On Monte Carlo Time-Dependent Variational Principles
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On Monte Carlo Time-Dependent Variational Principles Von der QUEST-Leibniz-Forschungsschule der Gottfried Wilhelm Leibniz Universit¨atHannover zur Erlangung des akademischen Grades Doktor der Naturwissenschaften { Doctor rerum naturalium { genehmigte Dissertation von FABIANWOLFGANGG UNTERTRANSCHEL¨ Geboren am 10.01.1987 in Gehrden 2016 0 Erstpr¨ufer : Prof. Dr. Reinhard F. Werner Zweitpr¨ufer : Prof. Dr. Klemens Hammerer Beratendes Mitglied des Pr¨ufungsausschusses : Prof. Dr. Tobias J. Osborne Vorsitzender des Pr¨ufungsausschusses : Prof. Dr. Rolf Haug Tag der Promotion: 17. Dezember 2015 On Monte Carlo Time-Dependent Variational Principles Fabian W. G. Transchel Abstract The present dissertation is concerned with the development and implementation of a novel scheme for quantitative, numeric approximation of the dynamics of quantum lattice systems based on the Time-Dependent Variational Principle together with Monte Carlo techniques in order to include dissipative interactions. The specific implementation is demonstrated on both common and not yet in-detail explored Heisenberg- and Fermi-Hubbard models in one and two dimensions. Additionally, the technical requirements regarding computational complexity and capacity are discussed, especially with regards toward parallelizable components of the imple- mentation. Concluding remarks include prospects with respect to application and extension of the presented methods. Keywords: Monte Carlo method, Dissipative Dynamics, Lindblad Equation iii Zusammenfassung Die vorliegende Dissertation befasst sich mit der Entwicklung und Umsetzung eines neuartigen Schemas zur quantitativen numerischen N¨aherung der Dynamik von Quantengittersystemen auf Grundlage des zeitabh¨angigenVariationsprinzips unter Zuhilfenahme von Monte-Carlo-Techniken zur Einbeziehung von dissipativen Wech- selwirkungen. Die Implementierung wird anhand von Beispielen f¨urHeisenberg- und Fermi-Hubbard-Modellen in einer und zwei Dimensionen gezeigt und erl¨autert. Erg¨anzenderfolgt eine Betrachtung der technischen Anforderungen an rechnerische Kapazit¨atenund komplexit¨atstheoretische Erw¨agungen,mit besonderem Augen- merk auf parallelisierbare Komponenten der Implementierung. Abschließend wird ein Ausblick hinsichtlich der weiteren M¨oglichkeiten von Einsatz und Erweiterung der pr¨asentierten Methoden vorgenommen. Schlagworte: Monte-Carlo-Simulation, Dissipative Dynamik, Lindblad-Gleichung v Selbst¨andigkeitserkl¨arung Hiermit best¨atigeich, die vorliegende Dissertation selbst verfasst zu haben, keine Textabschnitte von Dritten oder eigenen Pr¨ufungsarbeiten ohne Kennzeichnung ¨ubernommen und alle benutzten Hilfsmittel und Quellen in der Arbeit angegeben zu haben. Hannover, den 09. November 2015 Fabian W. G. Transchel vii Dedication What is a thesis But a 148-page love letter? To nature, to physics To every qubit in the universe What is a thesis but a ladder to climb on? For nature, for physics For dwarfs on the shoulders of a giant universe What is a thesis But an awe-struck observation? Of nature, of physics Of every qubit in the universe What is a thesis but a proposition that remains? When nature, when physics When everything dissipates To nature To physics To you And to everyone ix Acknowledgments Although an endeavor as bold as a dissertation in theoretical physics never can be fully understood in terms of a single effort of will that is merely assisted by others, this is exactly what one implies once it comes to the acknowledgments. Accordingly, one enumerates the \relevant" impacts, although in truth every little hint is worth acknowledging. Every conversation, every misplaced index, every syntax error spotted propels us toward the aim of making it a thesis. Much like mathematicians turn coffee into theorems, the whole lot of scientific experience that happens in the meantime is turned into a dissertation through means not fully understood by physicists. Ignoring the paradox to acknowledge the relevance, I proceed toward a (par- tially ordered) enumeration and thereby deeply thank Reinhard for the unique opportunity to push the boundary of scientific understanding as well as the pos- sibility to observing indisputable genius, Tobias for the tireless gauging of both possible and impossible research directions, Ash for the wonderful basis of evoMPS upon which the framework for open system could be built, Kais, Fabian F. and Robert for their patient (and particularly uncomplaining) office fellowship that led to much unanticipated insights as well as Bernhard, Ren´e,Lars, Tobias G., J¨org and all the other members of the QI group for all the corrections and annotations and entertainment opportunities that finally led to this thesis. Last but not least, of course it cannot be understated the amount of deprivation my family and especially my girlfriend Wiltrud had to endure. Where the group nourished me scientifically, she helped me keeping the pretense of normal function- ing and sanity nonetheless { on a ride that truly cannot be ventured alone. xi Contents Abstract iii Zusammenfassung v Selbst¨andigkeitserkl¨arung vii Dedication ix Acknowledgments xi List of abbreviations 1 Introduction 3 Part 1. Preliminaries 7 Chapter 1. Quantum Information 9 1. Bases 9 2. Products 9 2.1. Inner Product 9 2.2. Outer Product 9 3. Eigenproperties 10 4. Dynamics 11 5. The Qubit 12 5.1. Phase 12 6. Composition of quantum systems and entanglement 13 7. Density operators 13 7.1. Properties 14 7.2. Bloch sphere 15 7.3. No-cloning Theorem 17 7.4. Partial trace 17 7.5. Schmidt Decomposition 18 7.6. Purification 19 8. Entropy 19 8.1. Properties of the Von Neumann entropy 20 8.2. Entropies of composite systems 21 8.3. Monogamy of entanglement 21 9. Spin 22 10. Spins as logical qubits 23 11. Second Quantization 24 11.1. Creation and Annihilation operators 25 Chapter 2. Correlated Spin Systems and its Dynamics 27 1. Spin Lattices 27 2. Finitely Correlated States 29 2.1. Entanglement and Correlations 29 xiii xiv CONTENTS 2.1.1. Bell Inequality 30 2.1.2. Area laws 31 3. From FCS to MPS 32 3.1. Explicit MPS examples 33 3.1.1. GHZ state 33 3.1.2. W state 33 3.2. Properties and applications of MPS 34 3.2.1. Expectation values 34 3.2.2. MPS from slightly entangled states 34 3.2.3. Connection to DMRG 35 3.3. The canonical form 36 3.4. Translation invariance 38 3.5. Gapless and frustration-free Hamiltonians 39 3.6. AKLT model 41 3.7. Criticality 41 3.8. Heisenberg model 42 3.9. Hubbard model 44 3.9.1. Jordan-Wigner transformation 44 Chapter 3. Variational Ansatzes in numerical Many-body Physics 45 1. The variational principle 45 1.1. Derivation 46 2. DMRG 47 3. Matrix Product Operators 48 4. The Time-Dependent Variational Principle 50 4.1. Derivation 50 4.2. Properties 53 4.3. Examples and algorithms: evoMPS 53 4.3.1. Imaginary time evolution 55 4.3.2. Real time evolution 55 5. Variational Monte Carlo 56 5.1. Time-Dependent Variational Monte Carlo 57 Chapter 4. Stochastic Calculus 59 1. Markov Processes 59 1.1. The Fokker-Planck Equation 62 1.2. It¯o'sLemma 63 2. The Stochastic Integral 64 3. The Glivenko-Cantelli theorem 64 4. The Quantum Master Equation 65 Part 2. Monte Carlo Dissipation Models 69 Chapter 5. A Monte Carlo Time-Dependent Variational Principle 71 1. Motivation 71 2. Derivation of the method 72 3. Algorithm and implementation 76 3.1. Preparations 76 3.2. Iteration step 77 3.3. Distributed computation and sampling 79 3.4. Post-processing and analysis 81 4. Possible extensions 83 Chapter 6. Applications to the Heisenberg model 85 CONTENTS xv 1. Introduction 85 2. Results for edge-driven models 86 3. Probing the steady state 87 4. Quasi-homogenous models 88 4.1. Introduction 88 4.2. Homogenous dissipation 88 4.2.1. Results 88 4.3. Bihomogenous case 90 4.4. Results for bihomogenous case 90 5. Outlook 92 Chapter 7. Applications to the Fermi-Hubbard model 93 1. Introduction 93 2. Model-specific adaptions to the Monte Carlo TDVP 94 2.1. Jordan-Wigner transform 94 2.2. Algorithm 97 2.3. Results for the one-dimensional case 97 3. Treatment of higher spatial dimensions 99 3.1. The memory scaling problem 100 3.2. Results for the two-dimensional case 103 4. Outlook 105 4.1. One-dimensional super sites 105 4.2. Nested MPS 106 Conclusion 109 Appendix. List of tables 111 Appendix. List of figures 113 Appendix. Resources / Hilfsmittel 117 Appendix. Curriculum Vitae 119 Appendix. Bibliography 121 List of abbreviations ALS Alternating Least Squares method DMRG Density Matrix Renormalization Group BOINC Berkeley Open Infrastructure for Network Computing evoMPS Python many-body simulation package implementing the TDVP ITE Imaginary Time Evolution LAPACK Linear Algebra Package LCF Left Canonical Form LHS left-hand side MCTDVP Monte Carlo Time-Dependent Variational Principle MPS Matrix Product State MPO Matrix Product Operator NRG Numerical Renormalization Group NumPy Python extension for large array/matrix representations openBLAS Basic Linear Algebra Subprograms RCF Right Canonical Form RHS right-hand side RNG Random Number Generator RTE Real-Time Evolution SciPy Python extension for scientific calculations TDVP Time-Dependent Variational Principle TEBD Time-Evolving Block Decimation VMC Variational Monte Carlo 1 Introduction The quest of the physical sciences is to enhance our understanding of reality. While the scientific method has many different manifestations, its principles within will always stay the same: Based upon experience, the scientist formulates a hypothesis about empirically accessible experiments that can deepen a conviction about certain causal relations, be it about fundamentals like the law of gravity or its generalization in relativity as well as the advent of quantum theory. The role of theoretical work cannot be overstated as it provides a foundation for the inference step by precisely defining the necessary variables and interactions. Depending on the discipline and question, the relation between theory and experiment becomes more and more dialectic { without going into detail, we can already understand from the above definition that without theory, there is no ba- sis for a sound experiment, regardless of its sophistication { while, on the other hand, without experimental validation, any theory becomes hollow and scientifi- cally meaningless.