. In our work, we specifically search for these predicted phases, because they have not been reported in numerical simulations so far, other than in tiny slivers around the 푝/푞 = 1/3 crystal in a related, albeit different, model [227, 232, 233]. We find that floating phases feature rather prominently in the Rydberg chain phase diagram, even for experimentally accessible parameter ranges, see figure 6.1.

6.3 Method

In this work we use infinite matrix product states (iMPSs) [34] for studying the ground state of our model. An iMPS is a variational tensor network ansatz for an infinitely long chain with a periodically repeated 푁uc-site unit cell and a bond dimension 휒, which systematically limits the entanglement entropy 푆 to an area law, i. e. 푆 ≤ log(휒). We optimise the energy of the iMPS using the infinite density matrix renormalisation group (iDMRG) algorithm [151, 152] as provided by the TeNPy library [153]. iDMRG requires the Hamiltonian to be written in the form of a so-called infinite matrix product operator (iMPO). Unfortunately, one cannot directly formulate such an iMPO for power-law long-range interactions, but accurate decompositions into a sum of exponentials are a known way to proceed [132, 133, 242]. Here we use a decomposition comprising a sum of ten exponentials. The choice of 푁uc is crucial for iDMRG: If an incompatible unit cell is chosen, i. e. 푁uc is not a multiple of 푞 for a system where the true ground state is a 푝/푞 crystal, iDMRG will be numerically unstable or yield an iMPS in a different close-by physical phase. Therefore, this effect can be used to stabilise the twophases close to a phase transition with different choices of 푁uc and finally find the true ground state depending on the obtained energies for 휒 → ∞. For gapped phases the energy can easily be converged in the iMPS bond dimension 휒, but for gapless phases we have to compute the correlation length 휉 and extrapolate the energy density as a function of 1/휉 2, which is the expected Casimir scaling. Bond dimensions 휒 ranging from 4 to 256 are sufficient to obtain the energies reliably. 2 The 푝/푞 crystalline phases can be discriminated from the homogeneous, disordered phase

2We observed that particularly in simulations of crystalline phases, the smallest eigenvalue of the effective iDMRG-Hamiltonian does not coincide with the true energy density of the state, although all other observables converge properly. Therefore, as a cross-check, we evaluate the energy with interactions truncated to a few hundred sites.

97 Chapter 6 Floating Phases in One-Dimensional Rydberg Ising Chains

푥 = 4 100 휋

4휋 −1 5 10 2휋 3

−2 푘 휋 10 2 2휋 5

10−3 disordered 1/2 disordered 1/3 floating 1/4 floating 1/5 floating 0 10−4 1 1.2 2.1 2.2 3 3.2 3.9 4.2 4.75 5 푦 = Ω−1/6

Figure 6.3: Vertical strips show the structure factor 풮 (푘) as defined in equation6.3 ( ) for cuts through the phase diagram at 푥 = 4. As the structure factors are symmetric around the Brillouin zone boundary at 푘 = 휋, only one half of the zone is shown. In an ordered phase 푝/푞 one observes Bragg peaks at 푘 = 2휋푛/푞, 푛 = 0, … , 푞 −1. It should be noted that the finite width of these peaks in the plot are for improved readability. The floating phases show power-law diverging peaks at drifting wave vectors 푘, and weaker peaks at higher harmonics of the wave vector 푘푛 = 푛 푘, 푛 ∈ ℤ, which interpolate between the peaks of the crystalline phases. In the disordered phases, the structure factor is not divergent at finite wave vector and has broad peaks.

98 6.3 Method

푦 = 3.6 100 휋

1.0

4휋 −1 fit 5 푐 0.5 10 2휋 3 0.0 2.50 2.75 −2 푘 휋 푥 10 2 2휋 5

10−3 disordered floating 1/4 0 10−4 2.5 2.6 2.7 2.8 2.9 푥 = Δ/Ω

Figure 6.4: Vertical strips show the structure factor 풮 (푘) as defined in equation (6.3) for cuts through the phase diagram at 푥 = 4 (푦 = 3.6) in the upper (lower) panel. In addition to the characteristic features already discussed for figure 6.3, in the inset one can observe the extended central charge 푐 = 1 region, which characterizes the floating phase. by detecting a density imbalance within the unit cell (here 훿푛 ≳ 0.05), as the iDMRG algorithm generically converges to a minimally entangled instance within the 푞-fold degenerate ground state manifold, and Schrödinger cat-like superpositions are avoided. If a system is homogeneous, we still have to discriminate between disordered and floating phases. For the floating, gapless phase or isolated critical points, we can relate the correlation length 휉 and entanglement entropy 푆, 푐 푆 = log(휉) + const., (6.2) 6 and extract the central charge 푐 [135] as shown for two exemplary cases in figure 6.2. For automatic classification we use the criterion 푐 = 1 ± 5% to declare the system as floating. Due to the Kosterlitz-Thouless nature of the quantum phase transition from the disordered to the floating phase and the concomitant logarithmic corrections, the extent of the floating phase might be slightly overestimated using this criterion.

99 Chapter 6 Floating Phases in One-Dimensional Rydberg Ising Chains

푥 = 24 100 휋

4휋 −1 5 10 2휋 3

−2 푘 휋 10 2 2휋 5

10−3 1/2 disordered floating 2/5 floating 1/3 0 10−4 2.844 2.846 2.848 2.85 2.852 2.854 2.856 2.858 푦 = Ω−1/6

Figure 6.5: Vertical strips show the structure factor 풮 (푘) as defined in equation (6.3) for a cut through the phase diagram at 푥 = 24. In addition to the characteristic features already discussed for figure 6.3, for this value of 푥 also the 2/5 plateau appears.

We further define the structure factor,

푅 −푖푘푟 풮 (푘) ∝ |∑⟨푛0푛0+푟 ⟩푒 | , (6.3) 푟=0 where 푅 denotes the number of sites included in the Fourier transform. 3

6.4 Phase Diagram with Floating Phases

In figure 6.1 we show the phase diagram obtained with our simulations. The left panel shows the data using the coordinates, 푥 = Δ/Ω and 푦 = Ω−1/6, as introduced in reference [222]. The phase transition points are obtained by running a series of simulations for either constant 푥 or 푦 and the phase transition lines connecting these points are guides to the eye. In this first plot we have restricted ourselves to the region 1 ≤ 푥, 푦 ≤ 5. There we find ordered phases with

3Here we use 푅 = 300 in the crystalline phases and 푅 = 10000 for the remaining. phases.

100 6.5 Conclusion fractional fillings 1/푞, with 푞 = 2, 3, 4, 5 as indicated by the coloured lobes. We further find sizable floating phases indicated by the grey regions between the ordered phases, whichhave not been reported previously [222]. For larger values of 푦 the floating phase clearly approaches the tips of the ordered lobes and the 1/5-lobe appears fully coated by the floating phase. Based on this observation and the arguments in references [228, 229] we believe that all 1/푞-lobes for 푞 ≥ 5 are fully immersed into the floating phase. Note that there is a floating phase in between the 1/3 and the 1/4 lobe, which appears at experimentally accessible values of Ω and Δ [222], so they could be studied using the existing experiments. The two black lines in the left panel of figure 6.1 indicate two cuts through the phase diagram at fixed 푥 = 4 and fixed 푦 = 3.6, respectively, for which structure factors 풮 (푘) are presented in figures 6.3 and 6.4. Crystalline phases 푝/푞 exhibit infinitely sharp Bragg (i. e. 훿-function) peaks at wave vector 푘 = 2휋푛/푞, 푛 = 0, … , 푞 − 1. In the floating phases the real-space Rydberg density correlations decay as a power-law 1/푟훼 , with an exponent 훼 ≤ 1/4 [228, 229]. These correlations translate into power-law diverging peaks in the structure factor. Furthermore the wave vector of the peaks of the structure factor are not locked to commensurate values in the floating phase, and they indeed show a pronounced dependence on the parameters inthephase diagram. In figure 6.3 one can see how the floating phase interpolates between and touches the 1/3, 1/4 and 1/5 crystals. One expects the wave vector corresponding to the peaks in the structure factor to approach the commensurate values with a square-root singularity [239]. For the 1/3 crystal this is nicely visible (see also figure 6.5 below.). For higher 푞, this behaviour seems restricted to a more narrow window in the vicinity of the crystal, similar to observations in frustrated spin ladders [243]. On the scale presented, we do not observe a floating phase to the small 푦 side of the 1/3 crystal, although according to recent results in a related model, a very thin sliver is likely to occur [232]. For larger values of 푥 presented below, we also observe a sizeable floating phase on that side of the 1/3 crystal. Finally, note that we have not observed a floating phase touching the 1/2 lobe, as predicted in reference [229]. Indeed, in figure 6.2 we clearly observe the central charge 푐 = 1/2 of the Ising universality class expected for direct transition from the disordered phase into the 1/2 crystal. For the perpendicular cut at 푦 = 3.6, cf. figure 6.4, one can observe the successive sharpening of the primary peak as one moves from the gapped, disordered phase into the floating phase and finally into the 1/4 crystal phase. In the inset one can observe the extended central charge 푐 = 1 region, which characterises the floating phase. So far, we have only observed crystalline phases of type 1/푞 in the phase diagram. However, in the same range of 푦, but for larger values of 푥, one can also find other plateaux of the complete Devil’s staircase at Ω = 0. To illustrate this, we show the structure factor for a cut at 푥 = 24 in figure 6.5, which is beyond the scope of the phase diagram in the left panel of figure 6.1. For this value of 푥, the 2/5 plateau appears between the plateaux 1/2 and 1/3, which is coated by a sizeable floating phase, which however stops shortly before touching the 1/2 crystal lobe.

6.5 Conclusion

We have established the quantitative ground state phase diagram of a van der Waals interacting chain of Rydberg atoms. Our main result is to locate and characterise the so-called floating

101 Chapter 6 Floating Phases in One-Dimensional Rydberg Ising Chains phases, which can be seen as one-dimensional ‘crystalline’ phases which are not locked to the lattice. Our results pave the way for the experimental exploration of floating phases with existing Rydberg quantum simulators.

102 7 Finite Correlation Length Scaling in Lorentz-Invariant Gapless iPEPS Wave Functions

This chapter is a reprint of the article by Michael Rader and Andreas M. Läuchli published as Phys- ical Review X 8, 031030 on 30th July, 2018 under the terms of the Creative Commons Attribution 4.0 International license.

It is an open question how well tensor network states in the form of an infinite projected entangled-pair states (iPEPSs) tensor network can approximate gapless quantum states of matter. Here we address this issue for two different physical scenarios: (i) a conformally invariant (2 + 1)D quantum critical point in the incarnation of the transverse-field Ising model on the square lattice and (ii) spontaneously broken continuous symmetries with gapless Goldstone modes exemplified by the 푆 = 1/2 antiferromagnetic Heisenberg and XY models on the square lattice. We find that the energetically best wave functions display finite correlation lengths and we introduce a powerful finite correlation length scaling framework for the analysis of such finite-퐷 iPEPSs. The framework is important (i) to understand the mild limitations of the finite-퐷 iPEPS manifold in representing Lorentz-invariant, gapless many-body quantum states and (ii) to put forward a practical scheme in which the finite correlation length 휉(퐷) combined with field theory inspired formulae can be used to extrapolate the data to infinite correlation length, i. e. to the thermodynamic limit. The finite correlation length scaling framework opens the way for further exploration of quantum matter with an (expected) Lorentz-invariant, massless low-energy description, with many applications ranging from condensed matter to high-energy physics.

7.1 Introduction

The study of interacting quantum matter is of enormous current interest, with questions ranging from quantum spin liquids, topological matter, correlated electrons in solids, and ultracold atoms in optical lattices to strongly coupled quantum field theories.

103 Chapter 7 Finite Correlation Length Scaling in Lorentz-Invariant Gapless iPEPS Wave Functions

In this context numerical approaches play a very important role, with tensor networks being a central player. For problems in one spatial dimension methods, such as the density matrix renormalisation group [25, 27], (infinite) matrix product states [(i)MPSs] [34, 109, 151], and the multiscale entanglement renormalisation ansatz [118] have proven to be very powerful, both for gapped states and for quantum critical states with a low-energy conformal field theory (CFT) description [134, 135, 244, 245]. In two spatial dimensions, infinite projected entangled-pair states (iPEPSs) [38] have become a competitive numerical approach with successful applications to many problems in the field of quantum magnetism and for strongly correlated fermions [41, 43, 44, 246–248]. Furthermore, theoretical work has established how different forms of topological order can be represented and understood in iPEPS wave functions with finite bond dimension249 [ , 250]. It is, however, an open question how well tensor network states in the form of an iPEPS tensor network can approximate gapless quantum states of matter. Here we explore this issue for two distinct physical scenarios: (i) a conformally invariant (2 + 1)D quantum critical point in the incarnation of the transverse-field Ising model on the square lattice and (ii) spontaneously broken continuous symmetries with gapless Goldstone modes exemplified by the 푆 = 1/2 antiferromagnetic Heisenberg and XY models on the square lattice. We find that the best variational wave functions display finite correlation lengths and we introduce a powerful finite correlation length scaling framework for the analysis of 퐷such finite- iPEPSs. The outline of this paper is as follows. We start by providing a short introduction to the iPEPS framework and the energy optimisation strategies in section 7.2. In section 7.3, we study the critical properties of the (2 + 1)D transverse-field Ising model as an example ofa quantum critical point in the (2 + 1)D Ising universality class. In section 7.4, we present results for the 푆 = 1/2 antiferromagnetic Heisenberg model and the 푆 = 1/2 XY model, as examples for continuous symmetry breaking. In section 7.5, we provide an extensive discussion and interpretation of the results obtained, and we conclude in section 7.6.

7.2 Infinite PEPS

Considering a two-dimensional quantum many-body system consisting of 푝-level particles, placed on an infinite square lattice, one can make an ansatz for a wave function describing the system,

휎1 휎2 ⋯

|휓⟩ = ∑ |휎1, 휎2, …⟩ , (7.1) 휎 ,휎 ,… 1 2 퐷 which is commonly known as an infinite projected entangled-pair state (iPEPS) [38]. The graph in equation (7.1) is called a tensor network diagram, where nodes (edges) represent tensors (their corresponding indices). Edges connecting two tensors indicate summation indices and are of dimension 퐷, which we call the bond dimension of the iPEPS. The open indices 휎1, 휎2,… are of dimension 푝. In this work we consider only iPEPSs with a single-site unit

104 7.2 Infinite PEPS cell, i. e. iPEPSs where the same tensor is used for each site, but all the techniques described here can be generalised to arbitrary unit cells. iPEPSs are a straightforward generalisation of infinite system matrix product states for two spatial dimensions and obey an area law for the entanglement entropy as well [37, 38]. In contrast to iMPSs, iPEPSs can be constructed to have infinite correlation lengths, even for the simplest nontrivial case, 퐷 = 2 [192].

7.2.1 Contraction The key challenge in all iPEPS algorithms is the so-called contraction of the state. For instance, to evaluate single-site observables, one needs the single-site density matrix,

휌1 = x , (7.2)

which consists of an infinite sum of double-layer tensors,

∗ ∗ ∶= and x ∶= . (7.3)

The idea of a contraction is to find an approximation with controllable error for this infinitely large tensor network. There are several ways to go: finding an approximate environment in the form of a boundary matrix product state (bMPS) [103] or a corner transfer matrix (CTM) environment [179, 185] or by directly applying renormalisation group (RG) schemes such as tensor RG (TRG) [180], tensor-entanglement RG (TERG) [181], second RG (SRG) [182], higher- order TRG (HOTRG) [183] or tensor network renormalisation (TNR) [184]. Formally, a bMPS is nothing but an iMPS, with bond dimension 휒, that is an approximation for the dominant eigenvector of a transfer matrix of double-layer iPEPS tensors,

휒 휒 ≈ 퐶 , (7.4) 퐶 with corresponding eigenvalue 1. Note that the quality of this approximation can be improved by increasing 휒. The triangular tensors in equation (7.4) are isometric, i. e.

∗ ∗ = and = , (7.5)

105 Chapter 7 Finite Correlation Length Scaling in Lorentz-Invariant Gapless iPEPS Wave Functions so the bMPS is in a mixed canonical form. From the singular values 휁푗 of the central bond matrix 퐶 (Rényi) entropies,

(훼) 1 훼 푆bMPS = log (∑ 휁푗 ) , (7.6) 1 − 훼 푗 can be computed. In addition to the bMPS tensors, one has to determine a horizontal dominant eigenvector,

푦 + 1 푦 + 1 푦 = 푦 , (7.7)

푦 − 1 푦 − 1 푥 푥 + 1 푥 + 1 with corresponding eigenvalue 1 to be able to write the single-site density matrix as

휌1 ≈ x . (7.8)

The state-of-the-art method to find such a bMPS is described in reference [188] and has also been used in this work. Another way to contract a state is to find CTMs and half-line transfer tensors (HLTTs),

휒 ≈ and ≈ , (7.9) 휒 where again the contraction dimension 휒 is used to control the error of the approximation. With the CTMs and HLTTs, the single-site density matrix can be written as

휌1 ≈ x . (7.10)

A powerful numerical tool to find CTMs and HLTTs is the so-called CTM renormalisation group (CTMRG) [179, 185]. The specific CTMRG procedure introduced in reference [44] is used in this work, as it is a particularly stable variant of CTMRG. For this work both bMPS and CTMRG contractions have been implemented, and we observe that for equal 휒 both methods give almost identical results.

106 7.2 Infinite PEPS

7.2.2 Energy Optimisation As iPEPSs are especially well suited to describe ground states, energy optimisation algorithms for iPEPSs are of particular interest. For almost a decade there was only a single strategy for this task: imaginary time evolution [103] in various variants. Recently, references [81, 198] introduced new direct variational approaches, which achieved lower energies than imaginary time evolutions, even in the limit of vanishing Trotter step size. Both variational methods rely on interaction contractions, similar to the norm contractions described above, but including all interaction terms of the Hamiltonian. The first method [198] makes use of CTMRG interaction contractions to formulate a generalised eigenvalue problem for a given iPEPS, where the eigenvector corresponding to the lowest eigenvalue is used to propose the next iPEPS tensor in the minimisation run. The second method [81] computes the energy gradient of an iPEPS from a bMPS interaction contraction, such that any gradient minimisation technique can be used, e. g. conjugate gradient (CG) methods [154] or quasi-Newton methods such as the Broyden-Fletcher-Goldfarb-Shanno (BFGS) algorithm [155–158]. However, it should be noted that CTMRG interaction contractions can also be used to obtain energy gradients and, vice versa, bMPS interaction contractions to obtain the optimisation GEVP. For iPEPS energy optimisations in this work, bMPS interaction contractions in combination with the BFGS algorithm were primarily used, as this method turned out to be the most stable one. Also, the BFGS algorithm seems to be more stable close to the minimum compared to CG methods. Some states also have been optimised using a brute force minimization method, i. e. by computing finite difference energy gradients. All iPEPSs optimised in this workhave a single-site unit cell with a complex tensor which was forced to be invariant under spatial rotations. No symmetries at the virtual level were imposed. It turned out that starting with several random states and small contraction dimensions (16 ≲ 휒 ≲ 32) is the most economic way to bootstrap energy minimisations. The contraction dimension is then successively increased – in our optimisations up to values of 휒 = 512. For the transverse-field Ising model we observed that it can also be beneficial to use intermediate minimisation results to initialise minimisations for nearby values of the transverse field.

7.2.3 Correlation Length A correlation function 푐(푟) of two arbitrary observables can be written as

푟 − 1

푐(푟) = 푢 ( ) 푣 = 푢푇 퐴푟−1푣, (7.11)

using a bMPS contraction, where the tensors 푢 and 푣 contain the specific observables. It should be noted that the same correlation function can be expressed by replacing the bMPS tensors with HLTTs from a CTMRG contraction.

107 Chapter 7 Finite Correlation Length Scaling in Lorentz-Invariant Gapless iPEPS Wave Functions

We write the eigendecomposition of the transfer matrix as

푗 = 휆푗 × 푗 , (7.12)

with |휆0| ≥ |휆1| ≥ … and without loss of generality we assume 휆0 = 1. Then the dominant correlation length of the system 휉 can be extracted as 1 휉 = − . (7.13) log |휆1| With this method the dominant correlation length can be extracted without even knowing which observables lead to the corresponding correlation function. It should be noted that in contrast to local observables, the correlation length requires huge contraction dimensions 휒 to converge. Incorporating the ideas of reference [136] we observe the functional behaviour, 1 1 휆 (휒) = + 푘 log | 1 | , (7.14) 휉(휒) 휉(∞) 휆2(휒) which enables us to extract a converged value for the correlation length 휉(∞) precisely already for moderate values of 휒. In cases where 휆1 ≃ 휆2 (due to degeneracy, as observed e. g. in the 푆 = 1/2 Heisenberg model), one should use the largest eigenvalue different from 휆1 instead of 휆2 for the scaling in equation (7.14).

7.3 Quantum Critical Behaviour in (2 + 1)D Conformal Field Theory

In a first application we study the critical behaviour of our variationally optimised iPEPS tensors for a quantum critical point in the (2 + 1)D Ising universality class. In the following we call this universality class described by a (2 + 1)D, Lorentz-invariant CFT the 3D Ising CFT. Note that this critical behaviour is distinct from the one observed in the so-called Ising- PEPS [192], which amounts to promoting the thermal partition function of the 2D classical Ising model into a two-dimensional quantum many-body wave function in PEPS form with bond dimension 퐷 = 2. This wave function can be parametrised by the temperature 푇 entering the partition function, and at 푇 = 푇푐 describes a critical wave function with algebraically decaying correlation functions. However, the critical properties of this wave function are described by the 2D Ising CFT.

7.3.1 Overview The Hamiltonian studied in this section is the transverse-field Ising model on an infinite square lattice with an additional longitudinal magnetic field, 푧 푧 푥 푧 ℋ TFI = −퐽 ∑ 휎푗 휎푘 − ℎ ∑ 휎푗 − ℎ푧 ∑ 휎푗 , (7.15) ⟨푗,푘⟩ 푗 푗

108 7.3 Quantum Critical Behaviour in (2 + 1)D Conformal Field Theory

ℎ푧

푚푧 > 0 ↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑

(퐷=3) (퐷=2) ℎ푐 ℎ푐 ℎ푐 ℎ

푚푧 < 0 ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓

Figure 7.1: Phase diagram of the ferromagnetic (퐽 = 1) transverse-field Ising model with an ad- (퐷) ditional longitudinal field ℎ푧. The thick, coloured lines for ℎ < ℎ푐 or ℎ < ℎ푐 highlight the horizontal extent of the spontaneous symmetry-breaking line at ℎ푧 = 0.

Exponent 푑 = 2 푑 = 3 푑 = 4

Δ휎 1/8 0.518 148 9(10) 1 Δ휖 1 1.412 625(10) 2

휈휎 = 1/(푑 − Δ휎 ) 8/15 0.402 925 0(2) 1/3 휈휖 = 1/(푑 − Δ휖) 1 0.629 970(4) 1/2 훼휎 = Δ휖 × 휈휎 8/15 0.569 182(4) 2/3 훼휖 = Δ휖 × 휈휖 1 0.889 91(1) 1 훽휎 = Δ휎 × 휈휎 1/15 0.208 775 1(5) 1/3 훽휖 = Δ휎 × 휈휖 1/8 0.326 418(2) 1/2

Table 7.1: Upper two rows: Relevant scaling dimensions of the Ising CFT in 푑 ∈ {2, 3, 4} space- time dimensions; results for 푑 = 3 are from the most recent conformal bootstrap study [251]. The lower six rows denote the additional critical exponents probed in this work, which are derived from the scaling dimensions above.

109 Chapter 7 Finite Correlation Length Scaling in Lorentz-Invariant Gapless iPEPS Wave Functions

0.40 0.6

0.35 0.5 0.30 0.4 0.25 푧 푧

푚 0.20 푚 0.3

0.15 0.2 0.10 0.1 0.05 (a) (c) 0.00 0.0 3.000 3.025 3.050 3.075 3.100 0.00 0.05 0.10 0.15 ℎ ℎ푧 0.92 0.92 퐷 = 2 퐷 = 3 0.91 0.90

0.88 0.90 0.86

푥 퐷 = 2 푥

푚 0.89 푚 (퐷) ℎ < ℎ푐 0.84 (퐷) 0.88 ℎ > ℎ푐 퐷 = 3 0.82 (퐷) ℎ < ℎ푐 0.87 (퐷) ℎ > ℎ푐 0.80 (b) ℎ푐 (d) 0.86 0.78 3.000 3.025 3.050 3.075 3.100 0.00 0.05 0.10 0.15 ℎ ℎ푧

푧 푥 Figure 7.2: iPEPS results for the magnetisations per site 푚푧 = |⟨휎푗 ⟩| and 푚푥 = ⟨휎푗 ⟩ as a function of ℎ (ℎ푧 = 0) in (a) and (b) and as a function of ℎ푧 (ℎ = ℎ푐) in (c) and (d). Result for iPEPS bond dimension 퐷 ∈ {2, 3} are shown. The vertical dashed lines indicate (퐷) the critical values of ℎ푐 for both 퐷 values, as well as the literature reference for ℎ푐 [252].

110 7.3 Quantum Critical Behaviour in (2 + 1)D Conformal Field Theory where 퐽 = 1 denotes the strength of the ferromagnetic Ising interactions and sets the energy scale, ℎ denotes the transverse field while ℎ푧 parametrises the longitudinal magnetic field. The phase diagram of this model is sketched in figure 7.1. For ℎ푧 = 0 the model has a ℤ2 spin- inversion symmetry and features a quantum phase transition at ℎ/퐽 = ℎ푐 = 3.044 38(2) [252], 푧 which separates a paramagnetic phase with 푚푧 ≡ ⟨휎푗 ⟩ = 0 for ℎ > ℎ푐 from a symmetry broken phase with 푚푧 ≠ 0 for ℎ < ℎ푐. The quantum critical point at ℎ = ℎ푐, ℎ푧 = 0 is described by the 3D Ising CFT. For all finite ℎ푧 ≠ 0 there is no critical behaviour and the 푧-magnetisation 푚푧 is finite as a response to the finite ℎ푧. In the entire phase diagram with ℎ ≠ 0, the transverse 푥 푥-magnetization 푚푥 ≡ ⟨휎푗 ⟩ is finite. In the following we explore the physics in the vicinity of the critical point ℎ = ℎ푐, ℎ푧 = 0 using iPEPS simulations. In order to provide field theoretical predictions to compare with, we briefly review the properties of the 3D Ising CFT for our purposes. The 3D Ising CFT has two relevant perturbations, the 푂휎 and the 푂휖 field. Their scaling dimensions Δ휎 and Δ휖 are given in table 7.1. They clearly differ in 2D and 3D, which leads to distinct critical behaviour. In the transverse-field Ising model it is expected that in the continuum limit we can identify 푧 푥 휎푗 ∼ 푂휎 (퐫푗), while 휎푗 ∼ 푂휖(퐫푗) + const. We perturb a general CFT in 푑 space-time dimensions with a relevant perturbation 휙 with scaling dimension Δ휙:

푑−1 ℋ (휆) = ℋCFT + 휆 ∫ d푥 휙. (7.16)

1/(푑−Δ휙) Since the perturbation is relevant, i. e. Δ휙 < 푑, it opens a mass gap proportional to 휆 ; respectively, it leads to a finite correlation length 휉 ∼ 휆−1/(푑−Δ휙) ≡ 휆−휈휙 . In the transverse-field Ising model it is understood that the microscopic coupling ℎ − ℎ푐 couples to the field 푂휖, while ℎ푧 couples to the field 푂휎 . The 3D Ising CFT has two relevant perturbations, which translate into the two correlation length exponents, 휈휖 = 1/(푑 − Δ휖) and 휈휎 = 1/(푑 − Δ휎 ), which appear in the scaling relations −휈 −휈 휉 ∼ |ℎ − ℎ푐| 휖 and 휉 ∼ |ℎ푧| 휎 . We are in a situation where the perturbed theory can display magnetisations, and we thus define critical exponents 훼휙, 훽휙 which describe how the magnetisations grow as a function of the perturbing coupling for a perturbation 휙. 훼휙 is the exponent we use when measuring 푥 푧 푚푥,푐 − ⟨휎푗 ⟩ ∼ ⟨푂휖⟩, while we use 훽휙 for the ⟨휎푗 ⟩ ∼ ⟨푂휎 ⟩ observable. The subscript 휎, 휖 denotes the perturbing field as for the correlation length exponents before. For the transverse mag- 훼휙 netisation we expect |푚푥,푐 − 푚푥 | ∼ |휆| , while for the longitudinal magnetisation we expect 훽휙 푚푧 ∼ |휆| . The definitions and values for these exponents can be found intable 7.1. Some of the exponents we introduce here for clarity reasons are also more commonly known as 훽 ≡ 훽휖, 훿 ≡ 1/훽휎 , and 휈 ≡ 휈휖 in the statistical mechanics literature. Note that the 푑 = 4 critical exponents are equivalent to the mean-field exponents. This is because 푑 = 4 is the upper critical dimension for the Ising criticality, i. e. the dimension where mean-field theory becomes exact.

7.3.2 iPEPS Results for the Transverse-Field Ising Model

The transverse-field Ising model (with ℎ푧 = 0) has been studied extensively with both finite- size PEPSs and iPEPSs approaches in the past [81, 103, 179, 181, 187, 205, 253–255]. Variational

111 Chapter 7 Finite Correlation Length Scaling in Lorentz-Invariant Gapless iPEPS Wave Functions

100 100 퐷 = 2 (a) 퐷 = 2 (c) 퐷 = 3 퐷 = 3 훽 훽 6 × 10−1 휀 6 × 10−1 휎

4 × 10−1 4 × 10−1 푧 푧

푚 3 × 10−1 0.3 푚 3 × 10−1 0.2 ) ) 푧 ) 푧 − ℎ) 푧 푐 (푚 (ℎ 0.2 (푚 −1 (ℎ −1 0.1 log log 2 × 10 2 × 10 log 휕 log 휕 휕 휕 0.1 0.0 10−3 100 10−5 10−1 ℎ − ℎ ℎ 10−1 푐 10−1 푧 10−4 10−2 100 102 10−5 10−2 101 ℎ푐 − ℎ ℎ푧 100 100 퐷 = 2 (b) 퐷 = 2 (d) 퐷 = 3 퐷 = 3 훽휀 훼휎 (푀퐹) −1 훽휀 10 | 푥 푧 −1 0.6 − 푚

푚 10 −2 | 0.6 푥

푥,푐 10 ) − ℎ) 푧 ) |푚 푧 − 푚 (푚 (퐷) 푐 0.4 0.4 (ℎ 푥,푐 (ℎ log |푚 log 휕

log 0.2 휕 휕 0.2 −3 log

10 휕 0.0 10−3 100 10−5 10−1 (퐷) ℎ − ℎ ℎ푧 10−2 푐 10−4 10−2 100 10−5 10−2 101 (퐷) ℎ ℎ푐 − ℎ 푧

Figure 7.3: Analysis of the critical behaviour of the 푚푧 and 푚푥 magnetisations in the vicinity of the critical point at ℎ = ℎ푐, ℎ푧 = 0. (a) 푚푧 as a function of ℎ푐 − ℎ; inset shows (퐷) good convergence to the expected 훽휖 exponent. (b) 푚푧 as a function of ℎ푐 − ℎ; inset shows crossover to mean-field behaviour. (c) 푚푧 as a function of ℎ푧; inset shows good convergence to the expected 훽휎 exponent. (d) |푚푥 − 푚푥,푐| as a function of ℎ푧; inset shows good convergence to the expected 훼휎 exponent. The expected exponents are tabulated in table 7.1.

112 7.3 Quantum Critical Behaviour in (2 + 1)D Conformal Field Theory tensor network approaches have also been used to study the classical 3D Ising model, which is part of the same universality class [256, 257]. A common feature of all these simulations (퐷) is that for ℎ below a bond dimension 퐷 dependent value ℎ푐 the system shows a finite 푧 (퐷) magnetisation 푚푧. However, in the past the values of ℎ푐 and the functional behaviour of 푚푧 in its vicinity did depend significantly on the tensor optimisation methodology. We believe the newest generation of optimisation algorithms put forward in references [81, 198] does not suffer from these shortcomings anymore, so that a detailed analysis ofthe intrinsic iPEPS finite- 퐷 behaviour close to the 3D Ising CFT is finally possible.

Magnetisations in the Vicinity of the Critical Point

We start by presenting in figure 7.2 the behaviour of the 푧 and the 푥 magnetisation at ℎ푧 = 0 by varying ℎ in figures 7.2 (a) and (b) for bond dimensions 퐷 = 2 and 퐷 = 3. As previously (퐷) reported, we also observe a 퐷-dependent value ℎ푐 , where 푚푧 vanishes, while 푚푥 displays a (퐷=2) (퐷=3) kink. We find ℎ푐 ≈ 3.0893, while ℎ푐 ≈ 3.0476. Note that the 퐷 = 3 result differs only by about one part per thousand from the reference critical value ℎ푐 = 3.044 38(2) [252]. In an earlier study based on one-dimensional iMPSs for the (1 + 1)D transverse field Ising model, a bond dimension 퐷 > 10 was required to reach a comparable accuracy [253]. We have also tried to optimize 퐷 = 4 tensors, but albeit technically possible, it turns out to be extremely difficult to obtain energies which are lower than our best 퐷 = 3 results so far. We come back to this observation later in this section. Finally in panels figures 7.2 (c) and (d), we display 푚푧 and 푚푥 along an orthogonal cut at fixed ℎ = ℎ푐 with varying ℎ푧 > 0, i.e. along the violet axis in figure 7.1. While the plots in figure 7.2 seem to suggest large differences between 퐷 = 2 and 퐷 = 3 it should be noted that the ℎ and ℎ푧 ranges shown are already quite small. Shown on a scale ℎ ∈ [0, 4] it would be difficult to visually spot the differences between thetwo 퐷 values.

Critical Exponents In a next step we explore the critical behaviour contained in the presented data. In figure 7.3 (a), we plot 푚푧 as a function of ℎ푐 −ℎ on a log-log scale. For comparison we plot a straight line with the expected slope 훽휖 as a guide to the eye. In the corresponding inset we numerically calculate the derivative and find a collapse between the 퐷 = 2 and 퐷 = 3 data at larger distances from the critical point. The 퐷 = 2 running estimate for the critical exponent reaches a maximum of ∼0.29 and then drops to zero as ℎ푐 − ℎ goes to zero. The 퐷 = 3 running estimate rises to ∼0.32 before it also drops to zero as ℎ푐 − ℎ goes to zero. We expect that 퐷 > 3 would get even closer to the expected 훽휖 ≈ 0.326 418(2), before dropping to zero as ℎ푐 − ℎ goes to zero. The drop to zero is clearly due to the finite-퐷 remnant 푚푧 at the thermodynamic value ℎ푐. (퐷) Let us therefore investigate what happens when we analyse the data as a function of ℎ푐 − ℎ instead. figure 7.3 (b) displays the corresponding data. The 퐷 = 2 data clearly shows a limiting (MF) (퐷) mean-field behaviour 훽휖 = 1/2 at small ℎ푐 − ℎ (in the inset), as observed previously in reference [253]. The 퐷 = 3 data shows some hint of an intermediate plateau around the true 3D (MF) (퐷) Ising CFT value for 훽휖 before also crossing over to the mean-field value 훽휖 at small ℎ푐 − ℎ.

113 Chapter 7 Finite Correlation Length Scaling in Lorentz-Invariant Gapless iPEPS Wave Functions

The analysis of the 푚푥 magnetization is less clean and shown in figure 7.11 in appendix 7.A. In figures 7.3 (c) and (d), we present the analogous analysis for both 푚푧 and 푚푥 when staying at ℎ = ℎ푐 while tuning ℎ푧 > 0. The numerical derivatives provide running averages for 훽휎 and 훼휎 which converge nicely towards the expected values as we increase 퐷 from 2 to 3. The maximal values for 퐷 = 3 are only a few percent below the 3D Ising CFT results. So we learn that when the location of the critical point is known beforehand, the critical behaviour can be determined quite accurately already with a surprisingly small bond dimension of 퐷 = 3. In the vicinity of the finite-퐷 critical points we, however, observe mean-field behaviour, and due to the crossover it is more difficult to extract the genuine critical behaviour.

Correlation Lengths After having analysed the critical behaviour of local observables as a function of perturbing couplings, we now investigate the correlation lengths in our optimized iPEPS wave functions in the vicinity of the critical point. In the vicinity and at a quantum critical point in (1 + 1)D represented with a finite bond dimension iMPS, we know that only a finite correlation length can appear. Since an iPEPS is in principle able to represent wave functions with algebraically decaying correlations, i. e. states with infinite correlation lengths [192], it is not a priori clear what to expect in our optimised iPEPS wave functions. Let us note first that the correlation lengths for 퐷 = 1 (product states) vanish identically, even though the spontaneous (퐷=1) magnetisation shows critical mean-field behaviour at ℎ푐 = 4 (not shown). Based on the technology presented in section 7.2.3, we have determined the largest correlation lengths for 퐷 = 2 and 퐷 = 3 iPEPSs along the previously investigated cuts in the (ℎ, ℎ푧) plane. The results are shown in figures 7.4 (a) and (c). The observed correlation lengths for 퐷 = 2 do not grow (퐷=2) (퐷=2) beyond 휉max ≈ 1.67 lattice spacings, and they reach their maximum at ℎ = ℎ푐 , ℎ푧 = 0. We are quite confident that this short correlation length is not an artefact of incomplete optimisation, but is a genuine feature of optimised, translationally invariant, finite-퐷 iPEPS wave functions for Lorentz-invariant quantum critical systems with a 3D space-time description. For 퐷 = 3 we also observe finite correlation lengths, but now the maximum is substantially larger: (퐷=3) (퐷=3) 1 휉max ≈ 12.2 at ℎ = ℎ푐 , ℎ푧 = 0 . So both 퐷 values seem to indicate that our variational optima feature a finite correlation length. This is one of the key results of this paper, whose possible origin we are going to discuss later. We do, however, show in the following that the finite correlation length is also a blessing, as it helps us to understand and organize thefinite-퐷 effects in field theoretically motivated formulae basedon 휉(퐷). Before doing this, let us investigate the functional behaviour of the correlation length in the −휈 vicinity of the critical point. Depending on the cut in parameter space, we expect 휉 ∼ |ℎ푐 −ℎ| 휖 −휈 or 휉 ∼ |ℎ푧| 휎 , with the values of 휈휖 and 휈휎 given in table 7.1. Indeed, the data in figures 7.4 (b) and (d) shown on a log-log scale are (roughly) compatible with the expected correlation length exponents in some intermediate window of the couplings. This is expected since far away from the critical point we are outside the quantum critical regime, while very close to the critical point 휉 saturates. Still the agreement for the ℎ푧 detuning is much better than the |ℎ푐 −ℎ|

1Since the energy optimisation of states with such a large correlation length is computationally quite demanding, (퐷=3) there is a small uncertainty in the value of 휉max .

114 7.3 Quantum Critical Behaviour in (2 + 1)D Conformal Field Theory

14.0 (a) (b) 12.0 23 10.0 8.0 6.0 21 휉 휉 4.0 퐷 = 3 2.0 퐷 = 2 −1 (퐷) 2 ℎ < ℎ푐 1.8 (퐷) (퐷) ℎ < ℎ푐 ℎ > ℎ푐 1.6 (퐷) 1.4 ℎ > ℎ푐 −휈휀 1.2 2−3 3.0 3.02 3.04 3.06 3.08 3.1 10−4 10−3 10−2 10−1 100 101 ℎ (퐷) |ℎ푐 − ℎ|

10.0 (c) (d) 23 7.5 1 퐷 = 2 2 휉 5.0 퐷 = 3 휉 2−1 퐷 = 2 2.5 퐷 = 3 −휈휎 0.0 2−3 10−5 10−4 10−3 10−2 10−1 100 101 10−5 10−4 10−3 10−2 10−1 100 101 ℎ푧 ℎ푧 100 100 (e) (f) 6 × 10−1 | −1 −1 퐷 = 2 푥 10 4 × 10 (퐷) 푧

ℎ < ℎ − 푚 −1 푐 푚 3 × 10 ℎ푧 > 0 (퐷) 퐷 = 3 푥,푐 −2 (퐷) −1 (퐷) |푚 10 2 × 10 ℎ < ℎ푐 ℎ > ℎ푐 (퐷) ℎ푧 > 0 ℎ > ℎ푐 −Δ휎 −Δ휀 10−1 10−3 2−4 2−3 2−2 2−1 20 21 22 23 24 2−3 2−2 2−1 20 21 22 23 24 휉 휉

Figure 7.4: Correlation lengths of the variationally optimised iPEPSs for 퐷 = 2 and 퐷 = 3. (a) Correlation lengths as a function of the transverse field ℎ. (b) Correlation lengths as (퐷) function of |ℎ푐 − ℎ| on a log-log scale, including the theoretically expected slope −휈휖. (c) Correlation lengths as a function of the longitudinal field |ℎ푧|. (d) Correla- tion lengths as function of |ℎ푧| on a log-log scale, including the theoretically expec- (퐷) ted slope −휈휎 . Panels (e) and (f) display the expectation value of 푚푧 and |푚푥,푐 − 푚푥 |, respectively, as a function of the correlation length 휉, on a logarithmic scale. The expected slopes in the two cases are directly the (negated) scaling dimensions −Δ휎 and −Δ휖. The values of the exponents and scaling dimensions can be found in table 7.1.

115 Chapter 7 Finite Correlation Length Scaling in Lorentz-Invariant Gapless iPEPS Wave Functions

101 퐷 = 2 퐷 = 3 (퐷) (퐷) 100 ℎ < ℎ푐 ℎ < ℎ푐 (퐷) (퐷) ℎ > ℎ푐 ℎ > ℎ푐 ℎ > 0 ℎ > 0 10−1 푧 푧 ,∞)

−2 iPEPS

( 10 − 푒

,퐷) 10−3 −3.2336 iPEPS ( 푐 푒 ,퐷) 10−4 −3.2338

iPEPS −3.2340 ( 푒 −3.2342 10−5 0.0 0.1 0.2 0.3 1/휉 3 10−6 10−4 10−3 10−2 10−1 100 101 102 103 1/휉 3

(iPEPS,퐷) (iPEPS,∞) Figure 7.5: Variational excess critical energy density Δ푒푐 = 푒c − 푒c of transverse- field Ising iPEPS wave functions optimised for various (ℎ, ℎ푧) as a function of their 1/휉 3. The variational excess energy density corroborates the advocated 1/휉 3 scal- (iPEPS,∞) (iPEPS) ing. Inset: Fit according to equation (7.18), yielding 푒c and 훼3d Ising CFT in (7.19). detuning. Finally we plot the expectation values of the two magnetisations 푚푧 and |푚푥,푐 − 푚푥 | as a function of the measured correlation length 휉 (for both parameter cuts) in panels figures 7.4 (e) and (f). As discussed earlier, we expect this relation to be governed by the scaling dimensions Δ휎 and Δ휖, respectively. While the 퐷 = 2 results do not match too well, the 퐷 = 3 results for both parameter cuts and both observables are in good agreement with the expected scaling dimensions. Even though it seems that the critical exponents and scaling dimensions can be obtained more precisely based on the observables as a function of the couplings than of the correlation lengths, it is nevertheless rewarding to observe that the correlation lengths are also following the expected quantum critical behaviour with increasing 퐷, within the stated limitations.

Critical Energy Convergence

We now investigate the energy convergence of the TFI model at its critical point ℎ = ℎ푐, ℎ푧 = 0 for increasing bond dimension 퐷. It is one of the open problems in practical iPEPS calculations

116 7.3 Quantum Critical Behaviour in (2 + 1)D Conformal Field Theory to understand the convergence of energies as a function of 퐷. Here we advocate that the variational energy of an optimised iPEPS tensor at the critical point ℎ = ℎ푐, ℎ푧 = 0 can be understood as a particular type of a Casimir effect controlled by the correlation length 휉. It is well known that the ground state energy density 푒 = 퐸/푁sites of a 3D quantum critical system in a torus geometry with modular parameter 흉 [88] is given as (흉) 훼QCP × 푣 푒(퐿) = 푒(∞) − + … , (7.17) 퐿3 where 퐿 denotes the linear extent of the torus, 푣 is the ‘speed of light’, e. g. the critical spin wave (흉) velocity in a TFI model, and 훼QCP is a 흉-dependent Casimir amplitude which otherwise depends only on the universality class of the quantum critical point (QCP); for example, for the 3D Ising 2 (흉=퐢) CFT and a square torus (흉 = 퐢) with periodic boundary conditions, 훼3d Ising CFT = +0.35(2) according to reference [258]. We now postulate that our iPEPS setup can be considered as a new, distinct geometry with (iPEPS) its own Casimir amplitude, 훼3d Ising CFT, where, however, the length of the torus is replaced by the correlation length 휉, such that (iPEPS) 훼3d Ising CFT × 푣 푒(휉) = 푒(∞) − + … (7.18) 휉 3 We stress that this ansatz is in agreement with the expected scaling behaviour of the one-point function of the stress-energy tensor 푇 , whose scaling dimension in 푑 = 3 is Δ푇 = 3. In the past a similar scaling with a 1/휉 2 dependence has been suggested and numerically verified in the (1 + 1)D iMPS context in reference [135]. Since we only have two values of 퐷, it is a priori hard to determine the validity of the postulated energy convergence form. Let us nevertheless use the best variational energies at ℎ = ℎ푐, ℎ푧 = 0 for 퐷 = 2 and 퐷 = 3, together with the literature value of 푣 = 3.323(33) [88] to estimate (iPEPS) (iPEPS,∞) 훼3d Ising CFT ≈ −0.000 61, 푒TFI ≈ −3.234 262 3. (7.19) If correct, the scaling hypothesis in equation (7.18) combined with the very small iPEPS Casimir amplitude would explain the spectacular accuracy of the 퐷 = 3 results. The 퐷 = 3 correlation length of beyond 10 lattice sites gives an energy correction proportional to 1/휉 3 ≲ 10−3, while (iPEPS) the Casimir amplitude 훼3d Ising CFT is itself three orders of magnitude smaller than the square torus amplitude. So multiplying these two factors we are led to conjecture that the extrapolated iPEPS energies are accurate to about 10−6 already at 퐷 = 3. This result might explain why 퐷 = 4 simulations are so demanding, as the expected remaining energy gains are tiny and are accompanied with wave functions bestowed with large correlation lengths, which are even harder to contract accurately. In order to corroborate the advocated scaling ansatz in equation (7.18), we plot the critical, (iPEPS,퐷) i. e. evaluated at ℎ = ℎ푐, ℎ푧 = 0, variational energies 푒c of all available iPEPS wave functions in a common plot, cf. figure 7.5. In the inset we show the fit according to equation (7.18)

2A square torus is a quadratic system with periodic boundary conditions in both orthogonal spatial directions. For details on the definition of the modular parameter, see reference88 [ ].

117 Chapter 7 Finite Correlation Length Scaling in Lorentz-Invariant Gapless iPEPS Wave Functions of the two data points with the best 퐷 = 2 and 퐷 = 3 critical energies. The data in the main 3 plot is seen to approximately scale according to Δ푒푐 ∝ 1/휉 over several orders of magnitude, thus providing nontrivial empirical evidence in favour of the validity of the ansatz in equation (7.18) 3.

7.4 Continuous Symmetry Breaking

In this section we want to explore the properties of iPEPS wave functions as they represent or approximate quantum many-body states which exhibit continuous symmetry breaking in (2+1) space-time dimensions. This is a rather ubiquitous phenomenon, ranging from magnetic order in 푂(푁 ) symmetric quantum magnets with 푁 ≥ 2, to bosonic and fermionic superflu- ids, to superconductors and Goldstone phases in high-energy physics. In these systems the continuous symmetry is spontaneously broken, accompanied by the the appearance of a finite order parameter |M| > 0. Another hallmark is the required presence of gapless excitations (so-called Goldstone modes), which are the soft long-wavelength modes of order parameter variations. We study the 푆 = 1/2 Heisenberg antiferromagnet and the 푆 = 1/2 XY model, both on the square lattice, as paradigmatic examples for 푂(3) and 푂(2) continuous symmetry breaking with a three- and two-component vector order parameter.

7.4.1 Overview The field theoretical description of collinear magnetic order in 푂(푁 ) quantum magnets relies often on a quantum nonlinear sigma model (NLSM) formulation, or on a description in terms of the symmetry-breaking phase in an 푁 -component interacting 휙4 theory. The gapless Gold- stone modes are also known as spin waves in the magnetic context, and taken in isolation they behave as a collection of free massless scalar fields. These gapless modes are responsible e. g. for the algebraic decay of spin-spin correlations to their limiting value and the finite-size corrections of the energy or the order parameter. These linearly dispersing modes require a (2+1)D Lorentz-invariant description at low energies, similar to the Ising CFT discussed before. The quantum nonlinear 푂(푁 ) sigma model is described in detail in reference [259]. For our purpose it is sufficient to know that this is basically a hydrodynamic theory of quantum magnets with collinear order. Its description, being hydrodynamic, relies only on a handful of effective parameters entering the description: the spin stiffness 휌푠, the spin wave velocity 푣, the transverse susceptibility 휒⟂, and the squared order parameter in the thermodynamic limit 2 2 푚 (∞). The first three parameters are actually related via 푣 = 휌푠/휒⟂. Similar to the finite-size corrections to the ground state energy discussed in the quantum critical context, the finite- size corrections to the ground state energy and the order parameter have been derived for the quantum non-linear 푂(푁 ) sigma model in references [10, 260–263]. The finite-size corrections to the ground state energy 푒 = 퐸/푁sites in 푑 = 3 are as follows:

(shape/bc) 푁 − 1 1 (푁 − 1)(푁 − 2) 푣2 1 푒(퐿) = 푒(∞) − 훼 푣 + + 풪 . (7.20) [ NLSM ( ) ] 3 4 ( 5 ) 2 퐿 8 휌푠퐿 퐿 3We actually believe that the data in Fig. 7.5 ultimately organises in several families with different Casimir amplitudes, but all sharing a 1/휉 3 scaling.

118 7.4 Continuous Symmetry Breaking

−0.6675 ) −2 QMC

( 10

−0.6680 − 푒 10−3 ,퐷) 10−4 iPEPS ( 푒 10−2 10−1 100 푒 −0.6685 1/휉 3 퐷 = 2 퐷 = 3 퐷 = 4 −0.6690 퐷 = 5 fit QMC −0.6695 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 1/휉 3

퐷 = 2 0.60 퐷 = 3 퐷 = 4 퐷 = 5 0.55 2

| fit QMC ⟨흈⟩ |

= 0.50 2 푚 0.45

0.40

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1/휉

Figure 7.6: 푆 = 1/2 Heisenberg antiferromagnet: iPEPS data for the ground state energy per site 푒 (upper panel) and the order parameter squared 푚2 (lower panel). We plot the data as a function of the expected 1/휉 3 (for 푒) and 1/휉 (for 푚2) dependence. The linear fits to the 퐷 = 3, 4, 5 results in the upper and all 퐷 ≥ 2 in the lower panel extrapolate closely to the high-precision QMC reference results [10]. The inset in the upper panel highlights the overall 1/휉 3 convergence of the energy per site.

119 Chapter 7 Finite Correlation Length Scaling in Lorentz-Invariant Gapless iPEPS Wave Functions

(흉=퐢) For a square torus with periodic boundary conditions, 훼NLSM ≈ 1.437 745 has been obtained. The finite-size correction for the magnetic order parameter squared are as follows: 2 푚 (퐿) (shape/bc) 푁 − 1 푣 1 1 = 1 + 휇 + 풪 . (7.21) 2 [ NLSM ( ) ] ( 2 ) 푚 (∞) 2 휌푠 퐿 퐿 (흉=퐢) For a square torus with periodic boundary conditions, 휇NLSLM ≈ 0.620 75 has been found. As we see below, our variationally optimised iPEPS wave functions have a finite correlation length 휉, which depends on the model and on the bond dimension 퐷. We now conjecture the following 푑 = 3, finite-휉 corrections for the ground state energy density 푒 and the magnetic order parameter squared 푚2:

푁 − 1 1 1 푒(휉) = 푒(∞) − [훼(iPEPS) ( ) 푣] + 풪 ( ) (7.22) NLSM 2 휉 3 휉 4 and 푚2(휉) 푁 − 1 푣 1 1 = 1 + 휇(iPEPS) + 풪 . (7.23) 2 [ NLSM ( ) ] ( 2 ) 푚 (∞) 2 휌푠 휉 휉 The potential power of these formulae lies in the fact that one can extrapolate the results at finite 휉(퐷) to the limit 휉 → ∞ based on iPEPS data fits to the above formulae. Furthermore, itis possible to predict the finite-size corrections for other microscopic models, once the ‘universal’ (iPEPS) (iPEPS) values of 훼NLSM and 휇NLSM are determined. The knowledge of 푁 , 푣, and 푣/휌푠 allows us then to quantitatively predict the slope of the finite-size corrections. Conversely, it might become possible to estimate 푣 and 푣/휌푠 based on precise iPEPS data for a model with a known value for 푁 .

7.4.2 푆 = 1/2 Antiferromagnetic Heisenberg Model

The 푆 = 1/2 antiferromagnetic Heisenberg model has also been studied frequently using finite- size PEPS and iPEPS approaches in the past [81, 181, 198, 202, 205, 265, 266]. The ground state of this model has antiferromagnetic Néel order, which breaks the continuous 푂(푁 =3) rotation symmetry spontaneously down to a residual 푂(2) symmetry. The presence of 푁 − 1 = 2 Goldstone modes leads to an algebraic decay of the two-spin correlation function. The Hamiltonian is defined as

푥 푥 푦 푦 푧 푧 ℋ HB = 퐽 ∑ (푆푗 푆푘 + 푆푗 푆푘 + 푆푗 푆푘 ) , (7.24) ⟨푗,푘⟩

훼 with 퐽 = 1 and the 푆푗 are spin-1/2 operators. In order to be able to work with a single-site unit cell, we perform a spin rotation on one Néel sublattice, which negates the sign of the 푦 and 푧 parts of the interactions in the actual calculations. We proceed optimising the variational energies of iPEPS tensors for 퐷 ∈ {2, 3, 4, 5}. Then 2 2 we measure the order parameter squared 푚 = |⟨흈푗⟩| , i. e. the maximum possible for the order parameter squared amounts to one, as well as the correlation length 휉. These correlation lengths are finite and range from 휉(퐷 = 2) ≈ 0.7 to 휉(퐷 = 5) ≈ 4.5. The correlation lengths are thus substantially smaller than those of the critical transverse-field Ising model at 퐷 = 3.

120 7.4 Continuous Symmetry Breaking

−0.548700 ) 10−2 QMC −0.548725 ( 10−3 − 푒 10−4 ,퐷) 10−5 −0.548750 −6 iPEPS 10 ( 푒 10−3 10−2 10−1 100 푒 퐷 = 2 −0.548775 1/휉 3 퐷 = 3 퐷 = 4 −0.548800 퐷 = 5 fit NLSM prediction −0.548825 QMC

0.000 0.005 0.010 0.015 0.020 0.025 0.030 1/휉 3

퐷 = 2 퐷 = 3 0.82 퐷 = 4 퐷 = 5 2

| fit 0.80 NLSM prediction ⟨흈⟩ | QMC = 2 푚 0.78

0.76

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 1/휉

Figure 7.7: 푆 = 1/2 XY ferromagnet: iPEPS data for the ground state energy per site 푒 (upper panel) and the order parameter squared 푚2 (lower panel). We plot the data as a function of the expected 1/휉 3 (for 푒) and 1/휉 (for 푚2) dependence. The linear fits to the 퐷 = 4, 5 results in the upper and all 퐷 ≥ 2 in the lower panel extrapolate reasonably closely to the QMC results [264], and may well be more accurate than the somewhat antiquated QMC results. The inset in the upper panel highlights the overall 1/휉 3 convergence of the energy per site. In both panels we include a prediction based on the conjectured finite-휉 formulae for the NLSM, cf. equations (7.22) and (7.23) and main text.

121 Chapter 7 Finite Correlation Length Scaling in Lorentz-Invariant Gapless iPEPS Wave Functions

In figure 7.6, we present the energy per site 푒 as function of 1/휉 3 in the upper panel and 푚2 as a function of 1/휉 in the lower panel. This is the conjectured 휉 scaling form of equations (7.22) and (7.23). It is striking that for both observables a linear fit leads to accurate extrapolations to the limit 휉 → ∞, when compared to high-precision quantum Monte Carlo (QMC) reference (QMC) 2(QMC) values [10, 267], 푒HB = −0.669 437(5) and 푚 HB = 0.3770(8). We fit the largest three 퐷 values for the energy 푒 and all the 퐷 values for 푚2 and obtain the following iPEPS 휉 → ∞ (iPEPS) 2(iPEPS) estimates: 푒HB = −0.669 42(2) and 푚 HB = 0.380(4). Using the iPEPS fit slopes, the value 푁 = 3, and the known QMC values of the hydrodynamic parameters 푣 and 푣/휌푠 [10, 268], we can then proceed to determine

(iPEPS) (iPEPS) 훼NLSM ≈ −0.0029 and 휇NLSM ≈ 0.045. (7.25)

Note that similar to the Ising Casimir amplitude, the NLSM energy Casimir amplitude is three orders of magnitude smaller than the square torus one, highlighting that an iPEPS calculation at a certain 휉 should be considered three orders of magnitude more accurate than a square torus with 퐿 ∼ 휉. The iPEPS order parameter amplitude is however only one order of magnitude smaller than the square torus result.

7.4.3 푆 = 1/2 XY model

The 푆 = 1/2 XY model has also been investigated with PEPSs and iPEPSs in the past [81, 265, 269]. The ground state of this model has ferromagnetic order in the 푥-푦 spin plane, which breaks the continuous 푂(푁 =2) in-plane rotation symmetry spontaneously down to a residual discrete ℤ2 symmetry. The presence of 푁 − 1 = 1 Goldstone mode leads to an algebraic decay of the two-spin correlation function. The Hamiltonian is defined as

푥 푥 푦 푦 ℋ XY = −퐽 ∑ (푆푗 푆푘 + 푆푗 푆푘 ) , (7.26) ⟨푗,푘⟩ with 퐽 = 1. Note that for this model the two choices of the sign of 퐽 can be mapped into each other. Since we want to use a single-site iPEPS unit cell, we adopt the ferromagnetic sign convention. We proceed optimising the variational energies of iPEPS tensors for 퐷 ∈ {2, 3, 4, 5}. Then 2 2 we measure the order parameter squared 푚 = |⟨흈푗⟩| , i. e. the maximum possible for the order parameter squared amounts to one, as well as the correlation length 휉. These correlation lengths are again finite and range from 휉(퐷 = 2) ≈ 1.3 to 휉(퐷 = 5) ≈ 8.4. The correlation lengths for each 퐷 are roughly a factor two larger than for the Heisenberg model. So it seems that the smaller number of Goldstone modes has a beneficial effect on the growth ofthe correlation lengths. In figure 7.7, we present the energy per site 푒 as a function of 1/휉 3 in the upper panel and 푚2 as a function of 1/휉 in the lower panel. This is the conjectured 휉 scaling form of equations (7.22) and (7.23). We then fit the largest two 퐷 values for the energy 푒 and all the 퐷 values for 푚2 and

122 7.4 Continuous Symmetry Breaking

4 12 2 TFI@ℎ푐 (퐷) 23 TFI@ℎ푐 XY 22

10 휉 HB 21 20 8 2−1 2 3 4 5

휉 6

0 1 2 3 4 5 퐷

Figure 7.8: Overview of the finite-퐷 iPEPS correlation lengths 휉 observed in the critical TFI model and the continuous symmetry-breaking XY and Heisenberg models. The inset displays the same data on logarithmic scales.

(iPEPS) 4 2(iPEPS) obtain the following iPEPS 휉 → ∞ estimates: 푒XY ≈ −0.548 822 and 푚 XY = 0.7567(11). (QMC) These values compare well with the QMC results of reference [264], 푒XY = −0.548 824(2) 2(QMC) and 푚 XY = 0.764(7). Since the QMC results are relatively antiquated, it is not inconceivable that the iPEPS results are actually more precise than the QMC results. We are now also in a position to corroborate the 푂(푁 ) universality of the conjectured NLSM finite-휉 corrections. Using the hydrodynamic parameters 푣 and 푣/휌푠 from reference [264], the iPEPS amplitudes from equation (7.25) and inserting 푁 = 2, we arrive at the NLSM predictions, which are shown by solid lines in both panels of figure 7.7. The nice agreement between the linear fits and the NLSM predictions for the slopes provides further support for the validityand therefore power of the field theoretically inspired finite-휉 correction formulae.

4 (iPEPS) The extrapolation with 퐷 = 4 and 퐷 = 5 alone gives an energy density of 푒XY ≈ −0.548 822. However, from this extrapolation we cannot extract a numerical error.

123 Chapter 7 Finite Correlation Length Scaling in Lorentz-Invariant Gapless iPEPS Wave Functions

7.5 Discussion and Interpretation

After having studied the three different models we are confronted with the fact thatinall cases the correlation length 휉(퐷) was finite. While we have developed a powerful finite-휉 scaling framework, where many observables, including energies and order parameters, can be analysed and extrapolated to 휉 → ∞, we are still left in the dark both regarding the underlying origin of the finite 휉(퐷) in the first place and regarding the functional dependence of 휉 on 퐷 for a given model or universality class. While we do not yet have compelling answers to both questions, we can at least try to shed as much light as possible based on our numerical data. In figure 7.8, we have assembled the correlation lengths observed in the critical transverse-field Ising model, which is described by one of the simplest nontrivial 3D CFTs, together with two instances of 푂(푁 ) continuous symmetry-breaking phenomena with 푁 = 3 for the 푆 = 1/2 antiferromagnetic Heisenberg model and 푁 = 2 for the 푆 = 1/2 XY model. In the inset we also show the 퐷 ≥ 2 data in a log-log plot, yielding some rough estimates for a putative power-law relation 휉(퐷) ∼ 퐷휅 (it is not clear that such a law holds) [270]. With the two points for the critical transverse- field Ising model, we obtain 휅(3D Ising CFT) ≈ 5, while both continuous symmetry-breaking cases seem to share the same 휅NLSM ≈ 2. The latter two cases however differ by a factor 2 in the prefactor, which incidentally is also the inverse ratio of the number of Goldstone modes. This could mean that the XY model has twice as large a correlation length as the Heisenberg model because it only has half the number of Goldstone modes. It would be interesting to explore whether the known additive logarithmic correction to the entanglement area law in continuous symmetry breaking states [271] might be at the origin of this behaviour. There the prefactor of the log contribution is proportional to the number of Goldstone modes. This would also explain why the speculative values of 휅 are so different between the 3D Ising CFT and the symmetry-breaking cases. In terms of their low-energy degrees of freedom, the 3D Ising CFT and the XY model both contain a single real scalar field each – an interacting field in the Ising CFT case and a massless free field in the XY case. One could thus have expected the values of 휅 to be roughly similar. The crucial difference therefore seems to come from the broken continuous symmetry in the XY model, which is absent in the Ising CFT. The most pressing question remains, however, as to why we only find finite correlation lengths in the variationally optimised iPEPS wave functions for massless Lorentz-invariant (2 + 1)D scenarios. We believe that our observations are actually the generic result, and that the previously known examples of iPEPSs with algebraic correlations are fine-tuned and nongeneric. As shown as an illustration in the left-hand panel of figure 7.9, we think of our (2+1)D iPEPSs as wave functions whose correlation functions are represented by a path integral with a finite 휉휏 (퐷) extent in the (real or imaginary) time direction. The Lorentz invariance (or Euc- lidian invariance after a Wick rotation) of the fixed point we try to approximate then forcesthe spatial correlation lengths to be finite as well. The well-known iPEPSs with algebraic spatial correlations at finite 퐷 can actually be represented by a purely in-plane path integral, where the temporal extent 휉휏 is basically zero (right-hand panel). This is certainly true for the 2D classical partition function Ising-PEPS [192], quantum dimer Rokhsar-Kivelson states [249, 272], and certain quantum Lifshitz theories [273, 274], where there is a built-in space-time asymmetry. Some further evidence supporting this picture might be obtained from the entanglement en-

124 7.6 Conclusions

(2 + 1)D CFT/QFT (2 + 0)D Rokhsar-Kivelson 휏 휏

휉휏 ∼ 휉(퐷) 휉휏 ≈ 0

푦 푦 푥 푥 휉푥,푦 = ∞ 휉푥,푦 ∼ 휉휏

Figure 7.9: Illustration of the space-time volume sampled in a (2+1)D iPEPS versus the (2+0)D Rokhsar-Kivelson-type iPEPS wave functions.

tropies 푆bMPS of the bMPS resulting from the contractions of the iPEPSs, as it seems plausible that this entanglement entropy is amplified if the correlation volume extends into the 휏 direction. The data shown in figure 7.10 can indeed be interpreted that the entanglement entropy grows more rapidly with the correlation length 휉 in the genuine 3D space-time cases, compared to the quantitatively well understood logarithmic scaling of the (2 + 0)D wave functions, here exemplified by the Ising-PEPS at various temperatures 5. It would be interesting to study these bMPS entropies more systematically, as they roughly dictate how large the boundary 휒 value in the contraction schemes has to be. Coming back to the reason for the finite 휉휏 (퐷) in the first place: We speculate that finding the dominant eigenvector of a plane-to-plane transfer operator along the temporal direction combined with a projection to a finite-퐷 iPEPS leads invariably to a finite correlation length in the temporal direction. This view is also supported by a more formal argument stating that an entropic area law does not automatically imply an efficient iPEPS representation275 [ ]. To a first approximation we understand the finite-퐷 iPEPS to mimic a ground state wave function of the field theoretical fixed point perturbed by a certain amount of the most relevant allowed perturbation given the symmetry constraints imposed on the iPEPS. Such states are driven away from the gapless situation, leading to a finite correlation length 휉(퐷). In the examples studied here this always corresponds to a perturbation by coupling to the magnetic order parameter. Imposing symmetry constraints on the tensor might change the nature of the allowed relevant perturbation, and could also affect the values of some of the finite-휉 correction amplitudes introduced in this work.

7.6 Conclusions

In this paper we introduce a powerful finite correlation length scaling framework for the analysis of finite-퐷 iPEPSs which have been variationally optimised for Lorentz-invariant

5The XY model shows a somewhat irregular behaviour, whose origin we do not understand

125 Chapter 7 Finite Correlation Length Scaling in Lorentz-Invariant Gapless iPEPS Wave Functions

0.5 TFI, 퐷 = 2 ℎ < ℎ푐(퐷) ℎ > ℎ푐(퐷) 0.4 ℎ푧 > 0 TFI, 퐷 = 3 0.3 ℎ < ℎ푐(퐷) ℎ > ℎ푐(퐷) bMPS

푆 ℎ푧 > 0 0.2 XY HB Ising-PEPS 0.1 푇 < 푇푐 푇 > 푇푐 0.0 2−3 2−2 2−1 20 21 22 23 24 휉

0.5 TFI, 퐷 = 2 ℎ < ℎ푐(퐷) ℎ > ℎ푐(퐷) 0.4 ℎ푧 > 0 TFI, 퐷 = 3 0.3 ℎ < ℎ푐(퐷) ℎ > ℎ푐(퐷) bMPS

푆 ℎ푧 > 0 0.2 XY HB Ising-PEPS 0.1 푇 < 푇푐 푇 > 푇푐 0.0 0.0 2.5 5.0 7.5 10.0 12.5 15.0 17.5 20.0 휉

Figure 7.10: Half-chain von Neumann entropy 푆bMPS of the boundary MPS resulting from the contraction of the iPEPS tensors, plotted as a function of log 휉 (upper panel) and 휉 (lower panel). We display all available data sets. The transverse-field Ising model data is organized according to 퐷 and the parameter cut, while the Heisenberg and XY model connect all available 퐷 values. For comparison we also show entanglement entropies for the (2 + 0)D Ising-PEPS evaluated at different temperatures.

126 7.A Appendix: Further Results on the Transverse-Field Ising Model

(2 + 1)D quantum critical or continuous symmetry-breaking Hamiltonians. This framework is important (i) to understand the mild limitations of the finite-퐷 iPEPS manifold in representing Lorentz-invariant, gapless many-body quantum states and (ii) to put forward a practical scheme in which the finite correlation length 휉(퐷) and field theory inspired formulae can be employed to extrapolate the data to infinite correlation length, i. e. to the thermodynamic limit. We show that some of the amplitudes entering the equations have a field theoretical interpretation and are therefore universal to some degree. We demonstrate the power of the method for the energy convergence in all three considered models, and for order parameter extrapolations for the continuous symmetry-breaking models, where the previously employed 1/퐷 extrapolation schemes performed poorly. We believe that another advantage of this approach is also that the variational quality of an iPEPS tensor and the resulting correlation length (and other observables) seem to go hand in hand in the vicinity of a local optimum, in such a way that different data points still lie on the same휉 finite- extrapolation curve. We also carefully analyse the critical behaviour of the transverse-field Ising model in the vicinity of its critical point by measurements of local observables as a function of the two magnetic fields, and we are able to obtain critical exponents within a few percent ofthemost recent conformal bootstrap results already at bond dimension 퐷 = 3. The finite correlation length scaling framework opens the way for further exploration of quantum matter with an (expected) Lorentz-invariant, massless low-energy description. Within the quantum critical or CFT related questions one could explore the Wilson-Fisher 푂(푁 ) fixed points beyond the 푁 = 1 case studied here, Gross-Neveu-Yukawa universality classes arising in interacting Dirac fermion systems, or QED3 related behavior of gapless quantum spin liquids or deconfined criticality. In the context of continuous symmetry-breaking, various superfluid and superconducting phases in bosonic and fermionic systems should be described by the 푂(2) symmetry-breaking results established in this work. Further directions are non- collinear magnetic order or SU(푁 ) continuous symmetry-breaking. Ultimately one should also explore the occurrence of finite correlation lengths and their scaling in interacting systems with a Fermi surface. On the fundamental level it will be important to develop an understanding of the 휉(퐷) relation, that we started to explore here. Our preliminary results indicate a 휉 ∼ 퐷휅 scaling with values of 휅 which could be universal. In the (1 + 1)D iMPS context, the values of 휅 depend only on the central charge 푐 [135]. It would be interesting to understand in (2 + 1)D whether 휅 also depends only on some universal data, such as in the 퐹-theorem [276, 277]. Note: Similar results have been reported by P. Corboz, P. Czarnik, G. Kapteijns, and L. Tagliacozzo [194].

7.A Appendix: Further Results on the Transverse-Field Ising Model

In this appendix the critical behaviour of the 푚푥 magnetisations as a function of the transverse- field ℎ in the transverse field Ising model in the vicinity of the critical pointat ℎ = ℎ푐, ℎ푧 = 0 is presented. In the upper panel of figure 7.11 we plot |푚푥,푐 − 푚푥 | as a function of |ℎ푐 − ℎ| on a log-log scale together with a straight line showing the expected slope 훼휀 as a guide to the

127 Chapter 7 Finite Correlation Length Scaling in Lorentz-Invariant Gapless iPEPS Wave Functions

Model 퐷 푒/퐽 2 −3.233 573 421 TFI @ ℎ 푐 3 −3.234 260 711 2 −0.660 231 093 3 −0.667 974 240 푆 = 1/2 HB 4 −0.669 083 757 5 −0.669 378 064 2 −0.547 658 559 3 −0.548 706 183 푆 = 1/2 XY 4 −0.548 810 284 5 −0.548 818 968

Table 7.2: Best variational energies eye. In the corresponding inset we numerically calculate the derivative (excluding the hollow symbols as they cause divergences in the derivative). The data shows convergence towards the expected value for the critical exponent 훼휀 from 퐷 = 2 to 퐷 = 3, although the noise on this data is relatively large compared to the data presented in figure 7.3. This might be due to the uncertainty in the estimation of 푚푥,푐. (퐷) (퐷) In the lower panel of figure 7.11 |푚푥,푐 −푚푥 | is presented as a function of |ℎ푐 −ℎ| on a log-log scale and the inset contains the corresponding numerical derivatives. The 퐷 = 2 data clearly (퐷) shows mean-field behaviour at small |ℎ푐 − ℎ|, but the derivatives for 퐷 = 3 might already start to form an intermediate plateau. We expect data for 퐷 > 3 to give a better understanding of how this plateau is formed.

7.B Appendix: Variational iPEPS Energies

For reference, we provide in table 7.2 our best variational iPEPS energies for the three models considered in this paper. The transverse-field Ising model is at its critical point ℎ = ℎ푐, ℎ푧 = 0.

128 7.B Appendix: Variational iPEPS Energies

101 퐷 = 2 퐷 = 3 ℎ < ℎ푐 ℎ < ℎ푐 (퐷) (퐷) ℎ > ℎ푐 ℎ > ℎ푐 10−1 훼휀 | 푥 | 푥 − 푚 1.0 푥,푐 − 푚 − ℎ| 푐 |푚 푥,푐 −3 |ℎ 10 |푚

log 0.5 휕 log 휕

10−2 101

|ℎ푐 − ℎ| 10−5 10−4 10−3 10−2 10−1 100 101 |ℎ푐 − ℎ|

1 퐷 = 2 퐷 = 3 10 (퐷) (퐷) ℎ < ℎ푐 ℎ < ℎ푐 (퐷) (퐷) ℎ > ℎ푐 ℎ > ℎ푐 (MF) 10−1 훼휀 훼휀 | 푥 − 푚 |

−3 푥 (퐷) 푥,푐 10 − ℎ| − 푚

|푚 1.0 (퐷) 푐 (퐷) 푥,푐 |ℎ |푚 −5 log

10 log 0.5 휕 휕 10−5 10−2 101 |ℎ(퐷) − ℎ| 10−7 푐 10−5 10−4 10−3 10−2 10−1 100 101 (퐷) |ℎ푐 − ℎ|

Figure 7.11: Analysis of the critical behaviour of the 푚푥 magnetisations in the transverse-field Ising model in the vicinity of the critical point at ℎ = ℎ푐, ℎ푧 = 0 as a function of the transverse field. The upper panel shows |푚푥,푐 − 푚푥 | as a function of |ℎ푐 − ℎ| and the (퐷) (퐷) lower panel |푚푥,푐 −푚푥 | as a function of |ℎ푐 −ℎ|. In the insets numerical derivatives are presented (where the hollow symbols of the upper panel are excluded as they cause divergences in the derivative). The inset of the lower panel shows mean-field (퐷) behaviour of the 퐷 = 2 data at small |ℎ푐 − ℎ| while the 퐷 = 3 data is too noisy to allow firm conclusions. The expected exponents are tabulated in table 7.1.

129

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