QUANTUM MONTE CARLO METHODS

Transcription

1 QUANTUM MONTE CARLO METHODS Featuring detailed explanations of the major algorithms used in quantum Monte Carlo simulations, this is the first textbook of its kind to provide a pedagogical overview of the field and its applications. The book provides a comprehensive introduction to the Monte Carlo method, its use, and its foundations, and examines algorithms for the simulation of quantum many-body lattice problems at finite and zero temperature. These algorithms include continuous-time loop and cluster algorithms for quantum spins, determinant methods for simulating Fermions, power methods for computing ground and excited states, and the variational Monte Carlo method. Also discussed are continuous-time algorithms for quantum impurity models and their use within dynamical mean-field theory, along with algorithms for analytically continuing imaginary-time quantum Monte Carlo data. The parallelization of Monte Carlo simulations is also addressed. This is an essential resource for graduate students, teachers, and researchers interested in quantum Monte Carlo. j. e. gubernatis works at the Los Alamos National Laboratory. He is a Fellow of the American Physical Society (APS) and served as a Chair of the APS Division of Computational Physics. He represented the United States on the Commission of Computational Physics of International Union of Pure and Applied Physics (IUPAP) for nine years and chaired the Commission for three years. n. kawashima is a professor at the University of Tokyo. He is a member of the Society of Cognitive Science and has been a Steering Committee member for the public use of the supercomputer at the Institute for Solid State Physics (ISSP) for the last 15 years. He received the Ryogo Kubo Memorial Prize for his contributions to the development of loop and cluster algorithms in p. werner is a professor at the University of Fribourg. In 2010, he received the IUPAP Young Scientist Prize in computational physics for the development and implementation of quantum Monte Carlo methods for impurity models.

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3 QUANTUM MONTE CARLO METHODS Algorithms for Lattice Models J. E. GUBERNATIS Los Alamos National Laboratory N. KAWASHIMA University of Tokyo P. WERNER University of Fribourg

4 University Printing House, Cambridge CB2 8BS, United Kingdom Cambridge University Press is part of the University of Cambridge. It furthers the University s mission by disseminating knowledge in the pursuit of education, learning and research at the highest international levels of excellence. Information on this title: / Cambridge University Press 2016 This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 2016 Printed in the United Kingdom by TJ International Ltd. Padstow Cornwall A catalogue record for this publication is available from the British Library Library of Congress Cataloging in Publication data Gubernatis, J. E., author. Quantum Monte Carlo methods : algorithms for lattice models / J.E. Gubernatis (Los Alamos National Laboratory), N. Kawashima (University of Tokyo), P. Werner (University of Fribourg). pages cm Includes bibliographical references and index. ISBN (Hardback : alk. paper) 1. Monte Carlo method. 2. Many-body problem. I. Kawashima, N. (Naoki), author. II. Werner, P., 1975 author. III. Title. QC M64G dc ISBN Hardback Cambridge University Press has no responsibility for the persistence or accuracy of URLs for external or third-party internet websites referred to in this publication, and does not guarantee that any content on such websites is, or will remain, accurate or appropriate.

5 Contents Preface page xi Part I Monte Carlo basics 1 1 Introduction The Monte Carlo method Quantum Monte Carlo Classical Monte Carlo 6 2 Monte Carlo basics Some probability concepts Random sampling Direct sampling methods Discrete distributions Continuous distributions Markov chain Monte Carlo Markov chains Stochastic matrices Detailed balance algorithms Metropolis algorithm Generalized Metropolis algorithms Heat-bath algorithm Rosenbluth s theorem Entropy content 38 Exercises 40 3 Data analysis Equilibrating the sampling Calculating averages and estimating errors Correlated measurements and autocorrelation times 49 v

6 vi Contents 3.4 Blocking analysis Data sufficiency Error propagation Jackknife analysis Bootstrap analysis Monte Carlo computer program 61 Exercises 63 4 Monte Carlo for classical many-body problems Many-body phase space Local updates Two-step selection Cluster updates Swendsen-Wang algorithm Graphical representation Correlation functions and cluster size Worm updates Closing remarks 80 Exercises 82 5 Quantum Monte Carlo primer Classical representation Quantum spins Longitudinal-field Ising model Transverse-field Ising model Continuous-time limit Zero-field XY model Simulation with loops Simulation with worms Ergodicity and winding numbers Bosons and Fermions Bosons Fermions Negative-sign problem Dynamics 115 Exercises 117 Part II Finite temperature Finite-temperature quantum spin algorithms Feynman s path integral 121

7 Contents vii 6.2 Loop/cluster update General framework Continuous-time loop/cluster update XXZ models Correlation functions Magnetic fields Large spins ( ) S > High-temperature series expansion Stochastic series expansion Continuous-time limit Worm update Freezing problem Directed-loop algorithm Violation of the detailed balance condition Correlation functions XXZ model On-the-fly vertex generation Toward zero temperature Extrapolation to zero temperature Quantum phase transitions Finite-size scaling Applications to Bosonic systems 174 Exercises Determinant method Theoretical framework Hubbard-Stratonovich transformations Determinantal weights Single-particle Green s function Finite temperature algorithm Matrix representation Metropolis sampling The algorithm Measurements Hirsch-Fye algorithm Matrix product stabilization Comments 209 Exercises Continuous-time impurity solvers Quantum impurity models 214

8 viii Contents Chain representation Action formulation Dynamical mean-field theory Single-site effective model DMFT approximation DMFT self-consistency loop Simulation of strongly correlated materials Cluster extensions General strategy Weak-coupling approach Sampling Determinant ratios and fast matrix updates Measurement of the Green s function Multi-orbital and cluster impurity problems Strong-coupling approach Sampling Measurement of the Green s function Generalization Matrix formalism Generalization Krylov formalism Infinite-U limit: Kondo model Weak-coupling approach Strong-coupling approach Determinant structure and sign problem Combination of diagrams into a determinant Absence of a sign problem Scaling of the algorithms 258 Exercises 262 Part III Zero temperature Variational Monte Carlo Variational Monte Carlo The variational principle Monte Carlo sampling Trial states Slater-Jastrow states Gutzwiller projected states Valence bond states Tensor network states Trial-state optimization Linear method 293

9 Contents ix Newton s method Connection between linear and Newton methods Energy variance optimization Stabilization Summary of the linear and Newton s optimization methods 299 Exercises Power methods Deterministic direct and inverse power methods Monte Carlo power methods Monte Carlo direct power method Monte Carlo inverse power method Stochastic reconfiguration Green s function Monte Carlo methods Linear method Diffusion Monte Carlo Importance sampling Measurements Excited states Correlation function Monte Carlo Modified power method Comments 334 Exercises Fermion ground state methods Sign problem Fixed-node method Constrained-path method Estimators Mixed estimator Forward walking and back propagation The algorithms Constrained-phase method 360 Exercises 363 Part IV Other topics Analytic continuation Preliminary comments Dynamical correlation functions Bayesian statistical inference 373

10 x Contents Principle of maximum entropy The likelihood function and prior probability The best solutions Analysis details and the Ockham factor Practical considerations Comments 395 Exercises Parallelization Parallel architectures Single-spin update on a shared-memory computer Single-spin update on a distributed-memory computer Loop/cluster update and union-find algorithm Union-find algorithm for shared-memory computers Union-find algorithm for distributed-memory computers Back to the future 413 Appendix A Alias method 416 Appendix B Rejection method 418 Appendix C Extended-ensemble methods 420 Appendix D Loop/cluster algorithms: SU(N) model 425 Appendix E Long-range interactions 428 Appendix F Thouless s theorem 432 Appendix G Hubbard-Stratonovich transformations 435 Appendix H Multi-electron propagator 441 Appendix I Zero temperature determinant method 445 Appendix J Anderson impurity model: chain representation 449 Appendix K Anderson impurity model: action formulation 451 Appendix L Continuous-time auxiliary-field algorithm 455 Appendix M Continuous-time determinant algorithm 459 Appendix N Correlated sampling 462 Appendix O The Bryan algorithm 464 References 469 Index 484

11 Preface Fast computers enable the solution of quantum many-body problems by Monte Carlo methods. As computing power increased dramatically over the years, similarly impressive advances occurred at the level of the algorithms, so that we are now in a position to perform accurate simulations of large systems of interacting quantum spins, Bosons, and (to a lesser extent) Fermions. The purpose of this book is to present and explain the quantum Monte Carlo algorithms being used today to simulate the ground states and thermodynamic equilibrium states of quantum models defined on a lattice. Our intent is not to review all relevant algorithms there are too many variants to do so comprehensively but rather to focus on a core set of important algorithms, explaining what they are and how and why they work. Our focus on lattice models, such as Heisenberg and Hubbard models, has at least two implications. The first is obviously that we are not considering models in the continuum where extensive use of quantum Monte Carlo methods traditionally has focused on producing highly accurate ab initio calculations of the ground states of nuclei, atoms, molecules, and solids. Quantum Monte Carlo algorithms for simulating the ground states of continuum and lattice models, however, are very similar. In fact, the lattice algorithms are in many cases derived from the continuum methods. With fewer degrees of freedom, lattice models are compact and insightful representations of the physics in the continuum. The second implication is a focus on both zero and finite temperature algorithms. 1 On a lattice, it is natural to study phase transitions. In particular, the recent dramatic advances in quantum Monte Carlo lattice methods for the simulation of quantum spin models were prompted by a need for more efficient and effective ways to study finite-temperature transitions. While quantum Monte Carlo is profitably used to study zero temperature phase transitions (quantum critical phenomena), 1 Temperature zero is a finite temperature, but common usage separates T = 0 (zero temperature) from T > 0 (finite temperature) when classifying algorithms. Throughout, we adopt the common usage. xi

12 xii Preface some ground state algorithms have no finite temperature analogs and vice versa. In many respects, the lattice is where the current algorithmic action is. The book is divided into four parts. The first part is a self-contained, more advanced than average, discussion of the Monte Carlo method, its use, and its foundations. With the basics in place, this part then steps toward the more recent worm and loop/cluster Monte Carlo algorithms for simple classical models, and finally for simple quantum models. Our intent is to be as tutorial as possible and impart a good taste for what quantum Monte Carlo is like. In this introduction, we only briefly mention ground state simulations. The foundations for ground state simulations require less introduction, and we wanted to keep Part I reasonably sized so it can be used as teaching material for a course on computational classical and quantum many-body physics. Parts II and III present the main quantum Monte Carlo algorithms. Part II discusses finite-temperature methods for quantum spin and Fermion systems. The quantum spin chapter, plus its associated appendices, present a more extensive and sophisticated treatment of the worm and loop/cluster algorithms introduced in Part I. The Fermion chapter details the most important methods for the finitetemperature simulation of lattice Fermion models. Many of the formal techniques developed in this chapter are used in the discussion of the zero temperature Fermion methods in Part III. Besides well-established algorithms for Fermionic lattice models, we also discuss the more recent continuous-time Monte Carlo technique for quantum impurity models. This method is the dynamo driving today s lattice calculations based on the dynamical mean-field approximation, which in turn is being coupled to ab initio calculations of the electronic properties of solids. The chapter on impurity models hence connects the lattice and continuum modeling and simulations of many-electron physics. Part III discusses the two main zero temperature methods, the variational Monte Carlo and the power method. The power method is a more universal term for what is often called the Green s function Monte Carlo method. Part III also includes a special chapter on the use of the power method for the simulation of Fermion systems. It is in this chapter that the sign problem is discussed. This problem is the biggest inhibitor to quantum Monte Carlo reaching its full potential. The final part does not discuss quantum Monte Carlo algorithms but rather topics that accompany their use. First, we present a widely used method to analytically continue simulation data from imaginary time to real time, so dynamical properties can be extracted from finite-temperature simulations. Finally, we address the parallelization of Monte Carlo simulations. While Monte Carlo calculations per se are naturally parallel (the code, with different random number seeds, can be run on independent processors and the results combined when all calculations have finished), this chapter is about a relatively recent trend, namely, the complexity of

13 Preface xiii simulations is making the sharing of specific computational tasks among several processors desirable or mandatory, and about what is lying in the future, namely, running even more complex simulations on computers with orders of magnitude more processors. We happily thank Tom Booth, Yan Chen, Kenji Harada, Akiko Kato, Yasuyuki Kato, Tsuyoshi Okubo, Gerado Ortiz, Brenda Rubenstein, Richard Scalettar, Devinder Sivia, Synge Todo, and Cyrus Umrigar for helpful comments and suggestions about various parts of the manuscript. One of the coauthors (NK) thanks Yan Chen, in particular, for offering a quiet place for writing a part of the book. Another coauthor (PW) would like to thank Matthias Troyer, Emanuel Gull, and Andrew Millis for fruitful discussions and collaborations on quantum Monte Carlo and dynamical mean-field related topics. The third coauthor (JG) thanks his wife, Michele, for her proofreading of a manuscript that to say the least was not her cup of tea. Of course, for the errors that remain we take full responsibility. We also thank our editor, Simon Capelin. His polite, but yearly, inquiry about interest in writing such a textbook caused one of the coauthors eventually to cave in and to start writing this book. Simon s patience in waiting for something that took longer than just a few years is also appreciated. To the researchers, students, and teachers using this book: we hope that it helps you appreciate and understand the essence of quantum Monte Carlo algorithms. We also hope that it inspires you to develop even better ways to do quantum simulations. Without doubt, this important research tool has limitations needing mitigation to realize its full potential. Realizing this potential, however, will contribute to our understanding of quantum many-body physics, which in the long run is our attractor to this research area and our motivation to explore and develop new algorithms. J. E. Gubernatis N. Kawashima P. Werner