Projector Quantum Monte Carlo Methods for Linear and Non-linear Wavefunction Ansatzes Lauretta Rebecca Schwarz Trinity Hall University of Cambridge This dissertation is submitted for the degree of Doctor of Philosophy at the University of Cambridge October 2017 Projector Quantum Monte Carlo Methods for Linear and Non-linear Wavefunction Ansatzes Lauretta Rebecca Schwarz Abstract This thesis is concerned with the development of a Projector Quantum Monte Carlo method for non-linear wavefunction ansatzes and its application to strongly correlated materials. This new approach is partially inspired by a prior application of the Full Configuration Interaction Quantum Monte Carlo (FCIQMC) method to the three- band (p − d) Hubbard model. Through repeated stochastic application of a projector FCIQMC projects out a stochastic description of the Full Configuration Interaction (FCI) ground state wavefunction, a linear combination of Slater determinants spanning the full Hilbert space. The study of the p − d Hubbard model demonstrates that the nature of this FCI expansion is profoundly affected by the choice of single-particle basis. In a counterintuitive manner, the effectiveness of a one-particle basis to produce a sparse, compact and rapidly converging FCI expansion is not necessarily paralleled by its ability to describe the physics of the system within a single determinant. The results suggest that with an appropriate basis, single-reference quantum chemical approaches may be able to describe many-body wavefunctions of strongly correlated materials. Furthermore, this thesis presents a reformulation of the projected imaginary time evolution of FCIQMC as a Lagrangian minimisation. This naturally allows for the optimisation of polynomial complex wavefunction ansatzes with a polynomial rather than exponential scaling with system size. The proposed approach blurs the line between traditional Variational and Projector Quantum Monte Carlo approaches whilst involving developments from the field of deep-learning neural networks which can be expressed as a modification of the projector. The ability of the developed approach to sample and optimise arbitrary non-linear wavefunctions is demonstrated with several classes of Tensor Network States all of which involve controlled approximations but still retain systematic improvability towards exactness. Thus, by applying the method to strongly-correlated Hubbard models, as well as ab-initio systems, including a fully periodic ab-initio graphene sheet, many-body wavefunctions and their one- and two-body static properties are obtained. The proposed approach can handle and simultaneously optimise large numbers of variational parameters, greatly exceeding those of alternative Variational Monte Carlo approaches. For my parents Preface This dissertation contains an account of research carried out in the period between October 2013 and July 2017 in the Theoretical Sector of the Department of Chemistry in the University of Cambridge under the supervision of Prof. Ali Alavi, F.R.S. The following sections of this thesis are included in work which has either been published or is to be published: Chapter 4 : L. R. Schwarz, G. H. Booth and A. Alavi, Insights into the structure of many-electron wave functions of Mott-insulating antiferromagnets: The three-band Hubbard model in full configuration interaction quantum Monte ,Carlo Physical Review B, 91, 045139 (2015) Chapter 5 and 6 : L. R. Schwarz, A. Alavi and G. H. Booth, Projector Quantum Monte Carlo Method for Nonlinear Wave Functions, Physical Review Letters, 118, 176403 (2017) Chapter 7 : In preparation. This dissertation is the result of my own work and includes nothing which is the outcome of work done in collaboration except where indicated otherwise. This thesis is not substantially the same as any that I have submitted, or, is being concurrently submitted for a degree or diploma or other qualification at the University of Cambridge or any other University or similar institution. I further state that no substantial part of my thesis has already been submitted, or, is being concurrently submitted for any such degree, diploma or other qualification at the University of Cambridge orany other University or similar institution. This dissertation does not exceed 60, 000 words in length, including abstract, tables, and footnotes, but excluding table of contents, photographs, diagrams, figure captions, list of figures, list of abbreviations, bibliography, appendices and acknowledgements. Lauretta Rebecca Schwarz October 2017 Acknowledgements First, I would like to thank my supervisor, Ali Alavi, for all of his many ideas and enthusiasm. I am very grateful for the continual support that he provided, whilst giving me the space to explore my own ideas and creating an inspiring atmosphere to work in. The many inspiring discussions with him and his wide knowledge have been invaluable, and I appreciate all that he has done. Another person whom I cannot thank enough is George Booth. The huge amount of knowledge and advice that he has passed on to me has been invaluable. I am grateful not only for proof-reading this thesis and the many helpful discussions, but also for his guidance and support. I would also like to thank Charlie Scott for proof-reading parts of this thesis and Robert Thomas for many inspiring discussions, as well as the past and present members of the Alavi group for creating such a pleasant environment to work in. Moreover, it is a pleasure to acknowledge the Engineering and Physical Sciences Research Council for funding this work, and the Max Planck Society for the provision of high-performing computing facilities. Last but not least, I would like to thank my parents for their loving support which means so much to me. Their love and encouragement has continually accompanied and helped me on my way. Table of contents List of figures xv List of tables xvii 1 Introduction1 2 Theoretical Background9 2.1 Introduction . .9 2.2 An Alternative Formalism: Second Quantisation . 13 2.2.1 The Fock Space . 13 2.2.2 Elementary Operators . 15 2.3 Hartree-Fock Theory . 18 2.4 Electron Correlation . 22 2.5 Full Configuration Interaction . 25 2.6 Further Approximate Solution Methods . 28 2.7 Beyond Traditional Quantum Chemical Methods: Tensor Network States Approaches . 33 2.7.1 Entanglement Entropy and Area Laws . 33 2.7.2 Decomposing the FCI Wavefunction Ansatz . 34 2.7.3 Tensor Network States in One Dimension: Matrix Product States 35 2.7.4 The Area Law and Efficient Approximability . 38 2.7.5 Tensor Network States in Two Dimensions: Projected Entangled Pair States . 41 2.7.6 Optimisation of MPS, PEPS, ... 43 3 Zero-temperature Ground State Quantum Monte Carlo Methods 47 3.1 Variational Quantum Monte Carlo Approaches . 47 3.2 Projector Quantum Monte Carlo Approaches . 51 3.3 Full Configuration Interaction Quantum Monte Carlo: FCIQMC . 55 xii Table of contents 3.3.1 Derivation of FCIQMC Equations . 56 3.3.2 Stochastic Implementation of FCIQMC . 60 3.3.3 Energy Estimators . 62 3.3.4 The Fermion Sign Problem and the Importance of Annihilation 63 3.3.5 The Initiator Approximation . 66 3.3.6 Non-integer Real-weight Walkers . 71 3.3.7 The Semi-stochastic Adaptation . 72 3.4 Handling Stochastic Estimators . 73 4 Application of i-FCIQMC to Strongly Correlated Systems 75 4.1 Model Hamiltonians . 76 4.1.1 The Hubbard Model . 76 4.1.2 The three-band (p − d) Hubbard Model . 78 4.2 Geometries and Cluster Size . 83 4.3 The three-band Hubbard Model in FCIQMC . 83 4.3.1 The Choice of Single-Particle Basis Sets . 84 4.3.2 An i-FCIQMC Study of the Many-Body Ground State Wave- functions . 86 4.3.3 Orbital Occupation Numbers and Correlation Entropy . 90 4.3.4 Subspace Diagonalisations and CI Expansions . 93 4.3.5 Conclusions . 95 5 A Projector Quantum Monte Carlo Method for Non-linear Wave- functions 97 5.1 Non-linear Projector Quantum Monte Carlo Method: Combining Varia- tional and Projector Quantum Monte Carlo . 99 5.2 Derivation: Lagrangian Minimisation . 99 5.3 Sampling of the Lagrangian Gradient . 101 5.4 Momentum Methods: Nesterov’s Accelerated Gradient Descent . 111 5.5 Matrix Polynomials - Closing a Circle of Connections . 114 5.6 Step Size Adaptation: RMSprop . 118 5.7 Wavefunction Properties . 119 5.8 A Technical Point: Wavefunction Normalisation, Step Sizes and Param- eter Updates . 124 5.9 Comparison to State-of-the-art VMC Methods . 125 Table of contents xiii 6 Applications of the Projector Quantum Monte Carlo Method for Non-linear Wavefunctions to Correlator Product State Wavefunc- tions 127 6.1 The Correlator Product State Wavefunction Ansatz . 129 6.2 Wavefunction Amplitudes and Derivatives . 131 6.3 2D Hubbard Model . 132 6.4 1D Hubbard Model . 136 6.5 Linear H50 Chain . 141 6.6 Graphene Sheet . 143 6.7 Conclusions . 147 7 Projector Quantum Monte Carlo Method for Tensor Network States149 7.1 The Computational Challenge of PEPS Wavefunctions - Can It Be Tackled Stochastically ? . 151 7.2 The PEPS and MPS Wavefunctions Ansatz . 152 7.3 Wavefunctions Amplitudes and Derivatives . 154 7.4 A Sampling Approach for Wavefunction Amplitudes and Derivatives . 155 7.5 Investigation of TNS Projection Sampling Approach . 158 7.6 Applications for MPS and PEPS Wavefunctions . 165 7.7 Conclusions . 169 8 Concluding Remarks and Continuing Directions 171 Appendix A Comparison of Lagrangian and Ritz Functional Deriva- tives 177 References 179 List of figures 2.1 Excitation level classification of configurations . 27 2.2 Construction of MPS . 36 2.3 Construction of PEPS . 40 3.1 An example FCIQMC calculation . 65 3.2 A comparison of FCIQMC and i-FCIQMC simulations . 68 3.3 Convergence of initiator error . 70 4.1 Phase conventions for three-band (p − d) Hubbard model . 80 4.2 Clusters on 2D square lattice . 83 4.3 Convergence of ground state energy for three-band Hubbard model with total walker number in i-FCIQMC .