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Variational Monte Carlo Renormalization Group Variational Monte Carlo Renormalization Group Yantao Wu A Dissertation Presented to the Faculty of Princeton University in Candidacy for the Degree of Doctor of Philosophy Recommended for Acceptance by the Department of Physics Adviser: Roberto Car May 2021 © Copyright by Yantao Wu, 2021. All rights reserved. Abstract The renormalization group is an important method to understand the critical behaviors of a statistical system. In this thesis, we develop a stochastic method to perform the renormalization group calculations non-perturbatively with Monte Carlo simulations. The method is variational in nature, and involves minimizing a convex functional of the renormalized Hamiltonians. The variational scheme overcomes critical slowing down, by means of a bias potential that renders the coarse-grained variables uncorrelated. When quenched disorder is present in the statistical system, the method gives access to the flow of the renormalized Hamiltonian distribution, from which one can compute the critical exponents if the correlations of the renormalized couplings retain finite range. The bias potential again reduces dramatically the Monte Carlo relaxation time in large disordered systems. With this method, we also demonstrate how to extract the higher-order geometrical information of the critical manifold of a system, such as its tangent space and curvature. The success of such computations attests to the existence and robustness of the renormalization group fixed-point Hamiltonians. In the end, we extend the method to continuous-time quantum Monte Carlo simulations, which allows an accurate determination of the sound velocity of the quantum system at criticality. In addition, a lattice energy-stress tensor emerges naturally, where the continuous imaginary-time direction serves as a ruler of the length scale of the system. iii Acknowledgements The life in Princeton has been a long, rewarding, and fulfilling time. I would like to first thank my advisor, Roberto Car, for his guidance and mentorship. To me, he is a passionate, open-minded, and thoughtful advisor. He is interested and extremely knowledgeable in many areas of physics, and has shaped my interests as a young researcher. I am often moved by his passion, energy, extreme focus, and child-like curiosity of the unknowns. It has been a privilege and absolute pleasure to have worked with him. I would also like to thank members of the Car group and the Selloni group with whom I have spent countless hours in Frick: Hsin-Yu Ko, Linfeng Zhang, Fausto Martelli, Biswajit Santra, Marcos Calegari Andrade, Bo Wen, Xunhua Zhao, Yixiao Chen, Clarissa Ding, and Bingjia Yang. I want to give my special thanks to Hsin-Yu, Linfeng, Marcos, and Yixiao for stimulating scientific discussions. I would like to thank my friends and classmates in the physics department for friendship and the bits of physics that I have learned from them here and there: Jiaqi Jiang, Xinran Li, Junyi Zhang, Sihang Liang, Xiaowen Chen, Xue Song, Jingyu Luo, Huan He, Jie Wang, Zheng Ma, Zhenbin Yang, Yonglong Xie, Zhaoqi Leng, Yunqin Zheng, Jingjing Lin, Jun Xiong, Bin Xu, Erfu Su, and Wentao Fan. I would like to thank professor Sal Torquato for stimulating discussions and reading this thesis. I would like to thank professor David Huse for advices in research and serving in my committee. I thank professor Robert Austin for guiding me through the experimental project in biophysics. I thank Kate Brosowsky for being a patient helper as I navigate through the graduate student life in the physics department. I also would like to thank professor Lin Lin for hosting me in Berkeley, and professor Lin Wang for hosting me in Beijing. Life in Princeton would not have been this enjoyable without the many friends that I have made during graduate school. In particular, I would like to thank the following iv people for banters, encouragement, and lots of fun: Tianhan Zhang, Kaichen Gu, Lunyang Huang, Wenjie Su, Youcun Song, Wenxuan Zhang, Yezhezi Zhang, Yi Zhang, Yinuo Zhang, Pengning Chao, Lintong Li, Fan Chen, and the Princeton Chinese soccer team. I also thank the Terascale Infrastructure for Groundbreaking Research in Science and Engineering (TIGRESS) High-Performance Computing Center and Visualization Laboratory at Princeton University. In the end, I would like to thank my family and, in particular, my mom for their unconditional support, without whom none of this would be possible. I have always come to my mom for encouragement and advice in the lows of life. From her, I have learned sharing, humor, and never giving up. I dedicate this thesis to her. v To my mom. vi Contents Abstract..................................... iii Acknowledgements............................... iv List of Tables..................................x List of Figures.................................. xiv 1 Introduction1 2 Variational approach to Monte Carlo renormalization group7 2.1 An introduction to real-space renormalization group..........7 2.1.1 Scaling operators and critical exponents............ 10 2.2 Variational principle of the renormalized Hamiltonian........ 12 2.3 Monte Carlo sampling of the biased ensemble............. 15 2.4 Results of the renormalized couplings constants............ 16 2.5 Critical exponents in variational Monte Carlo renormalization group. 18 2.6 Appendix................................. 21 2.6.1 The coupling terms Sα ...................... 21 3 Tangent space and curvature to the critical manifold of statistical system 23 3.1 The critical manifold of a statistical system.............. 23 3.2 The critical manifold tangent space................... 25 vii 3.2.1 Critical manifold tangent space in the absence of marginal operators............................. 25 3.2.2 Critical manifold tangent space in the presence of marginal operators............................. 26 3.2.3 The Normal Vectors to Critical Manifold Tangent Space... 27 3.3 Numerical results for CMTS...................... 28 3.3.1 2D Isotropic Ising model..................... 28 3.3.2 3D Istropic Ising Model..................... 29 3.3.3 2D Anistropic Ising Model.................... 31 3.3.4 2D Tricritical Ising Model.................... 32 3.4 Curvature of the critical manifold.................... 34 3.5 Tangent space to the manifold of critical classical Hamiltonians repre- sentable by tensor networks....................... 36 3.5.1 Monte Carlo renormalization group with tensor networks... 37 3.5.2 Numerical Results........................ 40 4 Variational Monte Carlo renormalization group for systems with quenched disorder 47 4.1 The renormalization group of statistical systems with quenched disorder 48 4.2 Numerical results............................. 53 4.2.1 2D dilute Ising model...................... 53 4.2.2 1D transverse field Ising model................. 55 4.2.3 2D spin glass........................... 56 4.2.4 2D random field Ising model.................. 58 4.2.5 3D random field Ising model.................. 59 4.3 Time correlation functions in the biased ensemble........... 61 4.4 Appendix................................. 61 4.4.1 Optimization details for the 2D DIM.............. 61 viii 4.4.2 Couplings in the computation of critical exponents of the dilute Ising model............................ 64 4.4.3 Optimization details for the 3D RFIM............. 65 5 Variational Monte Carlo renormalization group for quantum sys- tems 68 5.1 Continuous-time Monte Carlo simulation of a quantum system.... 68 5.2 The MCRG for continuous-time quantum Monte Carlo........ 71 5.3 The sound velocity of a critical quantum system............ 72 5.3.1 Q = 2: The Ising model..................... 74 5.3.2 Q = 3 and 4 ............................ 76 5.4 The energy-stress tensor......................... 78 5.5 Appendix................................. 81 5.5.1 Spacetime correlator at large distance and low temperature.. 81 6 Outlook 83 Bibliography 85 ix List of Tables 0 2.1 Leading even (e) and odd (o) eigenvalues of @Kα at the approximate @Kβ fixed point found with VRG, in both the unbiased and biased ensembles. The number in parentheses is the statistical uncertainty on the last digit, obtained from the standard error of 16 independent runs. 13 (5) coupling terms are used for even (odd) interactions. See Sec. 2.6.1 for a detailed description of the coupling terms. The calculations used 106 MC sweeps for the 45 × 45 and 90 × 90 lattices, and 5 × 105 sweeps for the 300 × 300 lattice........................... 20 3.1 Pαβ for the isotropic Ising model. α indexes rows corresponding to the three renormalized constants: nn; nnn; and . The fourth row of the table at the Onsager point shows the exact values. β = 2; 3; and 4 respectively indexes the component of the normal vector to CMTS corresponding to coupling terms nnn; ; and nnnn. β = 1 corresponds to the nn coupling term and Pα1 is always 1 by definition. The simulations were performed on 16 cores independently, each of which ran 3 × 106 Metropolis MC sweeps. The standard errors are cited as the statistical uncertainty....................... 30 x 3.2 Pαβ for the odd coupling space of the isotropic Ising model. α indexes rows corresponding to the four renormalized odd spin products: (0, 0), (0, 0)-(0,1)-(1,0), (0, 0)-(1, 0)-(-1,0) and (0, 0)-(1,1)-(-1,-1), where the pair (i; j) is the coordinate of an Ising spin. The simulations were performed on 16 cores independently, each of which ran 3 × 106 Metropolis MC sweeps. The standard errors are cited as the statistical uncertainty................................. 30 3.3 Pαβ for the 3D isotropic Ising model. The two rows in the table correspond to the two different α which respectively index the nn and the nnn renormalized constants. β runs from 1 to 8, corresponding to (0) the following spin products, Sβ (σ): (0, 0, 0)-(1, 0, 0), (0, 0, 0)-(1, 1, 0), (0, 0, 0)-(2, 0, 0), (0, 0, 0)-(2, 1, 0), (0, 0, 0)-(1, 0, 0)-(0, 1, 0)-(0, 0, 1), (0, 0, 0)-(1, 0, 0)-(0, 1, 0)-(1, 1, 0), (0, 0, 0)-(2, 1, 1), and (0, 0, 0)-(1, 1, 1), where the triplet (i; j; k) is the coordinate of an Ising spin.
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