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

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Tensor Network State Methods and Applications for Strongly Correlated Quantum Many-Body Systems Bildquelle: Universität Innsbruck Tensor Network State Methods and Applications for Strongly Correlated Quantum Many-Body Systems 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. iii 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 exponen- tially 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 ob- jective 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 enorm- ous 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 spe- cific 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 simu- lators, 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 in- corporating field theoretical insights, a powerful finite correlation length scaling framework is established, resulting in formulae that enable extrapolations of observables to infinite correl- ation 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. v 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. vii 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
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