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

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 representable 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 systems 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. 16 independent simulations were run, each of which took 3 × 105 Metropolis MC sweeps. The simulations were performed at the

nearest-neighbor critical point with Knn = 0.22165...... 31

3.4 Pαβ for the 2D anisotropic Ising model. α indexes rows corresponding

to the four renormalized constants: nnx, nny, nnn, and . β = 2 − 6 respectively indexes the component of the normal vector to CMTS

corresponding to coupling terms nny, nnn, , nnnnx, and nnnny. β = 1

corresponds to the nnx coupling term and Pα1 is always 1 by deﬁnition. 32 3.5 The couplings used in the computation of CMTS for the 2D tricritical Ising model...... 33

3.6 Pαβ for the 2D tricritical Ising model. α indexes rows corresponding to the ﬁrst ﬁve renormalized couplings listed in Table 3.5, which also gives the couplings for β = 2 − 6...... 33

xi 3.7 au1 + bu2 computed from Table 3.6 for α = 3 − 5 and β = 2 − 6... 34

3.8 κβη at the same three critical points as in Table 3.1, calculated from

2 (n) (0) (0) ∂ Knn /∂Kβ ∂Kη . The exact curvature for β = nn and η = nnn at the Onsager point is also shown [28]...... 35 3.9 The tensor elements which are related to one another by symmetry.. 41 (n,0) Aαβ 2 3.10 The matrix Pαβ = (n,0) for the isotropic 2D square Ising model. A 256 Aα1 lattice was used with the renormalization level n = 5. The simulations were performed on 16 cores independently, each of which ran 3 × 106 Metropolis MC sweeps. The mean is cited as the result and twice the standard error as the statistical uncertainty...... 42 3.11 The tensor elements which belong to distinct symmetry classes. Only one representative of each class is listed...... 44 (n,0) Aαβ 3.12 The matrix Pαβ = (n,0) for the isotropic 2D square three-state Potts Aα1 model. A 2562 lattice was used with the renormalization level n = 5. The simulations were performed on 16 cores independently, each of which ran 9 × 105 Metropolis MC sweeps. The mean is cited as the result and the standard error as the statistical uncertainty...... 45 3.13 The tensor elements which belong to distinct symmetry classes. Only one representative of each class is listed...... 46

5.1 The couplings used for d = 1 TFIM. Note that when α = 2 and 3, the couplings are themselves isotropic between space and time...... 75 5.2 The renormalized constants for the d = 1 TFIM. For each n, L = P = 8 × 2n. VMCRG is done with 4000 variational steps. During each variaional step, the MC sampling is done on 16 cores in parallel, where each core does MC sampling of 20000 Wolﬀ steps. The optimization step is µ = 0.001. The number in the paranthesis is the uncertainty on the last digit...... 75 xii (4) 5.3 The renormalized constants for the d = 2 TFIM. L = P = 128. K1,x and (4) K1,yz are respectively the renormalized nearest neighbor spin constants along the time and the space direction at n = 4...... 76 5.4 The couplings used for d = 1, Q = 3 and 4 Potts model. Note that when α = 2, the coupling is itself isotropic between space and time.. 76 5.5 The renormalized constants for the d = 1, Q = 3 Potts model. For each n, L = P = 8 × 2n. When n = 0 to 3, the simulations are done with Metropolis local updates with 1000 variational steps. When n = 0, 1, 2, each variational step uses 100 sweeps of MC averaging in parallel on 8 cores. When n = 3, each variational step uses 500 sweeps. For n = 4 and 5, the simulation details are the same as in Table 5.2...... 77 5.6 The renormalized constants for the d = 1, Q = 4 Potts model. For each n, L = P = 8 × 2n. The simulation details are the same as in Table 5.5. 78

xiii List of Figures

2.1 Variation of the renormalized coupling constants over ﬁve VMCRG iterations on a 300 × 300 lattice. Each iteration has 1240 variational steps, each consisting of 20 MC sweeps. 16 multiple walkers are used for the ensemble averages in Eqs. 2.13 and 2.14. For clarity, we only show the four largest renormalized couplings after the ﬁrst iteration. Top:

Simulation starting with Knn = 0.4365. Bottom: Simulation starting

with Knn = 0.4355...... 18

2.2 (color online). Time correlation of the estimator A = S0(σ)S0(µ) on

45 × 45 and 90 × 90 lattices (Eq. 5.11). S0 is the nearest neighbor term in the simulations of Table 2.1...... 21

3.1 Left: part of a tensor network representing a 2D classical system. ijkl... represent tensor indices. Right: a single tensor in the network. Its four

tensor indices are labeled as i0i1i2i3. A grey circle represents a lattice site, or equivalently a tensor index. A green box represents a tensor.. 38 3.2 Optimization trajectory of the tensor network renormalized constants for the three-state Potts model on a 162 lattice at K = 1.005053. All 81 renormalized constants are independently optimized and shown. Each curve represents one coupling term...... 44 3.3 The tensor in cubic-lattice tensor network. It is associated with 8 spins. 45

xiv 0 4.1 Distribution of Knn for a DIM with KDI = 0.60 (left), 0.609377 (middle), and 0.62 (right). n denotes RG iteration. All ﬁgures have the same scale...... 55

4.2 Distribution of Knny for the Trotter approximation of the TFIM with

KTFIM = 0.935 (left), 1.035 (middle), and 1.135 (right)...... 56 4.3 Distribution of nearest neighbor couplings for a spin glass model with

KSG = 1.2...... 58 4.4 Distribution of the renormalized nearest neighbor (left) and local magne-

tization (right) coupling constants for the 2D RFIM with KRFIM = 0.8

and h0 = 1...... 59 4.5 Distribution of the renormalized coupling constants for the 3D RFIM

with h0 = 0.35, and KRFIM = 0.25 (left), 0.264 (middle), and 0.28 (right). The top row is for the nearnest neighbor couplings. The middle row is for the magnetization couplings. The bottom row is for the next nearest neighbor couplings...... 60

4.6 Distribution of Knn for the 3D random ﬁeld Ising model with h0 = 0.35

and KRFIM = 0.255, 0.257, 0.262, 0.266, 0.27 from left to right. The data are collected from one sample with L = 64...... 61

hM(t)M(0)i 4.7 Time correlation functions CM (t) = hM 2i of the system magneti- 1 P zation M = N i σi, in the biased and unbiased ensemble for various lattice sizes. For the spin glass model, we take the magnetization to

be in the ρ-system. In the biased ensemble, the bias potential VJmin is obtained by VMCRG for b = 2 with 1 RG iteration. The data are collected, shown from top to bottom, for the dilute Ising model with

KDI = KDI,c, the Trotter approximation of the transverse ﬁeld Ising

model with KTFIM = 1.0, the 2D spin glass with KSG = 1.2, and the

2D random ﬁeld Ising model with KRFIM = 0.8, h0 = 1.0,...... 62

xv 4.8 Time correlation functions CM (t) in the biased and unbiased ensemble for various lattice sizes for the 3D random ﬁeld Ising model with

KRFIM = 0.27, h0 = 0.35. The relaxation for the 3D RFIM in the unbiased ensemble is very slow, and an inset is placed in the center of the figure to show the unbiased time correlation functions more clearly. Also note that in the biased ensemble of the 3D RFIM, the time correlation function eventually converges to the average magnetization squared, hMi2, which in the random field Ising model is not necessarily zero due to the random fields...... 63 4.9 Renormalized nearest neighbor constants at criticality during the optimization of Ω[V ] for one bond realization of a 128 × 128 dilute Ising

model, for the ﬁrst 4 RG iterations. KDI = 0.609377...... 63 4.10 Renormalized nearest neighbor constants at criticality during the optimization of Ω[V ] for one bond realization of a 256 × 256 dilute Ising

model for the 5th RG iteration. KDI = 0.609377...... 64 4.11 Renormalized constants for the local magnetization coupling during the optimization of Ω[V ] for one bond realization of a 643 random ﬁeld

Ising model, for the ﬁrst 3 RG iterations. KRFIM = 0.264, h0 = 0.35 . 67 4.12 Renormalized constants for the nearest neighbor product coupling during the optimization of Ω[V ] for one bond realization of a 643 random

ﬁeld Ising model, for the ﬁrst 3 RG iterations. KRFIM = 0.264, h0 = 0.35 67

xvi Chapter 1

Introduction

This dissertation describes a variational method based on Monte Carlo (MC) sampling to carry out the renormalization group (RG) calculations non-perturbatively on lattice spin systems. We name this method the variational Monte Carlo renormalization group (VMCRG). RG theory originated in the context of quantum field theory to deal with the ultraviolet divergences of a continuum theory [1]. Its connection to statistical physics and critical phenomena was first realized and articulated by Wilson [2,3]. In a field theory where an ultraviolet momentum cutoff is required to make the theory well- defined, the precise value of the ultraviolet cutoff often does not affect the field theory’s prediction of physical results, such as a scattering amplitude. The independence of the physical observables on the cutoff values originates from the fact that the physical observables are often long-wave-length and low-energy quantities, while the cutoff value corresponds to the short-wave-length and high-energy details of a theory. The freedom in choosing any cutoff values gives a field theory predictive power. The momentum cutoff corresponds to the lattice spacing of a statistical system, and the freedom in choosing any lattice spacing is also present there. In both a field theory and a statistical system, the way by which to implement this freedom is the renormalization group.

1 First realized by Wilson, the implementation of this freedom itself gives non-trivial information on the macroscopic behavior of a statistical system. The experimental motivation for RG in statistical physics is critical phenomena. In the 1960s, it was discovered, both in experiments and in exactly solvable models, that at phase transitions, the thermodynamic variables often exhibit power-law singularities. The exponents of these power-law singularities were found to be very close in seemingly very different systems. This gave the notion of universality. In addition, it was found that many critical exponents satisfy very simple scaling relations among each other, leaving only a small number, most commonly two or three, of them independent. This pushed people’s attention to scaling relations. Motivated by these phenomenologies, Wilson was able to come up with RG to understand them at a deeper level. The Wilsonian RG, in a sense, is a theory of theories. The many microscopic theories are defined by different lattice spacings or microscopic Hamiltonians. An RG calculation computes how one theory gets transformed into another theory under a scale transformation. When a statistical system is posed at criticality, signaled by thermodynamic quantities becoming singular, invariance emerges when theories are transformed by scale dilations, i.e. the RG calculation exhibits a fixed-point. To explicitly carry out an RG calculation, however, is difficult. While Wilson spelled out the general procedure to carry out an RG calculation, his early calculations [2] were based on the diagrammatic expansions of partition functions, and were thus perturbative. Since Wilson’s seminal work, there has been strong interest in methods to compute the renormalized coupling constants and the critical exponents in a non-perturbative fashion. This goal has been achieved with the Monte Carlo renormalization group (MCRG) approach of Swendsen. In 1979, he introduced a method to compute the critical exponents, which did not require explicit knowledge of the renormalized Hamiltonian [4]. A few years later, he solved the problem of calculating the renormalized coupling constants for the Ising model, using

2 an equality due to Callen [5] to write the correlation functions in a form explicitly depending on the couplings. Swendsen was able to convert doing an RG calculation to measuring certain correlation functions in the system, which the Monte Carlo simulation was ready to do. A serious limitation of Swendsen’s MCRG is that the MC simulations can often be very difficult. In pure systems, i.e. systems with no quenched disorder, the MC simulations suffers from critical slowing down at criticality. In systems with quenched disorder, the MC simulations face challenges such as frustration in spin glasses and large field pinning down in random field models, even not at criticality. Inspired by the enhanced sampling techniques [6] in free energy sampling, we developed a variational approach to MCRG that alleviates these sampling difficulties greatly. In addition, for disordered systems, one needs the evolution of the distribution of the renormalized coupling constants, which requires their explicit computation. In our approach, the coupling constants are explicitly obtained by minimizing a certain variational principle. The critical exponents also derive from the same principle and Swendsen’s formulae emerge as a special case. Here we provide summaries of the individual chapters of this dissertation. In Chapter2, we first review the variational functional whose utility was initially demonstrated in free energy sampling. We then formulate it in a form that is useful to MCRG. We will identify the coarse-grained spins in RG as the order parameters whose Landau free energy profiles are often computed through statistical sampling [6]. The renormalized Hamiltonian will be numerically obtained through an MC simulation. A truncation error will occur and will be discussed in detail. The critical couplings of a system can be obtained through looking for the couplings that render the renormalized Hamiltonian of the system to flow into a non-trivial fixed-point Hamiltonian. The Jacobian matrix of the RG transformation will also be obtained in the same MC simulation, which, if computed at the fixed-point Hamiltonian, allows one to obtain

3 the critical exponents of a statistical system. We will demonstrate how the critical slowing down is greatly reduced in the variational MC simulation. The work in this chapter has been published previously as the following refereed journal article:

• Yantao Wu and Roberto Car. “Variational Approach to Monte Carlo Renor- malization Group”. In: Phys. Rev. Lett. 119 (22 2017), p. 220602. doi: 10.1103/PhysRevLett.119.220602.

In Chapter3, we show how to use VMCRG to extract higher order information of the critical manifold of a statistical system. Here, the zeroth order information of the critical manifold is the location of the critical couplings, and the ﬁrst order information is the tangent space of the critical manifold, and so forth. We pay special attention to the Jacobian matrix of the RG transformation. We will show that due to the reduction in critical slowing down, one is able to sample the entire Jacobian matrix well, for the models considered. The kernel of this matrix gives the tangent space of the critical manifold a statistical system. This determination will be free of truncation error. The curvature of the critical manifold can also be accessed with higher-order correlation functions in the MC and thus more statistical noise. The work in this chapter has been published previously as the following refereed journal article:

• Yantao Wu and Roberto Car. “Determination of the critical manifold tangent space and curvature with Monte Carlo renormalization group”. In: Phys. Rev. E 100 (2 2019), p. 022138. doi: 10.1103/PhysRevE.100.022138.

• Yantao Wu. “Tangent space to the manifold of critical classical Hamiltonians representable by tensor networks”. In: Phys. Rev. E 100 (2 2019), p. 023306. doi: 10.1103/PhysRevE.100.023306.

In Chapter4, we show how to use VMCRG to study statistical systems with quenched disorder. In disordered systems, one focuses on the renormalization of 4 the distribution of quenched Hamiltonians, and the scaling variables parametrize the renormalized distributions. We show how to extract these scaling information from VMCRG. We will also demonstrate that sampling diﬃculties associated with the quenched disorder can be greatly alleviated by VMCRG. The work in this chapter has been published previously as the following refereed journal article:

• Yantao Wu and Roberto Car. “Monte Carlo Renormalization Group for Classical Lattice Models with Quenched Disorder”. In: Phys. Rev. Lett. 125 (19 2020), p. 190601. doi: 10.1103/PhysRevLett.125.190601.

In Chapter5, we show how to use VMCRG to study quantum mechanical system simulatable by a continuous-time MC. One maps the partition function of a quantum system into its continuous-time path-integral representation. The quantity that can be extracted very accurately is the sound velocity of a critical quantum system with conformal symmetry. In addition, one can use the continuous nature of the time dimension to determine the energy stress tensor of a statistical system on a discrete lattice. The work in this chapter has been published previously as the following refereed journal article:

• Yantao Wu and Roberto Car. “Continuous-time Monte Carlo renormalization group”. In: Phys. Rev. B 102 (1 2020), p. 014456. doi: 10.1103/Phys- RevB.102.014456.

In addition to the work mentioned above, VMCRG has also indirectly inspired the following research work during this Ph.D:

• Yantao Wu and Roberto Car. “Quantum momentum distribution and quantum entanglement in the deep tunneling regime”. In: The Journal of Chemical Physics 152.2 (2020), p. 024106. doi: 10.1063/1.5133053.

5 • Yantao Wu. “Nonequilibrium renormalization group ﬁxed points of the quantum clock chain and the quantum Potts chain”. In: Phys. Rev. B 101 (1 2020), p. 014305. doi: 10.1103/PhysRevB.101.014305.

• Yantao Wu. “Dynamical quantum phase transitions of quantum spin chains with a Loschmidt-rate critical exponent equal to 12 ”. In: Phys. Rev. B 101 (6 2020), p. 064427. doi: 10.1103/PhysRevB.101.064427.

• Yantao Wu. “Time-dependent variational principle for mixed matrix product states in the thermodynamic limit”. In: Phys. Rev. B 102 (13 2020), p. 134306. doi: 10.1103/PhysRevB.102.134306.

• Yantao Wu. “Dissipative dynamics in isolated quantum spin chains after a local quench”. In: (2020). In Review. arXiv: 2010.00700 [cond-mat.stat-mech].

6 Chapter 2

Variational approach to Monte Carlo renormalization group

2.1 An introduction to real-space renormalization

group

As mentioned in Chapter1, the theory of RG is not only significant in statistical mechanics, but also in field theory. In field theory, the major goal of RG is to demonstrate the insensitivity of long-wave length quantities on the momentum cutoff. In statistical mechanics, the goal is very different. RG aims to reveal the underlying reason of the singular behavior of thermodynamic variables when a statistical system is fine-tuned to be at the special critical parameters. In doing so, it makes the notion of scale invariance explicit. It is remarkable that while the problems in field theory and in statistical mechanics look so different on the surface, they are only different facets of the same physical structure, i.e. the insensitivity of the low-energy quantities on the high-energy details of the theory in the ultraviolet limit for field theory, and in the thermodynamic limit for statistical mechanics.

7 Here we focus on statistical mechanics and brieﬂy review the theory of real-space RG in its context. The central quantity of a statistical system is its partition function. When at criticality, the partition function depends non-analytically on the system parameters, such as the temperature, pressure, volume, etc. To isolate the non-analytic nature of the system from its analytic background, the RG procedure extracts an analytic part from the partition function in each iteration while keeping its non- analytic part intact. After many iterations of peeling oﬀ the analytic parts, the non-analytic part of the partition function will be isolated out and its structure will become apparent. To be concrete, let us consider a statistical mechanical system in d spatial dimensions with spins σ and Hamiltonian H(0)(σ),

(0) X (0) H (σ) = Kβ Sβ(σ) (2.1) β

where Sβ(σ) are the coupling terms of the system, such as nearest neighbor spin

(0) (0) products, next nearest neighbor spin products, etc., and K = {Kβ } are the corresponding coupling constants. Here we call the original Hamiltonian before any RG transformation the zeroth level renormalized Hamiltonian, hence the notation (0) in the superscript. RG considers a ﬂux in the space of Hamiltonians (4.1) under scale transformations that reduce the linear size of the original lattice by a factor b, with b > 1. The motivation to consider a scale transformation is in part due to the early solutions of exactly solvable models [7], where it was discovered that the correlation functions at criticality exhibit power-law, instead of exponential, decay with distance. A power-law dependence is essentially scale-less. For example, consider a dimensionless quantity which decays as a power-law as a function of distance r: q(r) = 1/(ar)b = a−b1/rb, the quantity a with dimension inverse distance can always be factored out as a

8 trivial multiplicative factor. This is very different from an exponential function q(r) = exp(−ar) where a cannot be factored out and is intrinsic. Thus a critical statistical system is thought to be scale-invariant 1. In a real-space RG calculation, one defines coarse-grained spins σ0 in the renormalized system with a conditional probability T (σ0|σ) that effects a scale transformation with scale factor b. T (σ0|σ) is the probability of σ0 given spin configuration σ in the original system. The majority rule block spin in the Ising model proposed by Kadanoff [8] is one example of the coarse-grained variables. T (σ0|σ) can be iterated n times to define the nth level coarse-graining T (n)(µ|σ) realizing a scale transformation with scale factor bn:

X X T (n)(µ|σ) = .. T (µ|σ(n−1)) ··· T (σ(1)|σ) (2.2) σ(n−1) σ(1)

T (n) deﬁnes the nth level renormalized Hamiltonian H(n)(µ) up to a constant g(K(0)) independent of µ [9]:

X (0) H(n)(µ) ≡ − ln T (n)(µ|σ)e−H (σ) + g(K(0)) σ (2.3) X (n) (0) = Kα Sα(µ) + g(K ) α

(n) where {Kα } are the nth level renormalized coupling constants associated with the

(0) coupling terms Sα(µ) defined for the nth level coarse-grained spins. Here g(K ) is made unique by requiring that the identity coupling term in H(n)(µ) have a coupling constant of zero. g(K(0)) is the analytic-part of the free energy, or equivalently the partition function, mentioned in the second paragraph of this section. Repeated ad infinitum, the RG transformations generate a flux in the space of Hamiltonians, in which all possible coupling terms appear, unless forbidden by symmetry. For example, in an Ising model with no magnetic field, only even spin products appear. The

1A simplified understanding could be simply noting that at criticality, the correlation length diverges, and a scale is missing in the theory 9 space of the coupling terms is, in general, infinite. However, perturbative [3] and non-perturbative [4] calculations suggest that only a finite number of couplings should be sufficient for a given desired degree of accuracy.

2.1.1 Scaling operators and critical exponents

Because the RG procedure generates a flow in the coupling space, there are generally three possibilities for the asymptotic behaviors of the RG flow: i) flows into a fixed- point, ii) forms a periodic orbit, or iii) follows a chaotic trajectory. The last two possibilities do occur [10], but the first possibility is by far the most common scenario in equilibrium phase transitions, and we will focus exclusively on it in this thesis. A statistical system is critical when its RG flows goes into a fixed-point that requires the fine tuning of the system coupling constants for the RG flow to go into. In the simplest case, one only needs to fine-tune one parameter. Let us consider this case first, and denote the critical fixed-point by K∗ 2. Let the controlling coupling constant, for example the temperature, be T and its critical value be Tc. We also define t ≡ T − Tc. If t is different from but sufficiently close to 0, the RG flow will start from K(0), approach K∗, stay around K∗ for a while, and then eventually stray away to one of the non-critical fixed-points. When the RG flow is in the vicinity of K∗, its behavior can be linearized around K∗:

(n+1) (n+1) (n+1) (n+1) ∗ ∂K (n) ∗ ∂K (n) δK ≡ K − K = (n) (K − K ) = (n) δK (2.4) ∂K K∗ ∂K K∗

∂K(n+1) Let the eigenvalues and the eigenvectors of ∂K(n) be λi and φi, ordered by the K∗ (n) descending magnitude of λi. We also define the scaling variables ui as the coordinates 2The critical fixed-point is typically parametrized by an infinite number of coupling constants. But in practice one uses a finite approximations. K∗ is a vector in both cases.

10 (n) of δK in the eigenbasis {φi}:

(n) X (n) (n+1) (n) δK ≡ ui φi, ui = λiui (2.5) i

∗ When |λi| > 1, the RG ﬂow is repulsive against K and φi is called a relevant scaling

∗ operator. When |λi| < 1, the RG ﬂow is attractive to K and φi is called an irrelevant

scaling operator. When |λi| = 1, φi is called a marginal scaling operator.

When t = 0, all the relevant ui must be zero, because otherwise the RG flow will not end up on the critical fixed-point. This means that if one only needs to fine-tune

one parameter for criticality, there must only be one relevant ui. That is, when t = 0,

u1 = 0. If it takes a ﬁnite number, n, of RG transformations to bring the system

∗ (n) into the linear regime around Kc , one expects that ui depends analytically on t. (n) In particular, to the leading order of t, u1 = a δt, where a depends on n. Near

criticality, the free energy per site can be separated into a singular part, fs, which

depends non-analytically on t, and a regular part, fg, which depends analytically on t. When a ﬁnite number of RG iterations are done, the free energy per site has the following scaling relation [11]

−nd (n) fs(t) = b fs(u1 ) (2.6) −nd (n) −nd (n+1) −nd d (n) fs(λ1t) = b fs(λ1u1 ) = b fs(u1 ) = b b fs(u1 )

Thus,

−d −nd n fs(t) = b fs(λ1t) = b fs(λ1 t) (2.7) which implies that

d/y fs(t) = f±|t| , where y = logb λ1 (2.8)

11 where f+ is for t > 0, and f− for t < 0. Thus, d/y determines all the critical exponents controlled by the critical fixed-point. This explains the connection between the critical exponents and the Jacobian matrix of the RG transformation. Thus, in a numerical calculation of RG, one needs to determine first the critical value of the control parameter, and the RG Jacobian matrix at the critical fixed-point.

2.2 Variational principle of the renormalized Hamil-

tonian

To calculate accurately the renormalized Hamiltonian, H(n), is generally very difficult. Perturbation theory has been very successful [3], but at the same time very difficult and unable to take into non-perturbative effects. See [12] for a heroic effort in applying the perturbation theory to φ4 theory. A numerical but non-perturbative approach is to study the RG flow with MC sampling, where the correlation functions sampled should provide sufficient information to determine the RG flow. In the proximity of a critical point, the coarse-grained spins µ displays a divergent correlation length, originating critical slowing down of local MC updates. This can be avoided by modifying the distribution of the µ by adding to the Hamiltonian H(n)(µ) a biasing potential V (µ) to force the biased distribution of the coarse-grained spins,

pV (µ), to be equal to a chosen target distribution, pt(µ). For instance, pt can be the constant probability distribution. Then the µ have the same probability at each lattice site and act as uncorrelated spins, even in the vicinity of a critical point. It turns out that V (µ) obeys a powerful variational principle that facilitates the sampling of the Landau free energy [6], which follows from the variational principle of the generalized Legendre transformation. In the present context, we deﬁne the

12 functional Ω[V ] of the biasing potential V (µ) by:

P −[H(n)(µ)+V (µ)] µ e X Ω[V ] = log + p (µ)V (µ), (2.9) P e−H(n)(µ) t µ µ

where pt(µ) is a normalized known target probability distribution. As demonstrated in [6], the following properties hold:

1. Ω[V ] is a convex functional with a lower bound.

2. The minimizer, Vmin(µ), of Ω is unique up to a constant and is such that:

(n) H (µ) = −Vmin(µ) − log pt(µ) + constant (2.10)

3. The probability distribution of the µ under the action of Vmin is:

(n) e−(H (µ)+Vmin(µ)) pVmin (µ) = (n) = pt(µ) (2.11) P −(H (µ)+Vmin(µ)) µ e

The above three properties lead to the following MCRG scheme.

First, we approximate V (µ) with VJ(µ), a linear combination of a ﬁnite number

of terms Sα(µ) with unknown coeﬃcients Jα, forming a vector J = {J1, ..., Jα, ..., Jn}.

X VJ(µ) = JαSα(µ) (2.12) α

Then the functional Ω[V ] becomes a convex function of J, due to the linearity of the

expansion, and the minimizing vector, Jmin, and the corresponding Vmin(µ) can be found with a local minimization algorithm using the gradient and the Hessian of Ω:

∂Ω(J) = −hSα(µ)iVJ + hSα(µ)ipt (2.13) ∂Jα

13 ∂2Ω(J) = hSα(µ)Sβ(µ)iVJ − hSα(µ)iVJ hSβ(µ)iVJ (2.14) ∂Jα∂Jβ

Here h·iVJ is the biased ensemble average under VJ and h·ipt is the ensemble average

under the target probability distribution pt. The ﬁrst average is associated to the Boltzmann factor exp{−(H(n)(µ) + V (µ))} and can be computed with MC sampling

(see Sec. 2.3). The second average can be computed analytically if pt is simple enough.

h·iVJ always has inherent random noise, or even inaccuracy, and some sophistication is required in the optimization problem. Following [6], we adopt the stochastic optimization procedure of [13], and improve the statistics by running independent MC simulations, called multiple walkers, in parallel. For further details, consult [6].

(n) The renormalized Hamiltonian H (µ) is given by Eq. 2.10 in terms of Vmin(µ).

Taking a constant pt, we have modulo a constant:

(n) X H (µ) = −Vmin(µ) = (−Jmin,α)Sα(µ) (2.15) α

In this ﬁnite approximation the renormalized Hamiltonian has exactly the same terms

of Vmin(µ) with renormalized coupling constants

(n) Kα = −Jmin,α. (2.16)

The relative importance of an operator Sα in the renormalized Hamiltonian can be

estimated variationally in terms of the relative magnitude of the coeﬃcient Jmin,α.

When Jmin,α is much smaller than the other components of Jmin, the corresponding

Sα(µ) is comparably unimportant and can be ignored. The accuracy of this approx-

imation could be quantiﬁed by measuring the deviation of pVmin (µ) from pt(µ). In

the case of the Ising model, for example, if pt(µ) is the uniform distribution, any

spin correlators should vanish under pt and to determine how close pVmin , one simply

14 measures spin correlators in the fully optimized biased ensemble, and any deviation from zero would indicate an approximation.

2.3 Monte Carlo sampling of the biased ensemble

It is not entirely clear how to sample the observable O(µ) in the biased ensemble, so we explain here the details of the sampling.

P O(µ)e−(H(n)(µ)+V (µ)) hO(µ)i = µ V P e−(H(n)(µ)+V (µ)) µ (2.17) P O(µ)T (n)(µ|σ)e−(H(0)(σ)+V (µ)) = µ,σ P (n) −(H(0)(σ)+V (µ)) µ,σ T (µ|σ)e

Thus, the state space of the MC sampling should the product space of µ and σ: {(µ, σ)}. Let the proposal probability be:

g(µ0, σ0|µ, σ) = g(σ0|σ)T (n)(µ0|σ0) (2.18) with the acceptance probability

g(σ|σ0) A = min 1, e−(∆H+∆V ) (2.19) g(σ0|σ) where ∆H = H(0)(σ0) − H(0)(σ), ∆V = V (µ0) − V (µ). It is easy to prove that this Metropolis MC scheme satisfies the detailed balance. Then if one bases the sampling on the local Metropolis move [14], g(σ|σ0)/g(σ0|σ) = 1, and one uses an acceptance probability of e−(∆H+∆V ). If the MC sampling is based on the Wolff algorithm [15], then the acceptance probability is simply min(1, e−V ), because e−∆H g(σ|σ0)/g(σ0|σ) = 1 in the Wolff algorithm. Note that one can also define the expected value of hO(µ, σ)i according to the ensemble in the second line of Eq. 2.17. 15 2.4 Results of the renormalized couplings constants

There are two ways to carry out the VMCRG to compute the renormalized coupling constants for the nth RG iteration. The first way is to perform the VMCRG with T (1) to obtain H(1) and then use this H(1) as the starting Hamiltonian for another VMCRG calcualtion with T (1) to obtain H(2). This process is repeated n times to obtain H(n). The second way is to perform the VMCRG with T (n) once to obtain H(n). The drawback of the first way is that H(i) is truncated for all i < n, which gives a relatively large truncation error. But the comparative advantage is that in this scheme each VMCRG is done with a small block size b, and the critical slowing down is essentially eliminated. While in the second way the block size is bn and statistical correlation builds up within a block, so critical slowing down is only reduced partially. In this chapter, we focus exclusively on the first way, allowing for some truncation error. In Chapter3, we use the second way. To illustrate the method, we present a study of the Ising model on a 2D square lattice in the absence of a magnetic field. We adopt 3 × 3 block spins with the majority rule. 26 coupling terms were chosen initially, including 13 two-spin and 13 four-spin products. One preliminary iteration of VMCRG was performed on a 45 × 45 lattice starting from the nearest-neighbor Hamiltonian. The coupling terms with renormalized coupling constants smaller than 0.001 in absolute value were deemed unimportant and dropped from further calculations. 13 coupling terms, including 7 two-spin and 6 four-spin products, survived this criterion and were kept in all subsequent calculations. Each calculation consisted of 5 VMCRG iterations starting with nearest-neighbor

coupling, Knn, only. All the subsequent iterations used the same lattice of the initial iteration. Standard Metropolis MC sampling [14] was adopted, and the calculations

16 were done at least twice to ensure that statistical noise did not alter the results signiﬁcantly.

In Fig. 2.1, results are shown for a 300 × 300 lattice with two initial Knn, equal to

0.4355 and to 0.4365, respectively. When Knn = 0.4365, the renormalized coupling constants increase over the ﬁve iterations shown, and would increase more dramatically with further iterations. Similarly, they decrease when Knn = 0.4355. Thus, the critical

coupling Kc should belong to the window 0.4355 − 0.4365. The same critical window is found for the 45 × 45, 90 × 90, 150 × 150, and 210 × 210 lattices. Because each iteration is aﬀected by truncation and ﬁnite size errors, less iterations for the same rescaling factor would reduce the error. For example, 4 VMCRG iterations with a 2 × 2 block have the rescaling factor of a 16 × 16 block. The latter is computationally more costly than a calculation with 2 × 2 blocks, but can still be performed with modest computational resources. Indeed, with a 16 × 16 block, RG iterations on a 128 × 128 lattice gave a critical window 0.4394 − 0.4398, very close to the exact value,

Kc ∼ 0.4407, due to Onsager [16]. The statistical uncertainty of the calculated renormalized coupling constants is smaller with the variational method than with the standard method [17]. For example,

using VMCRG and starting with Knn = 0.4365 on a 300 × 300 lattice, we found a renormalized nearest-neighbor coupling equal to 0.38031 ± 0.00002 after one RG iteration with 3.968 × 105 MC sweeps. Under exactly the same conditions (lattice

size, initial Knn, coupling terms and number of MC sweeps) we found instead a renormalized nearest-neighbor coupling equal to 0.3740 ± 0.0003 with the standard method. In the VMCRG calculation we estimated the statistical uncertainty with the block averaging method [18], while we used the standard deviation from 14 independent calculations in the case of the standard method. A small diﬀerence in the values of the coupling constants calculated with VMCRG and the standard method is to be

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Figure 2.1: Variation of the renormalized coupling constants over ﬁve VMCRG iterations on a 300 × 300 lattice. Each iteration has 1240 variational steps, each consisting of 20 MC sweeps. 16 multiple walkers are used for the ensemble averages in Eqs. 2.13 and 2.14. For clarity, we only show the four largest renormalized couplings after the ﬁrst iteration. Top: Simulation starting with Knn = 0.4365. Bottom: Simulation starting with Knn = 0.4355.

expected, because the two approaches are diﬀerent embodiments of the truncated Hamiltonian approximation.

2.5 Critical exponents in variational Monte Carlo

renormalization group

As explained in Sec. 2.1.1, the critical exponents are obtained from the leading

∂K0 eigenvalues of ∂K K∗ , the Jacobian matrix of the RG transformation, at a critical ﬁxed point. Here K0 is the renormalized coupling constants from K after one RG iteration.

∂K0 In order to ﬁnd ∂K near a ﬁxed point, we need to know how the renormalized coupling 0 P constants Kα from a RG iteration on the Hamiltonian H = β KβSβ, change when

Kβ is perturbed to Kβ + δKβ, for ﬁxed target probability pt and operators Sα. The

18 minimum condition, Eq. 2.13, implies dΩ = 0, i.e. for all γ: dJα

P 0 P − β (Kβ Sβ (σ)−Kβ Sβ (µ)) σ Sγ(µ)e P 0 = hSγ(µ)ipt , (2.20) P − β (Kβ Sβ (σ)−Kβ Sβ (µ)) σ e

and P 0 0 P − β ((Kβ +δKβ )Sβ (σ)−(Kβ +δKβ )Sβ (µ)) σ Sγ(µ)e P 0 0 = hSγ(µ)ipt . (2.21) P − β ((Kβ +δKβ )Sβ (σ)−(Kβ +δKβ )Sβ (µ)) σ e

0 Expanding Eq. 2.21 to linear order in δKα and δKβ, we obtain

X ∂K0 A = α B , (2.22) βγ ∂K αγ α β where

Aβγ = hSβ(σ)Sγ(µ)iV − hSβ(σ)iV hSγ(µ)iV , (2.23)

and

Bαγ = hSα(µ)Sγ(µ)iV − hSα(µ)iV hSγ(µ)iV . (2.24)

Here h·iV denotes average under the biased ensemble in Eq. 2.17.

If we required the target average of Sγ(µ) to coincide with the unbiased average

P 0 under H(µ) = β KβSβ(µ), the bias potential would necessarily vanish and Eqs. 5.11-3.13 would coincide with Swendsen’s formulae [4]. If we used a uniform target probability, the µ at diﬀerent sites would be uncorrelated, and critical slowing down would be absent. In practice, in order to compute the critical exponents, we ﬁrst need to locate

Kc. From the above calculations on the 45 × 45, 90 × 90, and 300 × 300 lattices with a 3 × 3 block spin, we expect that Kc = 0.436 should approximate the critical nearest-neighbor coupling in our model. Then, we use Eqs. 4.12-3.13 to compute the Jacobian of the RG transformation by

setting Kc = 0.436. The renormalized coupling constants after the ﬁrst RG iteration

19 0 represent Kα, and those after the second RG iteration represent Kα. The results for biased and unbiased ensembles are shown in Table 2.1, which reports the leading

∂K0 even (e) and odd (o) eigenvalues of ∂K when including 13 coupling terms, listed in Sec. 2.6.1, for the three L × L lattices with L = 45, 90, and 300. As seen from the table, biased and unbiased calculations give slightly diﬀerent eigenvalues, as one should expect, given that the respective calculations are diﬀerent embodiments of the truncated Hamiltonian approximation. For L = 300 the results are well converged in the biased ensemble. By contrast, we were not able to obtain converged results for this lattice in the unbiased ensemble on the time scale of our simulation. The absence of critical slowing down in the biased simulation is demonstrated in Fig. 2.2, which displays the time decay of a correlation function in the biased and unbiased ensembles.

e o L λ1 λ1 unbiased 45 2.970(1) 7.7171(2) 90 2.980(3) 7.7351(1) biased 45 3.045(5) 7.858(4) 90 3.040(7) 7.870(2) 300 3.03(1) 7.885(5) Exact 3 7.8452

0 Table 2.1: Leading even (e) and odd (o) eigenvalues of ∂Kα at the approximate ﬁxed ∂Kβ 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.

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Figure 2.2: (color online). Time correlation of the estimator A = S0(σ)S0(µ) on 45 × 45 and 90 × 90 lattices (Eq. 5.11). S0 is the nearest neighbor term in the simulations of Table 2.1.

2.6 Appendix

2.6.1 The coupling terms Sα

We adopt the following notation for the coupling terms. Each spin product is deﬁned by its vertices on the square lattice. The vertices are labeled by 2 integers that represent their coordinates relative to the origin {0, 0}. For example, the nearest neighbor coupling is represented by the pair {0, 0} and {0, 1}. We include the 7 two-point products and the 6 four-point products listed below:

1. {0, 0}, {1, 0}

2. {0, 0}, {1, 1}

3. {0, 0}, {2, 0}

4. {0, 0}, {2, 1}

5. {0, 0}, {2, 2}

6. {0, 0}, {3, 0}

7. {0, 0}, {3, 1}

8. {0, 0}, {1, 0}, {0, 1}, {1, 1}

9. {0, 0}, {1, 1}, {2, 0}, {1, -1} 21 10. {0, 0}, {-1, 0}, {1, 0}, {0, 1}

11. {0, 0}, {-1, 0}, {1, 0}, {-1, 1}

12. {0, 0}, {0, 1}, {1, 0}, {-1, 1}

13. {0, 0}, {0, 1}, {1, 0}, {-1, -1}

Odd products are necessary to compute the leading odd eigenvalue of the Jacobian matrix. For that we use the total magnetization (1-point) and the 4 three-point spin products given below:

1. {0, 0}, {0, -1}, {-1, 0}

2. {0, 0}, {-1, 0}, {1, 0}

3. {0, 0}, {1, -1}, {-1, 0}

4. {0, 0}, {1, -1}, {-1, -1}

22 Chapter 3

Tangent space and curvature to the critical manifold of statistical system

In this chapter, we use VMCRG to study the geometry of the critical manifold of a statistical system. The determination of the tangent space and the curvature of the critical manifold requires the sampling of the full RG Jacobian matrix which is only practically doable in the biased ensemble where critical slowing down is reduced. We also would like the determination to be free of truncation errors, so in this chapter, we perform VMRG with T (n) to directly arrive at H(n) so that truncation errors are not introduced in between. As will be explained the study of the critical manifold will serve as a test of the fundamental assumptions of the RG theory.

3.1 The critical manifold of a statistical system

As explained in the last chapter, the introduction of RG theory in statistical physics by Wilson has greatly deepened our understanding of phase transitions. Our understanding of RG, however, is far from complete. The critical manifold of a lattice model is defined as the set of coupling constants for which the long range physics of the system is described by a unique underlying scale-invariant field theory. However, 23 the same lattice model may admit different critical behaviors described by different field theories, upon changing the coupling constants. This is the case, for instance, in the tricritical Ising model to be discussed later in the chapter. Thus, the critical manifold is always defined with respect to the field theory underlying the lattice model. It could be defined in any space of coupling constants associated with a finite number of coupling terms, with co-dimension in that space equal to the number of relevant operators of the system. General RG theory requires that the RG flow should go into a unique fixed-point Hamiltonian, if the starting point of the flow is on the critical manifold. There are various “natural” RG procedures where different points on a critical manifold do not go to the same critical fixed-point, the most well-known example being the decimation rule in dimension higher than one [11]. By contrast, when an RG procedure satisfies this requirement, the attractive basin of the critical fixed-point is the entire critical manifold, and a computational scheme should exist, at least in principle, to identify the critical manifold. Whether or not this approach can be successfully pursued would be a stringent test of the RG procedure under consideration. Conversely, the knowledge of the critical manifold provides a straightforward way to check the validity of any RG procedure: one could simply simulate the RG flow starting from two different points in the critical manifold and verify that they eventually land on the same fixed-point. This consideration alone should be enough motivation for developing a method to compute the critical manifold. Another issue for which the knowledge of the critical manifold would be of interest is the study of the geometry of the coupling constant space, i.e. the parameter manifold of a classical or quantum many-body system. How to define a Riemannian metric in the parameter manifold has been proposed since long time for both classical [19] and quantum systems [20]. Recently, there have been developments in understanding the significance of the geometry of the parameter manifold for both classical and quantum

24 systems [21, 22, 23, 24, 25]. One would expect knowledge of the critical manifold would ﬁt naturally into such developments. In the following sections, we present a method to determine the tangent space and curvature of the critical manifold at the critical points of a system with Variational Monte Carlo Renormalization Group (VMCRG) [26]. The method relies on taking derivatives of the minimizing conditions of the variational functional:

hSγ(µ)iVmin = hSγ(µ)ipt (3.1)

We will show that unlike the computation of the critical exponents with Monte Carlo Renormalization Group [4] or VMCRG, the determination of the critical manifold tangent space (CMTS) and curvature does not suffer truncation error no matter how few renormalized coupling terms are used. We discuss first the case where there are no marginal operators along the RG flow, and then the case where there are. The examples that we consider in this paper are all classical, but the method can be extended to quantum systems if a sign-free path integral representation of the quantum system would be available, as for example in Chapter5.

3.2 The critical manifold tangent space

3.2.1 Critical manifold tangent space in the absence of

marginal operators

(0) (0) (0) To compute the CMTS, let us suppose that Kβ and Kβ +δKβ belong to the critical manifold and apply the RG procedure starting from these two points. As the diﬀerence in the irrelevant directions becomes exponentially suppressed with progressively large n, the corresponding two renormalized Hamiltonians will tend to the same Hamiltonian H(n) in the absence of RG marginal operators. In particular, the truncated coupling

25 (n) (n) (n) (0) constants, Kα,truncate and Kα,truncate + δKα,truncate, renormalized respectively from Kβ (0) (0) and Kβ + δKβ , will be equal within deviations exponentially small with n, because they are the truncation approximation for two Hamiltonians, H(n) and H(n) + δH(n), whose diﬀerence is exponentially small in n. Thus, the spanning set of the CMTS,

(0) {δKβ }, satisﬁes the following equation for suﬃciently large n,

(n) (n) X ∂Kα,truncate (0) (n) Kα,truncate + (0) δKβ = Kα,truncate (3.2) β ∂Kβ

(0) for every α. That is, the CMTS {δKβ } is the kernel of the nth level RG Jacobian:

(n) (n,0) ∂Kα,truncate Aαβ ≡ (0) (3.3) ∂Kβ

(n) for any well-defined truncation scheme. In the following, we will use Kα to denote the truncated coupling constants. As shown in Chapter2, VMCRG provides an efficient way to compute the renormalized constants and the RG Jacobian matrix with MC under a given truncation scheme. Because the minimizer of Ω is unique, the truncation scheme is well-defined.

3.2.2 Critical manifold tangent space in the presence of

marginal operators

When there are marginal operators in the RG transformation, two different points on the critical manifold will converge to different fixed-point Hamiltonians. However, starting from any point on the critical manifold, at sufficiently large n, H(n) will be

(n+1) (n) equal to H , and so will the truncated renormalized constants Kα be equal to

(n+1) (0) (0) (0) Kα . Now suppose that both Kβ and Kβ + δKβ are on the critical manifold, (n) (n) (n) respectively giving rise to the truncated renormalized constants Kα and Kα + δKα .

(0) Then, the spanning set of CMTS, {δKβ }, instead of Eq. 3.2, satisﬁes the following 26 condition, (n) (n+1) (n) X ∂Kα (0) (n+1) X ∂Kα (0) Kα + (0) δKβ = Kα + (0) δKβ (3.4) β ∂Kβ β ∂Kβ

(n) (n+1) for every α. But Kα and Kα are already equal up to an exponentially small diﬀerence, because they are renormalized from the same point on the critical manifold. Thus, when marginal operators appear in the RG transformation, the CMTS is the kernel of the matrix,

(n+1,0) (n,0) Aαβ − Aαβ (3.5)

3.2.3 The Normal Vectors to Critical Manifold Tangent Space

Because of the spin-ﬂip symmetry, the renormalization of the even operators and of the odd operators are decoupled in the examples we consider here, so they can be considered separately. In the Ising models that we discuss later, the co-dimension of the critical manifold is one, and the tangent space is thus a hyperplane and the row vectors of A(n,0) or A(n+1,0) − A(n,0), for systems with or without marginal operators, are orthogonal to this hyperplane. This means that the row vectors of A(n,0) or A(n+1,0) − A(n,0) are all normal vectors to the CMTS and are parallel to one another. Thus, the P matrix deﬁned as

(n,0) (n+1,0) (n,0) Aαβ Aαβ − Aαβ Pαβ = (n,0) or (n+1,0) (n,0) , (3.6) Aα1 Aα1 − Aα1

that contains the normalized row vectors of A(n,0) or A(n+1,0) − A(n,0), should have identical rows. In the tricritical Ising model that we also discuss, the critical manifold in the even

subspace has co-dimension two [27]. In this case, we cannot expect all the rows of Pαβ to be equal. Instead, the rows should form a two-dimensional vector space to which the CMTS is orthogonal. This outcome can be checked, for example, by verifying

27 that all the row vectors of Pαβ lie in the vector space spanned by its first two rows. If such consistency checks can be satisfied, it is a testament of the validity of RG theory, which predicts that a critical fixed-point Hamiltonian exists and that the co-dimension of the critical manifold has precisely the assumed value for the models considered in this paper. In general, the CMTS computed from different renormalized couplings will have different statistical uncertainty because the sampling noise differs for different correlation functions in an MC simulation. One should, thus, trust the result with the least uncertainty and use the values computed from other renormalized constants as a consistency check.

3.3 Numerical results for CMTS

3.3.1 2D Isotropic Ising model

Consider the isotropic Ising model on a 2D square lattice with Hamiltonian H(σ)

(0) X (0) X H(σ) = −Knn σiσj − Knnn σiσj (3.7) hi,ji [i,j] where hi, ji denotes the nearest neighbor pairs and [i, j] the next nearest neighbor pairs.

(0) (0) Knn and Knnn are the corresponding coupling constants. This model is analytically

(0) (0) solvable when Knnn = 0 and is critical at the Onsager point with Knn = 0.4407... [16].

(0) (0) Four critical points are ﬁrst located with VMCRG in the coupling space of {Knn ,Knnn}.

(0) (0) This task can be achieved by ﬁxing Knnn and varying Knn while monitoring how the

(n) corresponding renormalized coupling constant Knn varies with n, the RG iteration

(0) (n) index. The largest value of the original coupling constant, Knn,1, for which Knn (0) (n) decreases with n, and the smallest value, Knn,2, for which Knn increases with n, deﬁne (0) (0) the best estimate, within statistical errors, of the interval [Knn,1,Knn,2] of location

28 (0) of the critical coupling, Knn,c. We notice that the calculated renormalized constants

(n) are truncated and we assume here that the truncated Knn increases or decreases

(n) monotonically with the exact Knn . This assumption is very natural and does not seem to be violated in the present study. Alternatively, the same procedure can be

performed by ﬁxing Knn and varying Knnn. In the following VMCRG calculations, we use n = 4,L = 256, and the b = 2 majority rule with a random pick on tie.

(n) We use three renormalized couplings: the nearest neighbor product Knn , the next

(n) (n) nearest product Knnn, and the smallest plaquette K . The model is known to have no marginal operators. The four critical points shown in Table 3.1 all belong to the same critical phase, as they all ﬂow into the same truncated ﬁxed-point renormalized Hamiltonian. The CMTSs are determined at these critical points in a four-dimensional

(0) (0) (0) coupling space spanned by Knn , Knnn, K , and the third nearest neighbor products, (0) Knnnn. The Pαβ is shown in Table. 3.1. In addition, we also show the CMTS at the Onsager point, which is analytically solvable [28]. The CMTS can also be computed in the odd coupling subspace, as we show here for the Onsager point. In this calculation, we take n = 5, L = 256, and again the b = 2 majority rule for coarse-graining. The CMTS in a space of four odd couplings, listed in the legend of Table 3.2, is calculated from the same four renormalized couplings. The result is shown in Table 3.2.

3.3.2 3D Istropic Ising Model

(0) Consider now the same model on a 3D square lattice with Knnn = 0, i.e. the 3D isotropic nearest neighbor Ising model. This model does not have an analytical

(0) solution, but is known to experience a continuous transition at Knn = 0.22165... [29]. To compute the CMTS at this nearest neighbor critical point, we use n = 3, L = 64, and the b = 2 marjority rule with a random pick on tie. The CMTS is computed in an eight-dimensional coupling space {K(0)} spanned by the nearest-neighbor and the

29 (0) (0) Knn Knnn Pα2 Pα3 Pα4 0.4407 0 1.4134(3) 0.5135(3) 1.7963(5) 1.4146(7) 0.5134(7) 1.799(2) 1.413(3) 0.511(3) 1.794(7) Exact 1.4142 0.5139 1.8006 0.37 0.0509 1.3717(4) 0.5242(3) 1.7664(8) 1.375(1) 0.5243(7) 1.773(2) 1.372(4) 0.527(3) 1.773(6) 0.228 0.1612 1.2529(7) 0.5303(4) 1.6545(8) 1.254(1) 0.5318(8) 1.659(2) 1.252(5) 0.535(3) 1.65(1) 0.5 -0.0416 1.4441(4) 0.5019(5) 1.816(1) 1.444(2) 0.503(2) 1.818(4) 1.441(7) 0.499(6) 1.80(1)

Table 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.

(0) (0) Knn Knnn Pα2 Pα3 Pα4 0.4407 0 3.31248(8) 1.65629(4) 1.49852(6) 3.296(2) 1.649(4) 1.479(2) 3.315(3) 1.658(2) 1.503(2) 3.32(5) 1.68(4) 1.51(3)

Table 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.

(n) (n) next nearest-neighbor renormalized coupling constants, Knn and Knnn, as shown in Table 3.3.

30 Pα2 Pα3 Pα4 Pα5 Pα6 Pα7 Pα8 2.642(8) 1.540(8) 6.61(3) 2.46(1) 0.788(3) 6.92(4) 1.99(1) 2.64(2) 1.55(2) 6.7(1) 2.50(2) 0.795(3) 7.0(1) 1.99(2)

Table 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 (0) constants. β runs from 1 to 8, corresponding to 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. 16 independent simulations were run, each of which took 3 × 105 Metropolis MC sweeps. The simulations were performed at the nearest-neighbor critical point with Knn = 0.22165.

3.3.3 2D Anistropic Ising Model

Consider then the anisotropic Ising model on a 2D square lattice with Hamiltonian H(σ)

(0) X (0) X H(σ) = −Knnx σiσj − Knny σiσj (3.8) hi,jix hi,jiy where hi, jix and hi, jiy respectively denote the nearest neighbor pairs along the

(0) (0) horizontal and the vertical direction. In the space of {Knnx ,Knny }, the model is exactly solvable and is critical along the line [7]

(0) (0) sinh(2Knnx ) · sinh(2Knny ) = 1 (3.9)

With the 2 × 2 majority rule, the system admits a marginal operator due to anisotropy in the RG transformation [30]. We performed VMCRG calculations on two crit-

(0) (0) ical points of the system with Knny /Knnx = 2, and 3, with four renormalized

(n) (n) (n) (n) couplings: Knn ,Knn ,Knnn,K . The CMTS is computed in the coupling space x y (0) (0) (0) (0) (0) (0) {Knn ,Knn ,Knnn,K ,Knnnn ,Knnnn } using Eq. 3.5, as shown by P in Table. 3.4. x y x y αβ

31 (0) Knnx Pα2 Pα3 Pα4 Pα5 Pα6 0.304689 0.653(8) 2.387(10) 0.814(8) 1.749(8) 1.21(1) 0.646(4) 2.381(5) 0.807(4) 1.755(4) 1.200(5) 0.647(8) 2.38(1) 0.808(12) 1.747(14) 1.20(1) 0.63(2) 2.37(3) 0.78(3) 1.76(4) 1.22(3) Exact 0.6478 0.240606 0.507(4) 2.241(5) 0.692(7) 1.74(1) 0.957(7) 0.498(2) 2.236(3) 0.681(3) 1.739(3) 0.946(4) 0.499(8) 2.24(1) 0.68(1) 1.736(14) 0.940(14) 0.500(16) 2.23(3) 0.67(3) 1.75(4) 0.94(2) Exact 0.5

Table 3.4: Pαβ for the 2D anisotropic Ising model. α indexes rows corresponding to the four renormalized constants: nnx, nny, nnn, and . β = 2 − 6 respectively indexes the component of the normal vector to CMTS corresponding to coupling terms nny, nnn, , nnnnx, and nnnny. β = 1 corresponds to the nnx coupling term and Pα1 is always 1 by deﬁnition.

3.3.4 2D Tricritical Ising Model

Finally, let us consider the 2D tricritical Ising model with the Hamiltonian

(0) X (0) X 2 H(σ) = −Knn σiσj − K4 σi (3.10) hi,ji i where σ = ±1, 0 and hi, ji denotes the nearest neighbor pairs. In the coupling space of

(0) (0) Knn and K4 , the model admits a line of Ising-like continuous phase transitions, which terminates at a tricritical point. At the tricritical point, the underlying conformal ﬁeld

1 theory (CFT) changes from the Ising CFT with central charge 2 to one with central 7 charge 10 [31]. Accompanying this phase transition is a change in the co-dimension of the even critical manifold, from 1 of the Ising case to 2 of the tricritical case [27]. We compute the CMTS at the tricritical point, which has been determined to occur at

(0) (0) Knn = 1.642(8) and K4 = −3.227(1) both by MCRG [27] and finite size scaling [32]. The coupling space we consider has six couplings, listed in Table 3.5. We use n = 5,L = 256 and the b = 2 majority-rule. The normal vectors to the CMTS are computed using the first five renormalized couplings, as the statistical uncertainty of 32 Coupling 2 1 σi 2 σiσj, i and j nearest neighbor 3 σiσj, i and j next nearest neighbor 4 σiσjσkσl, i, j, k, l in the smallest plaquette 2 5 (σiσj) , i and j nearest neighbor 2 6 (σiσj) , i and j next nearest neighbor

Table 3.5: The couplings used in the computation of CMTS for the 2D tricritical Ising model.

the sixth renormalized coupling is too large. The result is again represented by Pαβ and shown in Table 3.6. As can be seen, the rows of P are not equal within statistical

α Pα2 Pα3 Pα4 Pα5 Pα6 1 2.085(2) 2.100(5) 0.928(1) 2.079(1) 2.073(2) 2 2.200(2) 2.271(3) 1.046(2) 2.190(2) 2.232(2) 3 2.171(1) 2.2285(2) 1.0160(5) 2.163(1) 2.193(1) 4 2.214(1) 2.283(1) 1.04(1) 2.20(1) 2.24(1) 5 2.038(4) 2.03(1) 0.873(2) 2.03(1) 2.00(1)

Table 3.6: Pαβ for the 2D tricritical Ising model. α indexes rows corresponding to the ﬁrst ﬁve renormalized couplings listed in Table 3.5, which also gives the couplings for β = 2 − 6.

uncertainty, indicating that the co-dimension is higher than one. To verify that the co-dimension is two, one can check whether the row vectors for α = 3 − 5 are in the vector space spanned by the ﬁrst two row vectors. Let un be the nth row vector of P. If the hypothesis of co-dimension two were correct, one could write:

u3 = au1 + bu2 (3.11)

and ﬁnd a and b from the ﬁrst two components of the vectors u1, u2, and u3. We

could then check that the remaining components of u3 satisfy the linear relation in Eq. 3.11 with the so found a and b. A similar check can be carried out for the vectors

u4 and u5. The vectors u3, u4, and u5 calculated in this way are reported in Table

3.7. As we can see, the Pαβ for α = 3 − 5 and β = 2 − 6 in Table 3.7 are equal within 33 α Pα2 Pα3 Pα4 Pα5 Pα6 3 2.171 2.230 1.019 2.163 2.194 4 2.214 2.284 1.047 2.204 2.245 5 2.038 2.026 0.872 2.033 2.004

Table 3.7: au1 + bu2 computed from Table 3.6 for α = 3 − 5 and β = 2 − 6.

statistical uncertainty to the corresponding elements in Table 3.6, consistent with a co-dimension equal to two at the tricritical point.

3.4 Curvature of the critical manifold

Next, we compute the curvature of the critical manifold, using the isotropic Ising

(0) model as an example. For a change {δKβ } in the original coupling constants, we expand the corresponding change in the renormalized constants to quadratic order:

X (n,0) (0) 1 X (n,0) (0) δK(n) = A δK + B δK δK(0) (3.12) α αβ β 2 αβη β η β βη

(n,0) (n,0) where Aαβ and Bαβη can be determined by substituting Eq. 3.12 in Eq. 3.1 and (0) (n,0) enforcing equality to second order in δKα . Aαβ is already given in Eq. 4.12. The result for B is that for given β and η, for every γ, one requires

X (n,0) hhSγ(µ),Sα(µ)iiV Bαβη = hhSγ(µ),Sβ(σ)Sη(σ)iiV α X + AαβAνηhhSγ(µ),Sα(µ)Sν(µ)iiV (3.13) αν X − 2 AαηhhSγ(µ),Sβ(σ)Sα(µ)iiV α where the connected correlation functions are again sampled in the biased ensemble

h·iV . Note that Bαβη given above is not symmetric in β and η. In order for it to be

34 interpreted as a second-order derivative, it needs to be symmetrized:

∂2K(n) 1 α = B(n,0) + B(n,0) (3.14) (0) (0) 2 αβη αηβ ∂Kβ ∂Kη

(0) (0) In the coupling space of any pair β and η: {Kβ ,Kη }, the critical manifold of the

2D isotropic Ising model is a curve, and the curvature κβη of the critical curve can be computed from the curvature formula [33] of the implicit curve

(n) (0) (0) Kα (Kβ ,Kη ) = constant (3.15) with the second-order derivatives given in Eq. 3.14. Again, this curvature is determined separately by each renormalized constant α. The result is given Table 3.8. Here we

η K(0) nnn nnnn nn β 0.4407 nn 0.143(8) 0.27(2) 0.21(2) nnn 0.38(2) 0.341(8) 0.20(2) Exact (nn, nnn) 0.148 0.37 nn 0.18(1) 0.23(1) 0.30(3) nnn 0.35(2) 0.32(2) 0.18(3) 0.228 nn 0.35(2) 0.27(3) 0.49(3) nnn 0.35(4) 0.29(2) 0.20(4)

Table 3.8: κβη at the same three critical points as in Table 3.1, calculated from 2 (n) (0) (0) ∂ Knn /∂Kβ ∂Kη . The exact curvature for β = nn and η = nnn at the Onsager point is also shown [28].

(n) only quote the result calculated from the nearest neighbor renormalized constants Kα , α = nn. The curvature computed from other renormalized constants have statistical uncertainty much larger than the ones in Table 3.8. The diﬃculty in sampling the curvature, or generally any higher-order derivatives, compared to the tangent space, can be seen from Eq. 3.13. Note that on the left 35 side of Eq. 3.13, the connected correlation function hhSγ,Sαii is of order N, where N is the system size, but each of the terms on the right side is of order N 2. Thus, a delicate and exact cancellation of terms of order N 2 must happen between the terms on the right hand side of Eq. 3.13 to give a ﬁnal result only of order N. The variance due to the terms on the right hand side, however, will accumulate and give

3 an uncertainty typical for O(N ) quantities as each Sα is of order N. (For the CMTS, the connected correlation functions of interest are also of order N, but the statistical uncertainties are those typical of O(N 2) quantities, as seen in Eq. 4.12.) In general, as an m-th order derivative of the critical manifold is computed, the connected correlation functions of interest will always be of order N, but the correlation functions that need to sampled will be of order N m+1, giving an exceedingly large variance. Thus, although in principle arbitrarily high order information about the critical manifold is available by expanding Eq. 3.1, in practice only low-order knowledge on the critical manifold can be obtained with small statistical uncertainty from a simulation near a single critical point.

3.5 Tangent space to the manifold of critical classical

Hamiltonians representable by tensor networks

In dealing with classical lattice models, in addition to MCRG, there is another successful set of algorithms under the general name tensor network renormalization group (TNRG) [34, 35, 36, 37, 38, 39, 40, 41, 42]. These algorithms are based on the tensor network represetation of a system partition function, and the RG is carried out as a sequence of tensor contractions. To implement a correct RG transformation is not trivial. In a proper implementation of renormalization, the theory of RG requires that

A. the non-critical microscopic Hamiltonians flow into trivial fixed-points characteristic of the phases they represent, 36 B. different critical microscopic Hamiltonians flow into a unique non-trivial fixed-point in the absence of marginal RG operators.

If requirement B is satisfied in TNRG, one would expect that as a critical microscopic tensor is perturbed along the tangent space of the set of critical Hamiltonians representable by a tensor network, the change in the final renormalized tensor at a sufficiently large RG iteration level should change at most to quadratic order of the perturbation. To check this, however, prior knowledge on the behavior of the critical Hamiltonians representable by a tensor network would be necessary. In this section, as an interesting application of VMCRG, we describe how VMCRG can be performed with coupling contants encoded in a tensor network, and in so doing, determine the tangent space to the set of critical Hamiltonians representable by a tensor network. In the following, we will call the set of critical Hamiltonians representable by a tensor network the tensor network critical manifold (TNCM), which is a submanifold of the critical manifold.

3.5.1 Monte Carlo renormalization group with tensor net-

works

We ﬁrst review the tensor network representation of a classical partition function [34, 38]. Although the representation is general, for concreteness let us consider the two-dimensional Ising model on a square lattice with the Hamiltonian:

X H(σ) = −K σxσy (3.16) hx,yi

where hx, yi denotes nearest-neighbor pairs and σx = ±1 on each lattice site. K > 0 is the coupling constant. Its partition function has a tensor network representation

37 [38] shown in Fig. 3.1:

X a b c Z = AijklApqriAnojm ··· (3.17) ijk··· where the superscripts, a, b, c..., on A denote the distinct tensors located at diﬀerent positions on the lattice. The tensor indices, ijkl ··· , take values 0 or 1, labeling a tensor of bond dimension χ = 2. One can also label the spin associated with tensor

p q n o i i Ab Ac 0 1

r i j m

Aa A

e k l v

Ad Ae i2 i3 s t u w

Figure 3.1: Left: part of a tensor network representing a 2D classical system. ijkl... represent tensor indices. Right: a single tensor in the network. Its four tensor indices are labeled as i0i1i2i3. A grey circle represents a lattice site, or equivalently a tensor index. A green box represents a tensor.

a a a A by its relative position, x, to A with the notation σx. As shown in Fig. 3.1 (right panel), there can be four relative positions in a 2D square lattice: x = 0, 1, 2, 3. Note that each spin in the 2D square lattice is associated with two tensors, and can serve,

a b for example, both as σ0 and σ3 in the left panel of Fig. 3.1. We have also deﬁned the

tensor indices of A to be written as Ai0i1i2i3 where the tensor legs 0, 1, 2, 3 are labelled in Fig. 3.1 (right). For example, to describe the homogeneous Ising model in Eq. 3.16, the four-leg tensor Aa has tensor elements:

a K(ηi0 ηi1 +ηi1 ηi3 +ηi3 ηi2 +ηi2 ηi0 ) Ai0i1i2i3 = e (3.18)

where ηi is the Ising spin associated with the tensor index i:

  −1, i = 0 ηi ≡ (3.19)  1, i = 1 38 To perform MCRG, one needs to write the partition function as a conﬁguration sum in terms of a Hamiltonian H(σ):

X Z = e−H(σ) (3.20) {σ}

and the Hamiltonian needs to be written as a sum of Nc coupling terms, Sβ(σ):

N Xc H(σ) = KβSβ(σ) (3.21) β=1

where Kβ is the coupling constant of the corresponding coupling term labeled by β. Traditionally, the coupling terms have been chosen as spin products, such as the nearest-neighbor product. The partition function in Eq. 3.17 and Eq. 3.20 will be equal if the Hamiltonian is given by:

NA X a H(σ) = ln(Ai0i1i2i3 ) (3.22) a=1

a a a a when σ0 = ηi0 , σ1 = ηi1 , σ2 = ηi2 , σ3 = ηi3 . Here NA is the number of tensors in the network. Thus, the Hamiltonian which gives the same partition function as does the tensor network is the following:

NA X X a H(σ) = ln A δ a δ a δ a δ a (3.23) i0i1i2i3 σ0 ,ηi0 σ1 ,ηi1 σ2 ,ηi2 σ3 ,ηi3 a i0i1i2i3

a In translationaly invariant systems, ln Ai0i1i2i3 is independent from a, and

N X XA H(σ) = ln A δ a δ a δ a δ a i0i1i2i3 σ0 ,ηi0 σ1 ,ηi1 σ2 ,ηi2 σ3 ,ηi3 i i i i a=1 0 1 2 3 (3.24) N Xc ≡ KβSβ(σ) β=1

39 where we have identiﬁed the logarithm of each tensor element, ln Ai0i1i2i3 , as one

coupling constant Kβ with a corresponding coupling term Sβ(σ). Thus, the ordered

tuple i0i1i2i3 plays the role of β:

Kβ = Ki0i1i2i3 = ln Ai0i1i2i3 (3.25)

and N XA S (σ) = S (σ) = δ a δ a δ a δ a (3.26) β i0i1i2i3 σ0 ,ηi0 σ1 ,ηi1 σ2 ,ηi2 σ3 ,ηi3 a=1 For a tensor network with n legs and bond dimension χ on each leg, there are

n therefore Nc = χ coupling terms, for example 16 in the case of 2D square lattice Isng model. MCRG can then be performed with Eq. 3.24. Since we are interested in the tensors before any renormalization, which are element-wise positive, taking the logarithm does not pose a problem.

3.5.2 Numerical Results

2D square lattice Ising model

As demonstrated in Eq. 3.24, there is one coupling term for each tensor element. The space of Hamiltonians representable by a tensor network in Fig. 3.1 is thus spanned by 16 coupling terms for the 2D square lattice Ising model. However, if we are only interested in the tensor networks representing the Hamiltonians symmetric under spin ﬂip and the symmetry transformation of the underlying lattice, certain tensor elements should be restrained to equal one another, and there are only four truly distinct tensor elements, listed in Table 3.9. Accordingly there are also only four couplings terms.

The β = 1 coupling term, for example, will be deﬁned as the sum of i0i1i2i3 = 0000

40 β i0i1i2i3 1 0000, 1111 2 0100, 0010, 0001, 0111, 1011, 1101, 1110, 1000 3 0110, 1001 4 0101, 0011, 1010, 1100

Table 3.9: The tensor elements which are related to one another by symmetry.

and 1111 coupling terms:

N XA S (σ) = δ a δ a δ a δ a 1 σ0 ,η0 σ1 ,η0 σ2 ,η0 σ3 ,η0 a=1 (3.27) N XA + δ a δ a δ a δ a σ0 ,η1 σ1 ,η1 σ2 ,η1 σ3 ,η1 a=1 and

K1 = ln A0000 = ln A1111 (3.28)

The coupling terms for β = 2, 3, 4 are analagously summed with their respective symmetry partners according to Table 3.9. The four coupling terms thus formed, however, are not linearly independent, as evidenced by the equation

N =4 Xc Sβ(µ) = NA (3.29) β=1

Here we identify two Hamiltonians if they are diﬀerent only by a constant, so the constant function should be treated as the zero element of the vector space of Hamil- tonians. The vector space of Hamiltonians we will consider is therefore only three

41 dimensional:

3 X H(µ) = KβSβ(µ) + K4(NA − S1(µ) − S2(µ) − S3(µ)) β=1 3 X = (Kβ − K4)Sβ(µ) + constant (3.30) β=1 3 X 0 = KβSβ(µ) + constant β=1

The Jacobian matrix of the RG transformation which we will compute will be that of

0 Kβ: 0(n) (n,0) ∂Kα Aαβ = 0(0) (3.31) ∂Kβ

(n,0) Aαβ In Table 3.10, we report the matrix Pαβ = (n,0) , computed at the nearest-neighbor Aα1 critical tensor in Eq. 3.18 with K = 0.4406868. Its rows are the normal vector to the tangent plane of TNCM, normalized so that the first element of the vector is 1. The consistency among the different rows confirms our assumptions. The statistical

α Pα1 Pα2 Pα3 1 1 -0.522(1) -0.0184(3) 2 1 -0.522(7) -0.018(1) 3 1 -0.522(3) -0.0185(3)

(n,0) Aαβ 2 Table 3.10: The matrix Pαβ = (n,0) for the isotropic 2D square Ising model. A 256 Aα1 lattice was used with the renormalization level n = 5. The simulations were performed on 16 cores independently, each of which ran 3 × 106 Metropolis MC sweeps. The mean is cited as the result and twice the standard error as the statistical uncertainty.

uncertainties of the result, however, are different for different rows, because the connected correlation functions of different coupling terms α, β have different variance in an MC sampling. One should always cite the result with the least statistical

0 uncertainty. In converting the computed δKβ with β = 1, 2, 3 to the actual change in

the tensor elements, δKβ with β = 1, 2, 3, 4, one is free to choose the values of δKβ

42 0 so long as the resultant δKβ = δKβ − δK4 conforms to the computed value. Here we

0 take δKβ = δKβ for β = 1, 2, 3 and δK4 = 0. This freedom is the same multiplicative normalization freedom in a tensor network state. In the end, we present the tangent space to TNCM in matrix form by combining

i0i1 of Ai0i1i2i3 as a row index m = i0 + 2i1 and i2i3 as a column index n = i2 + 2i3. To

the linear order of δK2 and δK3, the set of all critical Hamiltonians representable by of a tensor network in Fig. 3.1 that respect the symmetry of the 2D square lattice is

      4 0 0 0 0.522(1) 1 1 0 0.0184(3) 0 0 0             0 0 −4 0  1 0 0 1   0 0 1 0        ln Ai0i1,i2i3 = Kc  + δK2  + δK3  0 −4 0 0  1 0 0 1   0 1 0 0              0 0 0 4 0 1 1 0.522(1) 0 0 0 0.0184(3) (3.32) where Kc = 0.4406868, and δK2 and δK3 are inﬁnitesimally small, but otherwise arbitrary.

2D square lattice three-state Potts mode

Next consider the three-state Potts model on a 2D square lattice:

X H(σ) = −K δσxσy (3.33) hx,yi where hx, yi denotes nearest-neighbor pairs and K > 0. The spin at each lattice

site takes on σx = 0, 1, 2 three possible values. The system experiences a continuous

phase transition at Kc = 1.005053... [43]. This model is also representable by a tensor network in Fig. 3.1 with bond dimension χ = 3. The map from tensor indices to spin variables is simply the identity map:

ηi = i, for i = 0, 1, 2 (3.34)

43 The tensor-representable Hamiltonian can again be written as in Eq. 3.24 with

4 Nc = 3 = 81. Unlike the 2D Ising model, the symmetry classes of the coupling terms are onerous to identify by hand. VMCRG, however, can be used to ﬁnd the symmetry partners of the many coupling terms. To perform this task, the renormalized constants after one iteration of 2 × 2 majority-rule is determined with all of the 81 couplings terms, shown in Fig. 3.2. The couplings with the same renormalized constants (up to some noise) are then the symmetry partners with one another. There are thus six symmetry

2.5 2 1.5 1

α 0.5 K 0 -0.5 -1 -1.5 0 200 400 600 800 1000 Variational Step Figure 3.2: Optimization trajectory of the tensor network renormalized constants for the three-state Potts model on a 162 lattice at K = 1.005053. All 81 renormalized constants are independently optimized and shown. Each curve represents one coupling term.

classes and coupling terms, listed in Table 3.11.

β i0i1i2i3 1 0000 2 1000 3 1100 4 2100 5 0110 6 2110

Table 3.11: The tensor elements which belong to distinct symmetry classes. Only one representative of each class is listed.

Eliminating the linear dependence, we use the ﬁrst ﬁve coupling terms to span the space of Hamiltonians representable by a tensor network, in which is embedded a four-dimensional critical manifold. (The codimension of the the critical manifold for 44 the 2D three-state Potts model is also one.) The tangent space to TNCM is again (n,0) Aαβ reported as the matrix Pαβ = (n,0) in Table 3.12, from which its matrix form can be Aα1 constructed as in Eq. 3.32.

1 -0.381(2) -0.363(1) -0.216(1) -0.0117(2) 1 -0.381(2) -0.363(1) -0.216(1) -0.0118(3) 1 -0.378(3) -0.364(2) -0.218(2) -0.0123(6) 1 -0.382(7) -0.361(4) -0.218(3) -0.012(1) 1 -0.39(5) -0.36(4) -0.21(2) -0.015(6)

(n,0) Aαβ Table 3.12: The matrix Pαβ = (n,0) for the isotropic 2D square three-state Potts Aα1 model. A 2562 lattice was used with the renormalization level n = 5. The simulations were performed on 16 cores independently, each of which ran 9 × 105 Metropolis MC sweeps. The mean is cited as the result and the standard error as the statistical uncertainty.

3D cubic lattice Ising model

In the end, we consider the Ising model Hamiltonian of Eq. 3.16 in the 3D cubic lattice. Although there has not been TNRG algorithms that generate a proper RG ﬂow for this model, we still present here the result of TNCM in anticipation of further advancement of TNRG in 3D. In the cubic lattice, the tensors have eight legs, shown in Fig. 3.3. They are placed in a network where each spin is associated with two tensors

i4 i6 i5 i7

i0 i2 i1 i3

Figure 3.3: The tensor in cubic-lattice tensor network. It is associated with 8 spins.

and each nearest-neighbor bond of the lattice is accounted once by the network, similar to the case in two dimension (Fig. 3.1, left). 28 = 256 coupling terms are present by Eq. 3.24. Among them are 13 symmetrized coupling terms, found with VMCRG, 45 listed in Table 3.13. Again, eliminating the linear dependence, the ﬁrst 12 coupling

β i0i1i2i3i4i5i6i7 β i0i1i2i3i4i5i6i7 1 00000000 8 11101000 2 10000000 9 10011000 3 11000000 10 11011000 4 01100000 11 01111000 5 11100000 12 00111100 6 11110000 13 10010110 7 01101000

Table 3.13: The tensor elements which belong to distinct symmetry classes. Only one representative of each class is listed.

terms are used to span the vector space of Hamiltonians representable by a 3D tensor (n,0) Aαβ network, which admits a 11-dimensional critical manifold. The matrix Pαβ = (n,0) is Aα,1 rather large, so we only cite here the row with the least statistical uncertainty in Eq. 3.35, and note that the consistency among the rows is indeed observed. A 643 lattice was used with the renormalization level n = 3. The simulations were performed on 464 cores independently, each of which ran 4.9 × 105 Metropolis MC sweeps. The mean is cited as the result and twice the standard error as the statistical uncertainty.

P1β = 1, 0.590(4), −0.151(3), −0.037(2), −0.621(3)

−0.164(2), −0.0241(8), −0.086(2), −0.195(2), (3.35)

−0.218(2), −0.076(1), −0.0176(6)

46 Chapter 4

Variational Monte Carlo renormalization group for systems with quenched disorder

Understanding the phase diagram of quench-disordered systems, such as glasses or materials with a disordered distribution of defects, is a major scientific goal. In random systems, the RG flow of the Hamiltonian distribution is of fundamental importance [44]. Its explicit calculation requires an average of the RG flows of many Hamiltonians, each with an extensive number of quench-disordered couplings. Moreover, in disordered systems MC relaxation times tend to be significantly longer than in pure systems. Although it is an old idea to study quench-disordered systems with real-space renormalization, the task is so computationally challenging that it has not been explicitly carried out within MCRG. So far, the challenge of dealing with many random couplings has been avoided, either by limiting the form of the disordered renormalized Hamiltonian, or by adopting techniques that do not require its explicit calculation [45, 46].

47 As explained above, VMCRG facilitates the calculation of the renormalized coupling constants and critical exponents by mitigating the effects of critical slowing down. In this chapter, we show that this approach makes possible to compute directly the evolution of the coupling distribution under scale transformations in classical quench disordered models, in addition to greatly alleviating sampling difficulties due to disorder. The method is particularly useful when dealing with finite disorder fixed-points whose critical distribution has a finite width that is difficult to estimate perturbatively. In these situations, VMCRG recovers the scaling law for the singular part of the free energy, and leads to a viable scheme for computing the critical exponents, when the evolving distribution can be parameterized in terms of local correlations between the renormalized couplings. The approach can also discern strong disorder fixed-points characterized by a diverging variance of the critical distribution, but in this case, it does not provide a way to compute the critical exponents. Strong disorder fixed points have often been associated to disordered quantum models that are amenable to exact solution with Strong Disorder Renormalization Group (SDRG) techniques [47, 48]. If the partition function of these systems has a sign-free path integral representation, the corresponding classical model can be studied numerically with VMCRG, which then provides an alternative way of assessing the strong disorder character of the critical distribution.

4.1 The renormalization group of statistical systems

with quenched disorder

In the following we consider a generic quench-disordered Hamiltonian with local interactions on a lattice of N sites:

N Ns(α) X X X i,s i,s HK(σ) = − Kα Sα (σ), K ∼ Pv(K) (4.1) α i=1 s=1 48 Here the index α speciﬁes the coupling type, such as nearest neighbor, next nearest neighbor, smallest plaquette, etc. The index i runs over the N lattice sites, while s

runs over the Ns(α) point group symmetry operations that generate distinct couplings of type α stemming from site i. For example, the nearest neighbor coupling has two

i,s terms at each lattice site, while the smallest plaquette has only one. Sα are products

i,s of spins in the neighborhood of i speciﬁed by α and s. The coupling constants Kα

−1 are made dimensionless by incorporating the factor (kBT ) in their deﬁnition. The

i,s vector K denotes the full set {Kα } of couplings corresponding to a disorder realization

drawn from the probability density Pv(K) speciﬁed by the parameter set v. Let the conditional probability T (µ|σ) represent a coarse-graining transformation

corresponding to a scale dilation that preserves the symmetry of Pv(K), such as Kadanoﬀ’s block spin transformation [49]. The corresponding renormalized couplings

0 0 K and Hamiltonian HK0 are related to their original counterparts by

0 X −HK(σ) HK0 (µ) + Ng(K) = − ln T (µ|σ)e (4.2) σ

Here δτ(σ),µ is the Kroneker delta function. g(K) indicates the “background” free energy

0 per site of a RG transformation [9] so that HK0 does not contain spin independent terms. Let R be the RG map of the coupling constants implicitly deﬁned by Eq. 4.2:

K0 = R(K) (4.3)

0 The distribution of the renormalized constants Pv0 (K ) is related to Pv(K) by

Z 0 0 Pv0 (K ) = dKPv(K)δ(K − R(K)) (4.4)

Thus, upon coarse-graining of σ, the renormalization of the coupling constants, from K to K0, induces a renormalization from v to v0.

49 The free energy of a quench-disordered system is

Z X −HK(σ) F (v) = − dKPv(K) ln e , (4.5) σ

and only depends on v. Upon a scale transformation it becomes

Z 0 X −H0 (µ) F (v ) = − dKPv(K) ln e R(K) µ (4.6) Z 0 0 0 X −H 0 (µ) = − dK Pv0 (K ) ln e K µ

By virtue of Eq. 4.5 and 4.6 and because g(K) in Eq. 4.2 is not singular, the following

scaling relation holds for the singular part of the free energy per site, fs(v):

1 f (v) = f (v0), (4.7) s bd s where b is the block size of the scaling transformation and d is the dimensionality of space. Thus, in disordered systems, v rather than K plays the role of scaling variable. In our procedure we calculate K0 = R(K) for a representative number of quenched realizations. Each map involves a large number of disordered coupling constants. Sampling is hampered by the rugged energy landscape and is slowed down by long- range correlations near criticality. VMCRG overcomes these diﬃculties by adding to

0 the renormalized Hamiltonian HK0 (µ) a bias potential V (µ) so that the distribution of

0 µ under the Hamiltonian HK0 (µ) + V (µ) becomes equal to a preset target probability

1 N 0 pt(µ). By choosing the uniform distribution for the latter, i.e. pt(µ) = ( 2 ) for Ising systems, the variables µ are uncorrelated. Thus, finite size effects are greatly reduced because, in the biased system, the correlation functions decay exponentially over a distance approximately equal to b, the linear size of the block spin, even at criticality. In practice, we adopt for V a finite representation that parallels the one of

50 the Hamiltonian in Eq. 4.1:

N Ns(α) X X X i i VJ(µ) = JαSα(µ) (4.8) α i=1 s=1

i The minimizing coeﬃcients, Jmin = {Jα,min}, can be found by minimizing Ω with a gradient descent procedure [6]. In disordered systems, the number of unknown

∂2Ω coeﬃcients is large and we use only the diagonal part of the Hessian i,s j,t in ∂Jα ∂Jβ ∂Ω addition to the gradient i,s in the minimization procedure. Convergence is facilitated ∂Jα by the convexity of Ω[V ] [6]. Empirically, we ﬁnd that the optimization cost increases

i,s linearly with the number of coeﬃcients Jα , making possible calculations on large lattices. We have by virtue of Eq. 2.16:

0 K = −Jmin (4.9)

By iterating this process we construct the map K0 = R(K). The procedure

0 is repeated for ND disorder realizations, generating many K vectors distributed

0 according to Pv0 (K ) at each RG iteration. This distribution can be visualized with histograms representing the marginal distribution of coupling type α:

ND N Ns(α) i,s X X X δ(Kα − (Kα )iD ) Qv(Kα) = (4.10) NDNαNs(α) iD=1 i=1 s=1

Here δ is a delta-function approximant with support , the histogram width. The RG

ﬂow of Qv(Kα) is very informative, because depending on the bare couplings of the

original Hamiltonian, Qv may flow toward a fixed distribution that is either trivial or critical. When the renormalized distribution has finite order asymptotically, this procedure also gives a viable method to compute the critical exponents. We indicate by v∗ the parameter set corresponding to the critical distribution. By Eq. 4.7, in order to

51 compute the critical exponents we should compute the leading eigenvalue(s) of the

∂v0 ∗ Jacobian ∂v at v . Assuming that the correlations between the couplings are short

ranged we may use a ﬁnite set of short-ranged basis functions Uβ(K) to represent

Pv(K) [45]: X − ln Pv(K) = C + vβUβ(K) (4.11) β Here C is a normalizing constant and the index β speciﬁes the coupling correlation

type, such as one-body, two-body, etc., associated to products of diﬀerent Kα or combinations thereof. The sum over β includes terms of increasing range up to some cutoﬀ distance on the lattice. The vector parameter v corresponds to the set of

amplitudes {vβ}. The coupling functions Uβ(K) are sums of local coupling products

i,s that play a role similar to that of the spin functions Sα (σ) in Eq. 4.1. For example, for

Hamiltonians with nearest neighbor (Knn) and next nearest neighbor (Knnn) couplings,

P i,s P i,s P i,s 2 the ﬁrst four Uβ(K) could be U1 = i,s Knn , U2 = i,s Knnn, U3 = i,s(Knn ) , and P i,s i,s ∂v0 ∗ U4 = i,s Knn Knnn. Taking the derivative ∂v in the close proximity of v , we obtain:

X ∂v0 A = α · B , (4.12) βγ ∂v αγ α β where

0 0 Aβγ = hUβUγi − hUβihUγi (4.13)

0 0 0 0 Bαγ = hUαUγi − hUαihUγi (4.14)

Here h·i denotes an average under the distribution Pv(K). Eq. 4.12-5.11 are analogous to the relations originally derived by Swendsen for the Jacobian of the RG ﬂow of K in homogeneous models [4]. In the systems that we studied, we found that a

relatively short cutoﬀ distance in the expansion over the Uβ functions was suﬃcient for convergence.

52 4.2 Numerical results

In this section, we present the numerical results of applying VMCRG on the 2D dilute Ising model, the 1D quantum transverse field Ising model, the 2D spin glass, the 2D random field Ising model, and the 3D random field Ising model. We also present the optimiazation details for the 2D dilute Ising model and the 3D random field Ising model in the appendix, which are similar for other models.

4.2.1 2D dilute Ising model

Let us consider the dilute Ising model (DIM), which in 2D is marginal for the Harris criterion [50] that is commonly used to characterize whether disorder is relevant at criticality. The Hamiltonian is

X ij HDI = −KDI k σiσj (4.15) hi,ji

ij 1 1 Here KDI > 0, hi, ji denotes nearest neighbors, and k = 1 or 2 with probability 2 .

The critical value of KDI is known to be KDI,c = 0.609377... by a duality argument [51]. We adopt the majority rule with a random tie-breaker on b × b blocks with b = 2. Three couplings are included in the renormalized Hamiltonian, namely nearest

neighbor (Knn), next nearest neighbor (Knnn), and smallest plaquette (K), which are the most important couplings in the pure Ising model. The calculations are done

2 on 128 lattices for 4 RG iterations for three values of KDI, i.e. KDI = KDI,c, 0.60,

2 and 0.62. In addition, for KDI = KDI,c, we carry out a 5th iteration on a 256 lattice. For n = 5, we deal with spin blocks of linear size bn = 32, for which spin correlations are significant. In this case, we find that sampling efficiency improves significantly by adopting the Wolff algorithm [15] instead of the Metropolis algorithm [14] used in all other simulations in this paper. For the simulation correlation time, τ ∼ ξz, the

53 cluster algorithm reduces the dynamical exponent z while the bias potential reduces the correlation length ξ.

We report in Fig. 4.1 the RG ﬂow of the marginal distribution Qv(Knn). See

Sec. 4.4.1 for the optimization details. The distribution initiating at KDI = KDI,c

converges to a ﬁxed distribution, whereas for KDI less and greater than KDI,c the distribution approaches the paramagnetic and the ferromagnetic ﬁxed points respec-

tively. The marginal distributions Qv(Knnn) and Qv(K) show similar behaviors. The RG evolution approaches a fixed critical distribution, which has finite width and is non-Gaussian indicating that the 2D DIM remains inhomogeneous at all length scales at criticality. Thus, the dilute and the pure Ising model in 2D do not share the same fixed-point. Indeed, although they have the same critical exponents, according to analytical [52, 53, 54] and numerical [55] studies, the singular dependence of the specific heat with respect to temperature is modified by logarithmic factors in the diluted model compared to the pure model [54].

As detailed in Sec 4.4.2, we use 17 coupling functions Uβ(K) to represent the

distribution Pv(K) in the computation of the critical exponents. With the adopted representation we ﬁnd a value of 2.018(6) for the leading even eigenvalue λe of the Jacobian matrix, to be compared with λe = 2 of the pure Ising model. Note that the relative error of the critical exponent is roughly 1%, on the same order as the relative error of the critical exponent of the pure Ising mode we found before. The error

of our estimate was not reduced by adding more Uβ functions, suggesting that the renormalized Hamiltonian should include more couplings than just nearest neighbor, next nearest neighbor, and square terms for better accuracy.

54 7 n=1 n=1 n=1 n=2 n=2 n=2 6 n=3 n=3 n=3 n=4 n=4 n=4 5 n=5 ) ' 4 K ( Q 3

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.1 0.2 0.3 0.4 0.5 0.6 0.7

K'nn K'nn K'nn

0 Figure 4.1: Distribution of Knn for a DIM with KDI = 0.60 (left), 0.609377 (middle), and 0.62 (right). n denotes RG iteration. All ﬁgures have the same scale.

4.2.2 1D transverse ﬁeld Ising model

Next we consider the random TFIM on a periodic chain of L spins with Hamiltonian:

ˆ X z z X x HTFIM = − kiσî σî+1 − hiσî (4.16) i i

z x where σˆ and σˆ are the Pauli matrices. Here ki and hi are independently drawn

from a Gaussian distribution with standard deviation 0.2, and mean equal to KTFIM and 1.0, respectively. By self-duality, the system experiences a ground-state quantum

phase transition when ki and hi are drawn from the same distribution, i.e. when

KTFIM = 1.0 [48]. The Trotter-approximation of this model at inverse temperature β is an anisotropic nearest-neighbor Ising model on an L × βm periodic 2D lattice with classical Hamiltonian [56]:

L βm X X ki H = − σ σ Trotter m i,j i+1,j i=1 j=1 (4.17) L βm X X 1 hi − ln coth σ σ 2 m i,j i,j+1 i=1 j=1

where σi,j is an Ising spin at the ith column and jth row, and m is the number of Trotter slices. As an approximation to m, β → ∞, we use m = 8, β = 16,L = 128. The 2 × 2 majority-rule block-spin is used despite the anisotropy, as in [56]. Four

55 renormalized coupling terms are included in our VMCRG computation: the nearest

neighbor coupling in the horizontal (Knnx ) and vertical (Knny ) directions, the next

nearest neighbor (Knnn) and smallest plaquette (K) couplings. In Fig. 4.2, we

report the RG ﬂow of the marginal distribution of Q(Knny ). As in the DIM, both paramagnetic and ferromagnetic ﬁxed-points are discovered. At the phase transition,

that we found to be at KTFIM = 1.035 with the adopted Trotter approximation, however, the critical ﬁxed-point is found to have increasing variance, in sharp contrast with the DIM, but consistent with the prediction of the strong disorder renormalization group [48]. This fact is at the basis of the application of SDRG, which establishes the strong disorder of a model in the energy space.

10 10 10 n=1 n=1 n=1 n=2 n=2 n=2 8 n=3 8 n=3 8 n=3 n=4 n=4 n=4 6 6 6 ) ' K ( Q 4 4 4

2 2 2

0 0 0 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1

Ky'nn Ky'nn Ky'nn

Figure 4.2: Distribution of Knny for the Trotter approximation of the TFIM with KTFIM = 0.935 (left), 1.035 (middle), and 1.135 (right).

4.2.3 2D spin glass

We now consider the spin glass Hamiltonian:

X ij HK = −KSG k σiσj (4.18) hi,ji where the nearest neighbor couplings kij take values equal to ±1 with equal probability. Spin glass order is signaled by non-vanishing values of the Edwards-Anderson order

56 parameter q [57]:

N Z X q ≡ dKP (K) hσ i2 (4.19) v i HK i=1 0 Z X X e−HK(σ)−HK(σ ) = dKP (K) σ σ0 (4.20) v i i Z2 σσ0 i K where ZK is the partition function for the Hamiltonian HK. By introducing a new

0 Ising variable ρi = σiσi, Eq. 4.20 deﬁnes an eﬀective Hamiltonian for the ρ spins:

P ij X −KSG k (1+ρiρj )σiσj Heﬀ(ρ) = − ln e hi,ji (4.21) σ

P so that q is equal to the quenched average of h i ρiiHeff(ρ), the total magnetization of the ρ-system. Thus, the emergence of spin glass order is signaled by the emergence of magnetic order in the ρ-system [58]. Note that the statistical weight associated to a configuration ρ requires tracing out the σ degrees of freedom: this is a coarse-graining process and we can use VMCRG to estimate the effective disordered Hamiltonian of the ρ spins. Then we can monitor the RG flow of the ρ spin Hamiltonian by making successive block spin transformations, again exploiting the variational principle. We have applied the above procedure involving two types of coarse graining transformations, one to obtain the effective Hamiltonian of the ρ-system and the other to perform iteratively RG scale transformation by blocking the ρ spins. We report in Fig. 4.4 the corresponding marginal distribution of the nearest neighbor couplings

2 obtained for a Hamiltonian with KSG = 1.2 on a 128 lattice by including 3 local

couplings: Knn,Knnn,K. In the ﬁgure, n = 0 indicates the coarse-graining transfor-

mation to generate Heﬀ(ρ), while n = 1, 2 indicate two successive RG transformations of the ρ-system. The marginal distribution keeps drifting toward lower coupling constants, consistent with the accepted view that there is no spin glass ordering at ﬁnite temperature in 2D [59].

57 10 n=0 9 n=1 8 n=2 7 6 5 4 3 2 1 0 0 0.2 0.4 0.6 0.8 1 1.2 1.4

K'nn (spin glass)

Figure 4.3: Distribution of nearest neighbor couplings for a spin glass model with KSG = 1.2.

We also applied the same method to a 3D spin glass with ±1 nearest neighbor couplings of size 643, which is expected to have a spin glass transition at ﬁnite

temperature, Tc = 1.1 ± 0.1 [60]. Our simulations did not use parallel tempering. We

found it very slow to converge the VMCRG calculation to block size of 4 around Tc without truncation. We did manage to do the calculation truncating ﬁrst at the ρ

system to obtain an explicit truncated Heﬀ(ρ) with eight coupling terms. We then

did further coarse-graining using the truncated Heﬀ(ρ), but the truncation error was very large, and the results were simply not reliable.

4.2.4 2D random ﬁeld Ising model

We then consider the RFIM in both 2D and 3D:

X X i HRFIM = −KRFIM σiσj − h0 h σi (4.22) hi,ji i

i with KRFIM positive, hi, ji nearest neighbor, and the h s independent unit Gaussian random variables. In both dimensions, we use lattices with linear size L = 64 and and adopt the majority rule with b = 2 for 3 RG iterations. In 2D, we also do a fourth-iteration calculation on a L = 128 lattice. 58 In the 2D RFIM, we use four couplings, the three even couplings of the DIM and

one odd coupling constant KM describing the strength of the local magnetization

i SM = σi to account for the random magnetic ﬁeld. As shown in Fig. 4.4 (left) when

KRFIM = 0.8, a coupling strength well above 0.4407, the critical coupling of the pure

Ising model, a random ﬁeld with strength h0 = 1.0 drives the spin-spin interactions to zero, in accord with the analytical result [61]. For the ﬁrst three iterations, the

distribution of KM broadens as RG iterates (Fig. 4.4, right), indicating the important role of disorder in suppressing the spin-spin interactions. When n = 4 as the spin-spin interactions have been greatly suppressed, the random ﬁelds start to decrease again. This must happen, because random ﬁelds in an interaction-free spin system renormalize to zero.

8 n=1 0.35 n=1 7 n=2 28 n=2 n=3 0.3 24 n=3 6 n=4 20 0.25 n=4 5 16 ) )

' 0.2 12 ' K K

( 4 8 ( Q Q 0.15 3 -0.05 0 0.05 0.1 0.1 2

1 0.05

0 0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 -10 -5 0 5 10 K'nn K'M

Figure 4.4: Distribution of the renormalized nearest neighbor (left) and local magnetization (right) coupling constants for the 2D RFIM with KRFIM = 0.8 and h0 = 1.

4.2.5 3D random ﬁeld Ising model

In the 3D RFIM, we use the six renormalized couplings, including the nearest neighbor

coupling (Knn) and the local magnetic ﬁeld (KM). See the details of VMCRG in Sec.

4.4.3. For fixed h0 = 0.35 and varying KRFIM the system has been analyzed extensively by finite size scaling for sizes up to L = 16, finding that a magnetic transition

occurs at KRFIM = 0.2705(3) [62]. We ﬁnd consistently that the mean value of the

Hamiltonian distribution starts drifting toward higher couplings when KRFIM = 0.27. The corresponding variance shows a divergent behavior, suggesting a strong-disorder 59 ﬁxed point. Within the number of RG iterations performed, this behavior suggests the presence, in the subcritical region, of magnetic clusters with a disordered distribution of magnetizations. The evolution of the renormalized couplings are illustrated in Fig.

4.5 for three values of KRFIM, i.e. KRFIM = 0.25 (well below the critical coupling),

KRFIM = 0.264 (slightly below the critical coupling), and KRFIM = 0.28 (well above the critical coupling). Interestingly, the distribution keeps a ﬁxed non-zero mean with

25 20 18 n=1 n=1 n=1 18 n=2 n=2 16 n=2 20 n=3 16 n=3 14 n=3 14 12 15 12 )

' 10 K

( 10

Q 8 10 8 6 6 5 4 4 2 2 0 0 0 0 0.05 0.1 0.15 0.2 0.25 0.3 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0 0.2 0.4 0.6 0.8 1 1.2 1.4

K'nn K'nn K'nn

1 1 0.9 n=1 n=1 n=1 0.9 0.9 n=2 n=2 0.8 n=2 0.8 n=3 0.8 n=3 0.7 n=3 0.7 0.7 0.6 0.6 0.6 )

' 0.5 K

( 0.5 0.5

P 0.4 0.4 0.4 0.3 0.3 0.3 0.2 0.2 0.2 0.1 0.1 0.1 0 0 0 -5 -4 -3 -2 -1 0 1 2 3 4 5 -8 -6 -4 -2 0 2 4 6 -15 -10 -5 0 5 10 15 K' K' K'

70 60 50 n=1 n=1 n=1 45 n=2 n=2 n=2 60 50 n=3 n=3 40 n=3 50 35 40 30

) 40 ' K

( 30 25 P 30 20 20 20 15 10 10 10 5 0 0 0 -0.02 0 0.02 0.04 0.06 0.08 0.1 -0.05 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 -0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 K' K' K' Figure 4.5: Distribution of the renormalized coupling constants for the 3D RFIM with h0 = 0.35, and KRFIM = 0.25 (left), 0.264 (middle), and 0.28 (right). The top row is for the nearnest neighbor couplings. The middle row is for the magnetization couplings. The bottom row is for the next nearest neighbor couplings.

increasing width even below the critical coupling, for 0.255 < KRFIM < 0.27, shown in Fig. 4.6. An RG calculation with n = 4 would require at least a system size of

60 25 25 25 20 20 n=1 n=1 n=1 n=1 n=1 n=2 n=2 n=2 18 n=2 18 n=2 20 n=3 20 n=3 20 n=3 16 n=3 16 n=3 14 14 15 15 15 12 12 ) ) ) ) ) ' ' ' ' ' K K K K K

( ( ( ( 10 ( 10 P P P P P 10 10 10 8 8 6 6 5 5 5 4 4 2 2 0 0 0 0 0 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0 0.1 0.2 0.3 0.4 0.5 0 0.1 0.2 0.3 0.4 0.5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

K'nn K'nn K'nn K'nn K'nn

Figure 4.6: Distribution of Knn for the 3D random ﬁeld Ising model with h0 = 0.35 and KRFIM = 0.255, 0.257, 0.262, 0.266, 0.27 from left to right. The data are collected from one sample with L = 64.

1283 and greater relaxation time, which would require a lot more simulation time than n = 3 and was thus not pursued.

4.3 Time correlation functions in the biased ensem-

ble

PN We focus on the magnetization, i.e. M = i=1 σi. We study the time correlation hM(t)M(0)i function CM (t) = hM 2i . (For the spin glass model, we study the magnetization in the ρ-system.) In Fig. 4.7 and 4.8, we plot CM (t) for biased and unbiased simulations for the models studied. A single disorder realization was considered in all the plots. While unbiased dynamics strongly depends on the lattice size, biased dynamics is fast and essentially size independent.

4.4 Appendix

4.4.1 Optimization details for the 2D DIM

For the ﬁrst 4 RG iterations of the dilute Ising model, the Metropolis algorithm is uses for the MC sampling. For n = 1, 2, 3, and 4, the nth RG iteration uses 1000 variational steps, where each variational step uses 100n MC sweeps per core on 16

61 1 0.8 biased 162 2 biased 32 ) 0.6 2 t unbiased 16 ( 0.4 unbiased 32 2 M

C 0.2 0 -0.2 0 500 1000 1500 2000 t/MC sweeps (dilute Ising)

1 0.8 biased 32x32

0.6

) biased 64x64 t ( 0.4 unbiased 32x32 M unbiased 64x64 C 0.2 0 -0.2 0 500 1000 1500 2000 t/MC sweeps (transverse field Ising)

0.8 biased 322 2 ) 0.6 biased 64 t unbiased 82 ( 0.4 unbiased 16 2 M

C 0.2 0 -0.2 0 1000 2000 3000 4000 5000 t/MC sweeps (spin glass)

1 biased 322 biased 642 0.8 2

unbiased 32 ) 0.6 unbiased 64 2 t ( 0.4 M

C 0.2 0 -0.2 0 500 1000 1500 2000 t/MC sweeps (2D random field Ising)

hM(t)M(0)i Figure 4.7: Time correlation functions CM (t) = hM 2i of the system magnetization 1 P M = N i σi, in the biased and unbiased ensemble for various lattice sizes. For the spin glass model, we take the magnetization to be in the ρ-system. In the biased

ensemble, the bias potential VJmin is obtained by VMCRG for b = 2 with 1 RG iteration. The data are collected, shown from top to bottom, for the dilute Ising model with KDI = KDI,c, the Trotter approximation of the transverse ﬁeld Ising model with KTFIM = 1.0, the 2D spin glass with KSG = 1.2, and the 2D random ﬁeld Ising model with KRFIM = 0.8, h0 = 1.0,

cores to compute the averages in the gradient and the Hessian of the functional Ω. The gradient descent step size is taken to be δ = 0.2 at each variational step. As mentioned in the main text, each calculation used three coupling terms: the nearest neighbor products, the next nearest neighbor products, and the smallest plaquette products. During each variational calculation, the number of variational coeﬃcients is of the order of the number of the lattice sites. We show in Fig. 4.9 only the

62 1

0.9

0.8 1 0.7 0.995 unbiased 8x8 unbiased 16x16 0.6 biased 8x8 0.99 biased 16x16

) t ( 0.5 M 0.985 C

0.4 0.98

0.3 0.975 0 100 200 300 400 500

0.2

0.1

0 0 100 200 300 400 500 t/MC sweeps (3D random field Ising)

Figure 4.8: Time correlation functions CM (t) in the biased and unbiased ensemble for various lattice sizes for the 3D random field Ising model with KRFIM = 0.27, h0 = 0.35. The relaxation for the 3D RFIM in the unbiased ensemble is very slow, and an inset is placed in the center of the figure to show the unbiased time correlation functions more clearly. Also note that in the biased ensemble of the 3D RFIM, the time correlation function eventually converges to the average magnetization squared, hMi2, which in the random field Ising model is not necessarily zero due to the random fields.

renormalized nearest neighbor coupling constants for four lattice sites, each associated with 2 nearest neighbor coupling terms.

0.6

0.5

0.4 n n i 0.3 K 0.2

0.1

0 0 500 1000 1500 2000 2500 3000 3500 4000 Variational Step

Figure 4.9: Renormalized nearest neighbor constants at criticality during the optimization of Ω[V ] for one bond realization of a 128 × 128 dilute Ising model, for the ﬁrst 4 RG iterations. KDI = 0.609377.

To allow a suﬃciently large renormalized cell, for n = 5, we use a 2562 lattice. Because in this case each spin block is quite large with a block length bn = 25 = 32,

63 correlation within each spin block becomes significant and slows down Monte Carlo (MC) relaxation. For better MC mixing time, we adopt the Wolff algorithm [15]. When a bias potential V (σ0) is added to the system Hamiltonian, at each MC step, a cluster is grown by the Wolff algorithm with the system Hamiltonian H(σ) and used as a trial move, which is accepted with the Metropolis probability: min{1, exp(−∆V (σ0)} (the detailed balance condition can be easily proved). Here ∆V (σ0) is the change in V due to the flip of the Wolff cluster. This MC scheme proves quite efficient in the present case with an acceptance probability of ∼ 0.45. For this calculation, 1000 variational steps are used, where 4000 Wolff steps per core on 16 cores are used for each variational step. The gradient descent step size is taken to be δ = 0.2 at each variational step. We show in Fig. 4.10 again only the renormalized nearest neighbor constants for four lattice sites on one sample.

0.6

0.5

0.4 n n i 0.3 K 0.2

0.1

0 0 200 400 600 800 1000 Variational Step

Figure 4.10: Renormalized nearest neighbor constants at criticality during the optimization of Ω[V ] for one bond realization of a 256 × 256 dilute Ising model for the 5th RG iteration. KDI = 0.609377.

4.4.2 Couplings in the computation of critical exponents of

the dilute Ising model

Let Knn({0, 0}, {1, 0}) denote the nearest neighbor coupling between lattice sites with coordinates {0, 0} and {1, 0}. We denote other coupling terms similarly. A basis

64 function Uβ(K) is a sum of local elementary terms induced by the action of the translational and point group symmetries of the Hamiltonian. Thus, it suﬃces to list only the elementary terms of each basis function, which we enumerate below.

• The ﬁrst four powers of each Knn,Knnn, and K. This amount to 12 U-basis functions.

• Knn({0, 0}, {1, 0}) · Knn({0, 0}, {0, 1})

• Knn({0, 0}, {1, 0}) · Knn({0, 0}, {−1, 0})

• Knnn({0, 0}, {1, 1}) · Knnn({1, 0}, {0, 1})

• Knnn({0, 0}, {1, 1}) · Knnn({1, 1}, {0, 2})

• K({0, 0}, {1, 0}, {0, 1}, {1, 1}) · K({1, 0}, {2, 0}, {1, 1}, {2, 1})

Thus, a total of 17 U-basis functions are used.

4.4.3 Optimization details for the 3D RFIM

Following the notation of Sec. 4.4.2, we list the renormalized couplings used in the 3D random ﬁeld Ising model:

1. local magnetic ﬁeld KM : ({0,0,0})

2. nearest neighbor product Knn: ({0,0,0}, {1,0,0})

3. ({0,0,0}, {1,1,0})

4. ({0, 0, 0}, {1, 0, 0}, {1, 1, 0}, {0, 1, 0})

5. ({0, 0, 0}, {1, 0, 0}, {0, 1, 0})

6. ({0, 0, 0}, {1, 0, 0}, {-1, 0, 0})

65 The optimization is done on 643 lattices with Metropolis sampling. Each RG iteration with n = 1, 2, 3 takes 1000 variational steps. Each variational step repectively takes 20, 50, 200 MC sweeps for n = 1, 2, 3 per core on 16 cores. The gradient descent step size is taken to be δ = 0.3 at each variational step. We show in Fig. 4.11 and 4.12 the

i optimization for the renormalized constants for the local magnetic ﬁeld, KM , and the

i nearest neighbor products, Knn for four lattice sites at KRFIM = 0.265 and h0 = 0.35.

66 2.5 2 1.5 1 n n i 0.5 K 0 -0.5 -1 -1.5 0 500 1000 1500 2000 2500 3000 Variational Step

Figure 4.11: Renormalized constants for the local magnetization coupling during the optimization of Ω[V ] for one bond realization of a 643 random ﬁeld Ising model, for the ﬁrst 3 RG iterations. KRFIM = 0.264, h0 = 0.35

0.6

0.5

0.4 n n i 0.3 K 0.2

0.1

0 0 500 1000 1500 2000 2500 3000 Variational Step

Figure 4.12: Renormalized constants for the nearest neighbor product coupling during the optimization of Ω[V ] for one bond realization of a 643 random ﬁeld Ising model, for the ﬁrst 3 RG iterations. KRFIM = 0.264, h0 = 0.35

67 Chapter 5

Variational Monte Carlo renormalization group for quantum systems

In this chapter, we map a sign-free quantum model into a classical model via the path-integral representation. In particular, we focus on the formulation where the path-integral is continous in the time direction. We will demonstrate that in addition to the determination of critical coupling and exponents, VMCRG can also extract the sound velocity and the energy-stress tensor of the lattice model.

5.1 Continuous-time Monte Carlo simulation of a

quantum system

We ﬁrst explain how to do the continous-time quantum Monte Carlo for the Q-state Potts model: Q−1 Ld Q−1 ˆ X X ˆ k ˆ Q−k X X ˆ k HQ = − Ωi Ωj − g Mi , (5.1) hi,ji k=0 i=1 k=1

68 where the system is in a d-dimensional hypercubic lattice with length L. i and j label diﬀerent lattice sites, and hi, ji denotes a nearest neighbor bond. The operators ˆ ˆ Ωi and Mi act on the Q states of the local Hilbert space at site i, which we label ˆ by |0ii, ..., |sii, ...|Q − 1ii. In this local basis, the Ωi is a diagonal matrix such that ˆ s i2π/Q ˆ Ωi|sii = ω |sii, where ω = e and s = 0, ··· ,Q − 1. Mi performs a cyclic

permutation: |0ii → |Q − 1ii, |1ii → |0ii, ··· , |Q − 1ii → |Q − 2ii, and acts as a transverse-ﬁeld. When d = 1, this model is self-dual and a quantum phase transition

occurs at the critical coupling gc = 1 for all Q [63]. When Q ≤ 4, the transition is continuous and is described by a CFT [64, 65]. When Q > 4, the transition is

ﬁrst-order and has a ﬁnite latent heat at gc [64]. To derive a path-integral representation of the system partition function Z =

ˆ Tr(e−βHQ ), one takes

Q−1 Ld Q−1 ˆ X X ˆ k ˆ Q−k ˆ X X ˆ k H0 = − Ωi Ωj , H1 = −g Mi (5.2) hi,ji k=0 i=1 k=1

and considers the partition function in the interaction picture

Z β ˆ ˆ ˆ Z = Tr[exp(−βH0)T {exp(− H1(τ)dτ)}], (5.3) 0

ˆ ˆ ˆ ˆ τH0 ˆ −τH0 where T is the time-ordering operator in imaginary time τ, and H1(τ) = e H1e ˆ is the H1, in the interaction picture. Eq. 5.3 can be written as a diagrammatic

69 expansion in the following way:

∞ n Z β Z β X X (−1) ˆ Z = hσ|e−βH0 Tˆ Hˆ (τ )dτ ··· Hˆ (τ )dτ |σi n! 1 n n 1 1 1 σ n=0 0 0 ∞ Z β Z τn Z τ2 X X n −(β−τn)Hˆ0 = g dτn dτn−1 ··· dτ1hσ|e σ n=0 0 0 0 Ld Q−1 ˆ ˆ X X ˆ k −(τn−τn−1)H0 −τ1H0 Mi e ··· e |σi (5.4) i=1 k=1 ∞ Z β Z τn Z τ2 X X n X −(β−τn)Hˆ0 = g dτn dτn−1 ··· dτ1hσ|e 0 0 0 σ n=0 i1···in Q−1 Q−1 ˆ ˆ X ˆ k −(τn−τn−1)H0 X ˆ k −τ1H0 Min e ··· Mi1 e |σi k=1 k=1

d partition function. For each i = 1, ··· ,L and each τ ∈ [0, β], a Potts spin, σi(τ), ranging from 0 to Q − 1, is assigned to an MC configuration. As the state |σi is propagated in imaginary time, spin flips can happen at any lattice site and at any time. Let τl and il denote the lth flip time and its associated lattice site. Here l could

− + be equal to 1, 2, ··· , or n. In addition, let τl and τl respectively denote the time immediately before and after the ﬂip time τl. If a spin ﬂip occurs at τl on site il,

− + − σil (τl ) will be made to switch to any σil (τl ) different from σil (τl ) by the action of PQ−1 Mˆ k. In Eq. 5.4, the earliest spin flip occurs at τ on site i ; the second one k=1 il 1 1 occurs at τ2 on site i2, etc.. The total number of spin flips, n, contributes a weight gn to the sampling of the diagrammatic expansion. In addition, the weight includes

+ −(τl+1−τl)h0(σ(τ )) factors equal to e l between two consecutive spin ﬂips at τl and τl+1. Finally, the periodicity of the trace requires σ(β) = σ(0), which in turn implies that n should be even. Thus, MC sampling does not have a sign problem even if g is negative.

70 The partition function in Eq. 5.4 is given by a sum of terms (diagrams) that entail summation over discrete variables and integration over continuous ones. The contribution of the different terms, which are associated to the weights detailed above, is evaluated stochastically with a MC algorithm that follows the protocols discussed in Refs. [66, 67, 68]. For the Q-state Potts model diagrammatic MC can use a continuous time cluster algorithm [67], based on the Wolff algorithm [15], which significantly reduces equilibration time. We will use both local and cluster MC algorithms in the following.

5.2 The MCRG for continuous-time quantum Monte

Carlo

Eq. 5.4 indicates that the thermodynamics of a d-dimensional quantum Potts model is described by a statistical ﬁeld theory in (d + 1) dimensions, where on each lattice site i

there is a worldline of length β described by the function σi(τ). We can coarse grain this worldline by placing P lattice points along the time direction through the majority-rule.

P P 2β (P −1)β That is, we partition the worldline into P intervals: [0, β ], [ β , P ], ··· , [ P , β]. In each MC configuration, to each interval, we assign the Potts spin which appears most often on that interval. By discretizing time in this way we represent each worldline with P discrete Potts spins and we end up with a (d+1)-dimensional hypercubic lattice that hosts PLd Potts spin. Each MC snapshot corresponds to a configuration of those spins. The probability distribution of the spin configurations on the discrete lattice is not known explicitly, but can be sampled by coarse-graining the configurations generated in the diagrammatic MC simulation [67]. We can then perform MCRG on the (d + 1)-dimensional hypercubic lattice. We designate the n-th RG iteration with a subscript (n), where n = 0, 1, 2, ... In the 0-th iteration the spin configuration, σ(0), is the one obtained from coarse graining the time direction of diagrammatic MC. The

71 probability distribution of σ(0) is described by the (unknown) lattice Hamiltonian H(0). The subsequent levels of coarse graining are generated by successive block spin RG transformations and are characterized by spin conﬁgurations σ(n) and Hamiltonians H(n). In all of the above coarse-graining transformations we use short-ranged coupling

terms Sα(σ) to parametrize the probability distribution P (σ) of the spin conﬁgurations σ:

−H(σ) X P (σ) ∝ e , where H(σ) = − KαSα(σ), (5.5) α with coupling constants Kα. The terms Sα(σ) include nearest neighbor (nn), next

nearest neighbor (nnn), smallest plaquette (), etc., interactions.

5.3 The sound velocity of a critical quantum system

The RG calculation of a continuous-time Monte Carlo simulation is most interesting when the system exhbits Lorentz invariance, i.e. when the energy of the low-lying excited states depends linearly on its momentum:

E(k) − E0 = vs|k|, (5.6)

where E0 is the ground state energy, and k is the momentum of a low-lying energy

eigenstate of energy E(k). The constant of proportionality, vs, is called the sound velocity and is a non-universal constant specific to a system. In the presence of Lorentz invariance, the path integral represents a statistical field theory in which isotropy between the time and space directions emerges at large distance. When Eq. 5.6 happens, the large-distance behavior of a lattice spin system is described by a massless quantum field theory, invariant under dilational coordinate transformations. The value

72 of the sound velocity is generally a real number, requiring that the coarse-graining either along the space or the time direction be continuous, as explained in Sec. 5.2. In systems with short-ranged Hamiltonians, one expects1 that the Lorentz invariance is elevated to the full conformal invariance [31]. Conformal invariance is very powerful in two-dimensions (2D), due to the infinite dimensionality of the local conformal algebra [69]. Conformal field theory (CFT) yields finite sizing scaling predictions of physical observables, such as the energy [70, 71] and the entanglement entropy [72]. The sound velocity is often a parameter in these predictions. When d = 1, one can compute the sound velocity with finite size scaling of the ground state energy or entanglement entropy, assuming validity of the CFT prediction [70, 71, 72], or one can directly compute the excitation spectrum of the system. The latter calculation can be done by exact diagonalization of the system Hamiltonian, which is limited to small lattice sizes, or it can be done with density matrix renormalization group (DMRG) techniques [73], which introduce truncation errors due to finite bond dimension. When d > 1, neither method works reliably, and one has to resort to QMC. In fact, the sound velocity has been calculated [74, 75] with continuous-time QMC by looking for a scale factor vs such that the correlation function C(x, τ) = hσˆ(0, 0)σˆ(x, τ)i, where

τHˆ −τHˆ σˆ(x, τ) = e σˆ(x)e , becomes isotropic between x and vsτ at large distance and low temperature. See Appendix 5.5.1 for a more detailed discussion of the space-time isotropy. In the following, we will use directly RG to compute the sound velocity and show that VMCRG can deal with rather large lattice sizes, up to at least L = 256, leading to very accurate estimates of the sound velocity. To compute the sound velocity, we perform a dilation transformation with b = 2 using the majority rule. In the VMCRG calculation, we take couplings along space and time directions to be independent, as the system is necessarily anisotropic between space and time. However, in the presence of Lorentz invariance, isotropy between

1There is no general proof of this expectation.

73 space and time is recovered at large distances up to a scale factor vs. One can thus

P vary β until the couplings along the space direction and those along the time direction P become equal for a large n. The β so determined is the sound velocity, vs.

5.3.1 Q = 2: The Ising model

ˆ When Q = 2, HQ=2 coincides up to an additive constant with the Hamiltonian of the transverse-ﬁeld Ising model (TFIM),

Ld ˆ X z z X x HIsing = − σˆi σˆj − g σˆi , (5.7) hi,ji i=1

z,x where σˆi are the Pauli matrices at site i. We carry out the discussion using the terminology of the Ising model, i.e. instead of Potts spin σ = 0, 1, we will speak of Ising spin σ = −1, 1. The sound velocity for the one-dimensional TFIM is known exactly from its

fermionic solution, and is vs = 2 [76]:

p 2 E(k) = 2 1 − 2gc cos k + gc = 2|k| + o(|k|), (5.8)

where gc = 1. Note that sometimes the Ising Hamiltonian is written with spin ˆ 1 1 (n) operators S = 2 σˆ, in which case vs would be 2 . The K s calculated with VMCRG

for the d = 1 TFIM, simulated at gc, are presented in Table 5.2, for the coupling terms listed in Table 5.1. We present the data for all the RG iterations from n = 0 to 5 to illustrate the method. As clearly seen, the convergence to isotropy occurs with

P (n) (n) increasing n when β = vs. Note that the relative magnitude of K1,x and K1,y switches P when β crosses 2.0. This gives an estimation of vs in the interval (1.997, 2.003), to be compared, for example, with the DMRG result of 2.04 found in [77].

74 α Coupling term 1, x nearest neighbor product along the time direction 1, y nearest neighbor product along the space direction 2 2nd neighbor product 3 product of spins in the smallest plaquette 4, x 3rd neighbor product along the time direction 4, y 3rd neighbor product along the space direction

Table 5.1: The couplings used for d = 1 TFIM. Note that when α = 2 and 3, the couplings are themselves isotropic between space and time.

(n) (n) (n) (n) n K1,x K1,y K4,x K4,y 0 0.46708(6) 0.30104(7) -0.0360(1) 0.0179(1) 1 0.3593(1) 0.32367(5) -0.01224(3) -0.0027(1) 2 0.34310(5) 0.33548(4) -0.01065(5) -0.0089(1) 3 0.3411(2) 0.33906(8) -0.0109(1) -0.0106(1) 4 0.3411(3) 0.3401(1) -0.0111(2) -0.0112(2) 5 0.3411(2) 0.3402(1) -0.0112(1) -0.0114(2)

P (a) β = 2.0031299 (n) (n) (n) (n) n K1,x K1,y K4,x K4,y 0 0.46681(5) 0.30126(8) -0.0362(1) 0.0180(1) 1 0.3590(1) 0.3242(1) -0.0123(1) -0.0027(1) 2 0.3430(1) 0.33586(6) -0.01074(5) -0.0088(1) 3 0.3407(2) 0.3394(3) -0.0110(1) -0.0106(1) 4 0.3408(3) 0.3406(2) -0.0112(2) -0.0110(2) 5 0.3410(3) 0.3408(2) -0.0112(1) -0.0112(1)

P (b) β = 2

n K1,x K1,y K4,x K4,y 0 0.46579(6) 0.30175(7) -0.0361(1) 0.0179(1) 1 0.3584(1) 0.3245(1) -0.0124(1) -0.0026(1) 2 0.3423(1) 0.3366(1) -0.0107(1) -0.0088(1) 3 0.3405(2) 0.3398(2) -0.0110(1) -0.0105(1) 4 0.3404(2) 0.3409(2) -0.0113(2) -0.0110(2) 5 0.3402(3) 0.3409(2) -0.0113(2) -0.0111(2)

P (c) β = 1.99688 Table 5.2: The renormalized constants for the d = 1 TFIM. For each n, L = P = 8×2n. VMCRG is done with 4000 variational steps. During each variaional step, the MC sampling is done on 16 cores in parallel, where each core does MC sampling of 20000 Wolﬀ steps. The optimization step is µ = 0.001. The number in the paranthesis is the uncertainty on the last digit.

75 P (4) (4) β K1,x K1,yz 3.42246 0.1603(3) 0.1597(2) 3.40426 0.1594(3) 0.1602(2)

(4) Table 5.3: The renormalized constants for the d = 2 TFIM. L = P = 128. K1,x and (4) K1,yz are respectively the renormalized nearest neighbor spin constants along the time and the space direction at n = 4.

α Coupling term

1, x δσiσj for 1st neighobor i, j along the time direction

1, y δσiσj for 1st neighobor i, j along the space direction

2 δσiσj for 2nd neighbor i, j

3, x δσiσj for 3rd neighobor i, j along the time direction

3, y δσiσj for 3rd neighobor i, j along the space direction

Table 5.4: The couplings used for d = 1, Q = 3 and 4 Potts model. Note that when α = 2, the coupling is itself isotropic between space and time.

For d = 2 TFIM model, we look for the value of P/β where switching of the renormalized constants in space and time occurs at large n. The comparison is only done for the nearest neighbor coupling, which has the smallest statistical uncertainty. Thus, we present in the tables only the nearest neighbor coupling constants that correspond to the last RG iteration. The K(n)s calculated by VMCRG for the d = 2

TFIM, simulated at g = gc = 3.04438 [67], are reported in Table 5.3. They lead to an estimate of the sound velocity in the interval (3.40, 3.42).

5.3.2 Q = 3 and 4

When d = 1, and Q = 3 or 4, the Potts model experiences a continuous phase

transition at gc = 1, exhibiting conformal invariance [64]. A sound velocity is thus well-deﬁned at criticality. The spin variable is σ = 0 or 1, and we use coupling terms listed in Table 5.4. We report the calculated renormalized constants K(n)s in Table 5.5 and 5.6. The sound velocity is determined with the nearest neighbor coupling at the last RG iteration.

76 (n) (n) (n) (n) n K1,x K1,y K3,x K3,y 0 1.068(2) 0.6701(4) -0.0651(6) 0.0343(9) 1 0.8158(4) 0.7323(5) -0.0267(1) -0.0076(4) 2 0.766(1) 0.749(2) -0.0260(8) -0.022(1) 3 0.754(1) 0.750(1) -0.025(1) -0.025(1) 4 0.7477(4) 0.7468(4) -0.0255(6) -0.0251(6) 5 0.7452(2) 0.7446(2) -0.0252(4) -0.0250(4)

P (a) β = 2.5995 (n) (n) (n) (n) n K1,x K1,y K3,x K3,y 0 1.067(2) 0.6714(5) -0.0653(5) 0.0350(8) 1 0.8150(4) 0.7350(5) -0.0278(2) -0.0079(5) 2 0.765(1) 0.751(2) -0.0252(7) -0.022(1) 3 0.752(1) 0.750(1) -0.025(1) -0.024(1) 4 0.7469(4) 0.7479(5) -0.0250(6) -0.0248(3) 5 0.7441(3) 0.7456(3) -0.0253(4) -0.0251(2)

P (b) β = 2.5942 Table 5.5: The renormalized constants for the d = 1, Q = 3 Potts model. For each n, L = P = 8 × 2n. When n = 0 to 3, the simulations are done with Metropolis local updates with 1000 variational steps. When n = 0, 1, 2, each variational step uses 100 sweeps of MC averaging in parallel on 8 cores. When n = 3, each variational step uses 500 sweeps. For n = 4 and 5, the simulation details are the same as in Table 5.2.

This estimates vs in the interval (2.594, 2.600) for Q = 3, and (3.137, 3.146) for

Q = 4, to be compared with the analytical result vs = π when Q = 4 [78]. For comparison, ﬁtting the ﬁnite size behavior of the critical free energy against the CFT

prediction [70, 71] leads to vs = 2.598 for Q = 3 and vs = 3.156 for Q = 4 [65]. Fitting the ﬁnite size behavior of the critical entanglement entropy against the CFT prediction

[72] leads to vs = 2.513 for Q = 3 and vs = 2.765 for Q = 4 [79]. We observe that the (approximate) space-time isotropy occurs before the fixed- point Hamiltonian is reached. For example, in the Q = 4 Potts model, it is known that a logarithmic scaling operator is present around the fixed-point Hamiltonian, which makes the approach to the fixed-point Hamiltonian very slow. This is indeed what one sees in Table 5.6. However, as along as this scaling operator is isotropic, one expects that the slow approach to the fixed-point Hamiltonian should not affect the

77 (n) (n) (n) (n) n K1,x K1,y K3,x K3,y 0 1.171(2) 0.7188(5) -0.0615(4) 0.033(1) 1 0.872(2) 0.777(1) -0.027(1) -0.0084(4) 2 0.801(1) 0.781(2) -0.024(1) -0.021(1) 3 0.770(1) 0.765(1) -0.023(1) -0.023(1) 4 0.7519(5) 0.7498(4) -0.0225(2) -0.0225(2) 5 0.7374(3) 0.7355(5) -0.0217(3) -0.0215(3)

P (a) β = 3.146 (n) (n) (n) (n) n K1,x K1,y K3,x K3,y 0 1.167(2) 0.7206(4) -0.0612(4) 0.033(2) 1 0.872(2) 0.779(1) -0.0255(9) -0.0081(4) 2 0.800(1) 0.782(2) -0.022(1) -0.019(1) 3 0.769(1) 0.765(1) -0.022(1) -0.024(1) 4 0.7500(4) 0.7508(3) -0.0226(2) -0.0221(2) 5 0.7355(3) 0.7372(3) -0.0217(4) -0.0216(4)

P (b) β = 3.137 Table 5.6: The renormalized constants for the d = 1, Q = 4 Potts model. For each n, L = P = 8 × 2n. The simulation details are the same as in Table 5.5.

convergence to isotropy. This is also what one sees. The sound velocity can therefore be obtained with less RG iterations than requried for computing, say, the critical exponents of the model.

5.4 The energy-stress tensor

P As one changes the parameter β in the zeroth RG iteration, one also changes the ﬁxed- point Hamiltonian reached by the RG procedure, as shown, for example, in Table 5.1. Since dilational transformations are isotropic, there is a line of ﬁxed-point Hamiltonians

P reflecting the different extent of anisotropy in the system [11]. A change of β generates a movement along this line of fixed-point Hamiltonians. In fact, fixing P , the change

0 δβ 0 β → β + δβ induces a coordinate transformation: x0 → x0 = (1 − β )x0, x1 → x1 = x1, where x0 and x1 are time and space coordinates, respectively. Here we have taken the

0 0 0 coordinate transformation to be passive, i.e. x = (x0, x1) and x = (x0, x1) denote

78 the number of lattice spacings needed to describe the same physical length, before and after the transformation. Thus, a time dilation generates a change in the system Hamiltonian. In ﬁeld theory, the response of the system Hamiltonian to a generic coordinate transformation, xµ → x0µ = xµ + µ(x), is described by the energy-stress tensor, T µν, deﬁned by

Z µ 1 ∂ D δH = − D−1 Tµνd x (5.9) (2π) ∂xν where D is the space-time dimension of the system. As β is conjugate to Hˆ in the

action, we identify T00 as the energy operator in the path integral. To appreciate the novelty brought by VMCRG in this context, let us consider, for example, the two-dimensional classical Ising model with the Hamiltonian

X X HIsing(σ) = −K0 σiσj − K1 σiσj (5.10)

hi,ji0 hi,ji1

where hi, ji0 and hi, ji1 are nearest neighbor spins along the x0 and the x1 direction, re-

spectively. The system is isotropic and critical when K0 = K1 = Kc = arcsinh(1)/2 =

0.4407 ··· [7]. An inﬁnitesimal change in the coupling constant, K0 = Kc − δJ and

K1 = Kc + δJ, turns on anisotropy yet the system still maintains its criticality. That is, the deviation from the isotropic Hamiltonian,

X δH(σ) = Am,n(σ)δJ m,n (5.11) X ≡ (σm,nσm+1,n − σm,nσm,n+1)δJ m,n

0 0 generates a length scale transformation x0 → x0 = (1−δλ)x0 and x1 → x1 = (1+δλ)x1,

1 where δλ = γ δJ with an unknown proportionality constant γ. Continuous-time

79 VMCRG provides a way to determine γ directly, and in that sense, it is a ruler of anisotropy. To determine γ, we invoke the universality of the ﬁxed-point Hamiltonians. Let H∗(µ) be the ﬁxed-point Hamiltonian that VMCRG eventually reaches, starting from

P the critical d = 1 TFIM with β = vs and from the critical isotropic d = 2 classical Ising model. In practice, we approximate H∗(µ) with H(n)(µ) for some large n. For the TFIM, the change in the action βHˆ → (β + δβ)Hˆ generates a change in the

(n) (n) P ∂Kα fixed-point Hamiltonian δH (µ) = − α ∂β Sα(µ)δβ with an anisotropy of the 0 x1 x1 x1 δβ x1 extent δ( ) = 0 − = . For the classical d = 2 Ising model, the change in the x0 x0 x0 β x0 P unrenormalized Hamiltonian HIsing → HIsing + m,n Am,nδJ generates a change in the (n) (n) P ∂Kα fixed-point Hamiltonian δH (µ) = − α ∂J Sα(µ)δJ, with an anisotropy of the extent δ( x1 ) = 2δλ x1 . The δH(n)(µ) for the TFIM and for the d = 2 classical Ising x0 x0 model should be multiples of each other, because the line of fixed-point Hamiltonians is universal. In particular, when they are equal, the anisotropies that they represent should coincide. This means that

(n) δJ β ∂Kα /∂β γ = = 2 (n) (5.12) δλ ∂Kα /∂J

(n) (n) ∂Kα ∂Kα for all α. The Jacobians of the RG transformation, ∂J and ∂β , can be readily computed with VMCRG, where the ﬁrst Jacobian is calculated for the TFIM, and the second one for the classical Ising model. For the operator A(σ) deﬁned in Eq. 5.11, γ √ 2 is analytically known and is 2 = 0.7071 ··· [80]. A VMCRG calculation using Eq. 5.12 with n = 4 gives γ = 0.708 ± 0.001, so the ruler works.

0 0 With the coordinate transformation x0 → x0 = (1−δλ)x0 and x1 → x1 = (1+δλ)x1, a part of the energy-stress tensor can now be read oﬀ from Eq. 5.9:

X 1 Z A (σ)δJ = δH(σ) = − (−δλT + δλT )d2x. (5.13) m,n 2π 00 11 m,n

80 P R 2 We take the lattice spacing to be 1, and m,n is equivalent to d x. This gives

1 γA = (T + T¯), (5.14) π where, in 2D, T and T¯ are respectively the holomorphic and the antiholomorphic

1 component of the energy-stress tensor, and are deﬁned as T = 4 (T00 − 2iT01 − T11) ¯ 1 and T = 4 (T00 + 2iT01 − T11). While the argument is developed for the Ising model, it also generalizes to other 2D systems. A consequence of Eq. 5.14 is that one obtains a prediction of the ﬁnite-size dependence of hAi, due to CFT. For example, if one simulates a critical system

inﬁnitely long along the x0 direction but periodic of size L along x1, CFT predicts that ¯ 2π 2 c 1 2π 2 c hT i = hT i = −( L ) 24 [70, 71], and thus hAi = − γπ ( L ) 12 , where c is the central charge of the underlying CFT. This prediction on A has been veriﬁed in [80].

5.5 Appendix

5.5.1 Spacetime correlator at large distance and low temper-

ature

Let us consider one space dimension. Let a system with Hamiltonian Hˆ have translational symmetry, i.e. there is a translation operator Tˆ(x) that commutes with Hˆ . Consider the ﬁnite-temperature correlator C(x, τ) at inverse temperature β:

X ˆ C(x, τ) = hk|ψˆ(x, τ)ψˆ(0, 0)e−βH |ki (5.15) k where ψˆ(x) is some generic local operator that is labeled by the position x. The |kis are the common eigenstates of Hˆ and Tˆ(x), and ψˆ(x, τ) = eτHˆ ψˆ(x, 0)e−τHˆ . Also,

81 ψˆ(x, 0) = Tˆ(x)ψˆ(0, 0)Tˆ(−x) and Tˆ(x)|ki = e−ikx|ki. Then,

At zero temperature β = ∞, we keep only the ground state for |k = 0i. At large

0 0 distance τ, we keep only the low-lying states for k , where Ek0 = vs|k |. We also change the summation variable k0 to k. Thus,

Z C(x, τ) ≈ dkeikxe−|k|vsτ |h0|ψˆ(0, 0)|ki|2 (5.17)

The function g(k) ≡ |h0|ψˆ(0, 0)|ki|2 depends on the details of ψˆ and |kis, and is generally diﬃcult to compute. One should note that C(x, τ) is not generically isotropic ˆ between x and vsτ for any ψ. For example, in the exactly solvable transverse-ﬁeld Ising model, if one takes ψˆ to be the Fermionic operator after the Jordan-Wigner transformation, then g(k) does not depend on k [81], and one can evaluate

Z ikx −|k|vsτ |vsτ| C(x, τ) ∼ dke e ∼ 2 2 (5.18) x + (vsτ)

which is not isotropic between x and vsτ. In our VMCRG calculation, we are interested in ψˆ = σˆz, which turned out to have space-time isotropy. The calculation for σz is not trivial [82].

82 Chapter 6

Outlook

In MCRG, variational or not, the coarse-graining T (µ|σ) is typically chosen heuristi- cally, by physical intuition, such as the block majority rule. As we see in Sec.3, the majority rule coarse-graining is remarkably successful in q-state Potts models with low q, not only able to accurately determine the critical coupling and exponents, but also the tangent space and the curvature of the critical manifold. It is also extraordinarily successful for the dilute Ising model. On the one hand, one would like to understand why this is so. On the other hand, one would like to have an automatic way of deter- mining the coarse-graining kernel T (µ|σ) for more complicated systems. A necessary criterion for a correct coarse-graining kernel is that the regular part of the free energy after the renormalization must be a regular function of the unrenormalized coupling constants. This seems to be satisﬁed by the majority-rule for the Ising models, but there is no general proof. This criterion is also diﬃcult to implement in practice when one needs to design the coarse-graining kernel for a complicated system. Some progress has been made recently regarding designing the coarse-graining kernel. In [83], the kernel is represented as a neural network and the mutual information between the renormalized system and the unrenormalized system is maximized. For the Ising model, it was able to “discover” the majority rule. However, there does

83 not seem to be any fundamental reason why maximizing the mutual information should yield a good coarse-graining kernel. In [84], another neural network-based method is proposed to determine the coarse-graining kernel. While the results on the Ising models are very encouraging, this method is also an ansatz, requiring further understanding whether there is any fundamental truth contained in the method.

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