Variational Monte Carlo Renormalization Group

Variational Monte Carlo

Renormalization Group

Yantao Wu

A Dissertation

Presented to the Faculty

of Princeton University

in Candidacy for the Degree

of Doctor of Philosophy

Recommended for Acceptance

by the Department of

Physics

Adviser: Roberto Car

May 2021

Abstract

The renormalization group is an important method to understand the critical behaviors

of a statistical system. In this thesis, we develop a stochastic method to perform the

renormalization group calculations non-perturbatively with Monte Carlo simulations.

The method is variational in nature, and involves minimizing a convex functional of the

renormalized Hamiltonians. The variational scheme overcomes critical slowing down,

by means of a bias potential that renders the coarse-grained variables uncorrelated.

When quenched disorder is present in the statistical system, the method gives access

to the flow of the renormalized Hamiltonian distribution, from which one can compute

the critical exponents if the correlations of the renormalized couplings retain finite

range. The bias potential again reduces dramatically the Monte Carlo relaxation time

in large disordered systems.

With this method, we also demonstrate how to extract the higher-order geometrical

information of the critical manifold of a system, such as its tangent space and curvature.

The success of such computations attests to the existence and robustness of the

renormalization group fixed-point Hamiltonians.

In the end, we extend the method to continuous-time quantum Monte Carlo

simulations, which allows an accurate determination of the sound velocity of the

quantum system at criticality. In addition, a lattice energy-stress tensor emerges

naturally, where the continuous imaginary-time direction serves as a ruler of the length

scale of the system.

iii

Acknowledgements

The life in Princeton has been a long, rewarding, and fulfilling time. I would like to

first thank my advisor, Roberto Car, for his guidance and mentorship. To me, he is

a passionate, open-minded, and thoughtful advisor. He is interested and extremely

knowledgeable in many areas of physics, and has shaped my interests as a young

researcher. I am often moved by his passion, energy, extreme focus, and child-like

curiosity of the unknowns. It has been a privilege and absolute pleasure to have

worked with him.

I would also like to thank members of the Car group and the Selloni group with

whom I have spent countless hours in Frick: Hsin-Yu Ko, Linfeng Zhang, Fausto

Martelli, Biswajit Santra, Marcos Calegari Andrade, Bo Wen, Xunhua Zhao, Yixiao

Chen, Clarissa Ding, and Bingjia Yang. I want to give my special thanks to Hsin-Yu,

Linfeng, Marcos, and Yixiao for stimulating scientific discussions.

I would like to thank my friends and classmates in the physics department for

friendship and the bits of physics that I have learned from them here and there: Jiaqi

Jiang, Xinran Li, Junyi Zhang, Sihang Liang, Xiaowen Chen, Xue Song, Jingyu Luo,

Huan He, Jie Wang, Zheng Ma, Zhenbin Yang, Yonglong Xie, Zhaoqi Leng, Yunqin

Zheng, Jingjing Lin, Jun Xiong, Bin Xu, Erfu Su, and Wentao Fan. I would like to

thank professor Sal Torquato for stimulating discussions and reading this thesis. I

would like to thank professor David Huse for advices in research and serving in my

committee. I thank professor Robert Austin for guiding me through the experimental

project in biophysics. I thank Kate Brosowsky for being a patient helper as I navigate

through the graduate student life in the physics department.

I also would like to thank professor Lin Lin for hosting me in Berkeley, and professor

Lin Wang for hosting me in Beijing.

Life in Princeton would not have been this enjoyable without the many friends that

I have made during graduate school. In particular, I would like to thank the following

iv

people for banters, encouragement, and lots of fun: Tianhan Zhang, Kaichen Gu,

Lunyang Huang, Wenjie Su, Youcun Song, Wenxuan Zhang, Yezhezi Zhang, Yi Zhang,

Yinuo Zhang, Pengning Chao, Lintong Li, Fan Chen, and the Princeton Chinese soccer

team.

I also thank the Terascale Infrastructure for Groundbreaking Research in Science

and Engineering (TIGRESS) High-Performance Computing Center and Visualization

Laboratory at Princeton University.

In the end, I would like to thank my family and, in particular, my mom for their

unconditional support, without whom none of this would be possible. I have always

come to my mom for encouragement and advice in the lows of life. From her, I have

learned sharing, humor, and never giving up. I dedicate this thesis to her.

v

To my mom.

vi

Contents

Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii

Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv

List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . x

List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiv

1 Introduction 1

2 Variational approach to Monte Carlo renormalization group 7

2.1 An introduction to real-space renormalization group . . . . . . . . . . 7

2.1.1 Scaling operators and critical exponents . . . . . . . . . . . . 10

2.2 Variational principle of the renormalized Hamiltonian . . . . . . . . 12

2.3 Monte Carlo sampling of the biased ensemble . . . . . . . . . . . . . 15

2.4 Results of the renormalized couplings constants . . . . . . . . . . . . 16

2.5 Critical exponents in variational Monte Carlo renormalization group . 18

2.6 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.6.1 The coupling terms Sα . . . . . . . . . . . . . . . . . . . . . . 21

3 Tangent space and curvature to the critical manifold of statistical

system 23

3.1 The critical manifold of a statistical system . . . . . . . . . . . . . . 23

3.2 The critical manifold tangent space . . . . . . . . . . . . . . . . . . . 25

vii

3.2.1 Critical manifold tangent space in the absence of marginal

operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

3.2.2 Critical manifold tangent space in the presence of marginal

operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

3.2.3 The Normal Vectors to Critical Manifold Tangent Space . . . 27

3.3 Numerical results for CMTS . . . . . . . . . . . . . . . . . . . . . . 28

3.3.1 2D Isotropic Ising model . . . . . . . . . . . . . . . . . . . . . 28

3.3.2 3D Istropic Ising Model . . . . . . . . . . . . . . . . . . . . . 29

3.3.3 2D Anistropic Ising Model . . . . . . . . . . . . . . . . . . . . 31

3.3.4 2D Tricritical Ising Model . . . . . . . . . . . . . . . . . . . . 32

3.4 Curvature of the critical manifold . . . . . . . . . . . . . . . . . . . . 34

3.5 Tangent space to the manifold of critical classical Hamiltonians repre-

sentable by tensor networks . . . . . . . . . . . . . . . . . . . . . . . 36

3.5.1 Monte Carlo renormalization group with tensor networks . . . 37

3.5.2 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . 40

4 Variational Monte Carlo renormalization group for systems with

quenched disorder 47

4.1 The renormalization group of statistical systems with quenched disorder 48

4.2 Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

4.2.1 2D dilute Ising model . . . . . . . . . . . . . . . . . . . . . . 53

4.2.2 1D transverse field Ising model . . . . . . . . . . . . . . . . . 55

4.2.3 2D spin glass . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

4.2.4 2D random field Ising model . . . . . . . . . . . . . . . . . . 58

4.2.5 3D random field Ising model . . . . . . . . . . . . . . . . . . 59

4.3 Time correlation functions in the biased ensemble . . . . . . . . . . . 61

4.4 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

4.4.1 Optimization details for the 2D DIM . . . . . . . . . . . . . . 61

viii

4.4.2 Couplings in the computation of critical exponents of the dilute

Ising model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

4.4.3 Optimization details for the 3D RFIM . . . . . . . . . . . . . 65

5 Variational Monte Carlo renormalization group for quantum sys-

tems 68

5.1 Continuous-time Monte Carlo simulation of a quantum system . . . . 68

5.2 The MCRG for continuous-time quantum Monte Carlo . . . . . . . . 71

5.3 The sound velocity of a critical quantum system . . . . . . . . . . . . 72

5.3.1 Q = 2: The Ising model . . . . . . . . . . . . . . . . . . . . . 74

5.3.2 Q = 3 and 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

5.4 The energy-stress tensor . . . . . . . . . . . . . . . . . . . . . . . . . 78

5.5 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

5.5.1 Spacetime correlator at large distance and low temperature . . 81

6 Outlook 83

Bibliography 85

ix

List of Tables

2.1 Leading even (e) and odd (o) eigenvalues of ∂K′α∂Kβ

at the approximate

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

the following spin products, S(0)β (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

∂2K(n)nn /∂K

(0)β ∂K

(0)η . 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

3.10 The matrix Pαβ =A(n,0)αβ

A(n,0)α1

for the isotropic 2D square Ising model. A 2562

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

3.12 The matrix Pαβ =A(n,0)αβ

A(n,0)α1

for the isotropic 2D square three-state Potts

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

5.3 The renormalized constants for the d = 2 TFIM. L = P = 128. K(4)1,x and

K(4)1,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

4.1 Distribution of K ′nn 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

4.7 Time correlation functions CM(t) = 〈M(t)M(0)〉〈M2〉 of the system magneti-

zation M = 1N

∑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, 〈M〉2, 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 opti-

mization 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 opti-

mization 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

Chapter 2, we first review the variational functional whose utility was initially demon-

strated 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 Chapter 3, 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 Chapter 4, we show how to use VMCRG to study statistical systems with

quenched disorder. In disordered systems, one focuses on the renormalization of4

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 Chapter 5, 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 Chapter 1, 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 dimen-

sions with spins σ and Hamiltonian H(0)(σ),

H(0)(σ) =∑β

K(0)β Sβ(σ) (2.1)

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

products, next nearest neighbor spin products, etc., and K(0) = K(0)β 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 σ′ in the renormalized system with a conditional

probability T (σ′|σ) that effects a scale transformation with scale factor b. T (σ′|σ) is

the probability of σ′ 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 (σ′|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:

T (n)(µ|σ) =∑

σ(n−1)

..∑σ(1)

T (µ|σ(n−1)) · · ·T (σ(1)|σ) (2.2)

T (n) defines the nth level renormalized Hamiltonian H(n)(µ) up to a constant g(K(0))

independent of µ [9]:

H(n)(µ) ≡ − ln∑σ

T (n)(µ|σ)e−H(0)(σ) + g(K(0))

=∑α

K(n)α Sα(µ) + g(K(0))

(2.3)

where K(n)α are the nth level renormalized coupling constants associated with the

coupling terms Sα(µ) defined for the nth level coarse-grained spins. Here g(K(0)) 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. The1A 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∗:

δK(n+1) ≡ K(n+1) −K∗ =∂K(n+1)

∂K(n)

∣∣∣∣K∗

(K(n) −K∗) =∂K(n+1)

∂K(n)

∣∣∣∣K∗δK(n) (2.4)

Let the eigenvalues and the eigenvectors of ∂K(n+1)

∂K(n)

∣∣∣∣K∗

be λi and φi, ordered by the

descending magnitude of λi. We also define the scaling variables u(n)i 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

of δK(n) in the eigenbasis φi:

δK(n) ≡∑i

u(n)i φi, u

(n+1)i = λiu

(n)i (2.5)

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

into the linear regime around K∗c , one expects that u(n)i depends analytically on t.

In particular, to the leading order of t, u(n)1 = 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]

fs(t) = b−ndfs(u(n)1 )

fs(λ1t) = b−ndfs(λ1u(n)1 ) = b−ndfs(u

(n+1)1 ) = b−ndbdfs(u

(n)1 )

(2.6)

Thus,

fs(t) = b−dfs(λ1t) = b−ndfs(λn1 t) (2.7)

which implies that

fs(t) = f±|t|d/y, 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:

Ω[V ] = log

∑µ e−[H(n)(µ)+V (µ)]∑µ e−H(n)(µ)

+∑µ

pt(µ)V (µ), (2.9)

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:

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

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

pVmin(µ) =e−(H(n)(µ)+Vmin(µ))∑∑∑µ e−(H(n)(µ)+Vmin(µ))

= pt(µ) (2.11)

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.

VJ(µ) =∑α

JαSα(µ) (2.12)

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)

∂Jα= −〈Sα(µ)〉VJ + 〈Sα(µ)〉pt (2.13)

13

∂2Ω(J)

∂Jα∂Jβ= 〈Sα(µ)Sβ(µ)〉VJ − 〈Sα(µ)〉VJ〈Sβ(µ)〉VJ (2.14)

Here 〈·〉VJ is the biased ensemble average under VJ and 〈·〉pt 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.

〈·〉VJ 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].

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

Taking a constant pt, we have modulo a constant:

H(n)(µ) = −Vmin(µ) =∑α

(−Jmin,α)Sα(µ) (2.15)

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

of Vmin(µ) with renormalized coupling constants

K(n)α = −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.

〈O(µ)〉V =

∑µO(µ)e−(H(n)(µ)+V (µ))∑

µ e−(H(n)(µ)+V (µ))

=

∑µ,σ O(µ)T (n)(µ|σ)e−(H(0)(σ)+V (µ))∑

µ,σ T(n)(µ|σ)e−(H(0)(σ)+V (µ))

(2.17)

Thus, the state space of the MC sampling should the product space of µ and σ:

(µ,σ). Let the proposal probability be:

g(µ′,σ′|µ,σ) = g(σ′|σ)T (n)(µ′|σ′) (2.18)

with the acceptance probability

A = min(

1, e−(∆H+∆V ) g(σ|σ′)g(σ′|σ)

)(2.19)

where ∆H = H(0)(σ′)−H(0)(σ),∆V = V (µ′)− 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|σ′)/g(σ′|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−∆Hg(σ|σ′)/g(σ′|σ) = 1 in the Wolff algorithm.

Note that one can also define the expected value of 〈O(µ,σ)〉 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 Chapter 3, 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

17

Figure 2.1: Variation of the renormalized coupling constants over five VMCRGiterations on a 300 × 300 lattice. Each iteration has 1240 variational steps, eachconsisting of 20 MC sweeps. 16 multiple walkers are used for the ensemble averages inEqs. 2.13 and 2.14. For clarity, we only show the four largest renormalized couplingsafter 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

eigenvalues of ∂K′∂K

∣∣K∗

, the Jacobian matrix of the RG transformation, at a critical fixed

point. Here K′ is the renormalized coupling constants from K after one RG iteration.

In order to find ∂K′

∂Knear a fixed point, we need to know how the renormalized coupling

constants K ′α from a RG iteration on the Hamiltonian H =∑

βKβSβ, change when

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

18

minimum condition, Eq. 2.13, implies dΩdJα

= 0, i.e. for all γ:

∑σ Sγ(µ)e−

∑β(KβSβ(σ)−K′βSβ(µ))∑

σ e−

∑β(KβSβ(σ)−K′βSβ(µ))

= 〈Sγ(µ)〉pt , (2.20)

and ∑σ Sγ(µ)e−

∑β((Kβ+δKβ)Sβ(σ)−(K′β+δK′β)Sβ(µ))∑

σ e−

∑β((Kβ+δKβ)Sβ(σ)−(K′β+δK′β)Sβ(µ))

= 〈Sγ(µ)〉pt . (2.21)

Expanding Eq. 2.21 to linear order in δK ′α and δKβ, we obtain

Aβγ =∑α

∂K ′α∂Kβ

Bαγ, (2.22)

where

Aβγ = 〈Sβ(σ)Sγ(µ)〉V − 〈Sβ(σ)〉V 〈Sγ(µ)〉V , (2.23)

and

Bαγ = 〈Sα(µ)Sγ(µ)〉V − 〈Sα(µ)〉V 〈Sγ(µ)〉V . (2.24)

Here 〈·〉V 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

under H(µ) =∑

βK′βSβ(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

represent Kα, and those after the second RG iteration represent K ′α. The results for

biased and unbiased ensembles are shown in Table 2.1, which reports the leading

even (e) and odd (o) eigenvalues of ∂K′

∂Kwhen 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.

L λe1 λo1unbiased 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

Table 2.1: Leading even (e) and odd (o) eigenvalues of ∂K′α∂Kβ

at the approximate fixedpoint found with VRG, in both the unbiased and biased ensembles. The number inparentheses is the statistical uncertainty on the last digit, obtained from the standarderror 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 used106 MC sweeps for the 45×45 and 90×90 lattices, and 5×105 sweeps for the 300×300lattice.

20

Figure 2.2: (color online). Time correlation of the estimator A = S0(σ)S0(µ) on45 × 45 and 90 × 90 lattices (Eq. 5.11). S0 is the nearest neighbor term in thesimulations 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, -121

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 under-

standing 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 consider-

ation. 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:

〈Sγ(µ)〉Vmin = 〈Sγ(µ)〉pt (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 Chapter 5.

3.2 The critical manifold tangent space

3.2.1 Critical manifold tangent space in the absence of

marginal operators

To compute the CMTS, let us suppose that K(0)β and K(0)

β +δK(0)β 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

constants, K(n)α,truncate and K

(n)α,truncate + δK

(n)α,truncate, renormalized respectively from K

(0)β

and K(0)β + δK

(0)β , 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,

δK(0)β , satisfies the following equation for sufficiently large n,

K(n)α,truncate +

∑β

∂K(n)α,truncate

∂K(0)β

δK(0)β = K

(n)α,truncate (3.2)

for every α. That is, the CMTS δK(0)β is the kernel of the nth level RG Jacobian:

A(n,0)αβ ≡

∂K(n)α,truncate

∂K(0)β

(3.3)

for any well-defined truncation scheme. In the following, we will use K(n)α to denote

the truncated coupling constants.

As shown in Chapter 2, VMCRG provides an efficient way to compute the renor-

malized 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

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

K(n+1)α . Now suppose that both K(0)

β and K(0)β + δK

(0)β are on the critical manifold,

respectively giving rise to the truncated renormalized constants K(n)α and K(n)

α + δK(n)α .

Then, the spanning set of CMTS, δK(0)β , instead of Eq. 3.2, satisfies the following

26

condition,

K(n)α +

∑β

∂K(n)α

∂K(0)β

δK(0)β = K(n+1)

α +∑β

∂K(n+1)α

∂K(0)β

δK(0)β (3.4)

for every α. But K(n)α and K

(n+1)α 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,

A(n+1,0)αβ −A(n,0)

αβ (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

Pαβ =A(n,0)αβ

A(n,0)α1

orA(n+1,0)αβ −A(n,0)

αβ

A(n+1,0)α1 −A(n,0)

α1

, (3.6)

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 cor-

relation 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)

H(σ) = −K(0)nn

∑〈i,j〉

σiσj −K(0)nnn

∑[i,j]

σiσj (3.7)

where 〈i, j〉 denotes the nearest neighbor pairs and [i, j] the next nearest neighbor pairs.

K(0)nn and K(0)

nnn are the corresponding coupling constants. This model is analytically

solvable when K(0)nnn = 0 and is critical at the Onsager point with K(0)

nn = 0.4407... [16].

Four critical points are first located with VMCRG in the coupling space of K(0)nn , K

(0)nnn.

This task can be achieved by fixing K(0)nnn and varying K(0)

nn while monitoring how the

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

index. The largest value of the original coupling constant, K(0)nn,1, for which K

(n)nn

decreases with n, and the smallest value, K(0)nn,2, for which K

(n)nn increases with n, define

the best estimate, within statistical errors, of the interval [K(0)nn,1, K

(0)nn,2] of location

28

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

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

monotonically with the exact K(n)nn . 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.

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

nearest product K(n)nnn, and the smallest plaquette K(n)

. 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

coupling space spanned by K(0)nn , K(0)

nnn, K(0) , and the third nearest neighbor products,

K(0)nnnn. 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

Consider now the same model on a 3D square lattice with K(0)nnn = 0, i.e. the 3D

isotropic nearest neighbor Ising model. This model does not have an analytical

solution, but is known to experience a continuous transition at K(0)nn = 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

K(0)nn K

(0)nnn 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 1.80060.37 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 threerenormalized constants: nn, nnn, and . The fourth row of the table at the Onsagerpoint shows the exact values. β = 2, 3, and 4 respectively indexes the componentof 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. Thesimulations were performed on 16 cores independently, each of which ran 3 × 106

Metropolis MC sweeps. The standard errors are cited as the statistical uncertainty.

K(0)nn K

(0)nnn 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 rowscorresponding 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 anIsing spin. The simulations were performed on 16 cores independently, each of whichran 3× 106 Metropolis MC sweeps. The standard errors are cited as the statisticaluncertainty.

next nearest-neighbor renormalized coupling constants, K(n)nn and K(n)

nnn, 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 correspondto the two different α which respectively index the nn and the nnn renormalizedconstants. β runs from 1 to 8, corresponding to the following spin products, S(0)

β (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 MCsweeps. The simulations were performed at the nearest-neighbor critical point withKnn = 0.22165.

3.3.3 2D Anistropic Ising Model

Consider then the anisotropic Ising model on a 2D square lattice with Hamiltonian

H(σ)

H(σ) = −K(0)nnx

∑〈i,j〉x

σiσj −K(0)nny

∑〈i,j〉y

σiσj (3.8)

where 〈i, j〉x and 〈i, j〉y respectively denote the nearest neighbor pairs along the

horizontal and the vertical direction. In the space of K(0)nnx , K

(0)nny, the model is

exactly solvable and is critical along the line [7]

sinh(2K(0)nnx) · sinh(2K(0)

nny) = 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-

ical points of the system with K(0)nny/K

(0)nnx = 2, and 3, with four renormalized

couplings: K(n)nnx , K

(n)nny , K

(n)nnn, K

(n) . The CMTS is computed in the coupling space

K(0)nnx , K

(0)nny , K

(0)nnn, K

(0) , K

(0)nnnnx , K

(0)nnnny using Eq. 3.5, as shown by Pαβ in Table. 3.4.

31

K(0)nnx 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 0.64780.240606 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 correspondingto the four renormalized constants: nnx, nny, nnn, and . β = 2 − 6 respectivelyindexes the component of the normal vector to CMTS corresponding to coupling termsnny, 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

H(σ) = −K(0)nn

∑〈i,j〉

σiσj −K(0)4

∑i

σ2i (3.10)

where σ = ±1, 0 and 〈i, j〉 denotes the nearest neighbor pairs. In the coupling space of

K(0)nn and K(0)

4 , 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

theory (CFT) changes from the Ising CFT with central charge 12to one with central

charge 710

[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

K(0)nn = 1.642(8) and K(0)

4 = −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

Coupling1 σ2

i

2 σiσj, i and j nearest neighbor3 σiσj, i and j next nearest neighbor4 σiσjσkσl, i, j, k, l in the smallest plaquette5 (σiσj)

2, i and j nearest neighbor6 (σiσj)

2, i and j next nearest neighbor

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

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 thefirst five renormalized couplings listed in Table 3.5, which also gives the couplings forβ = 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 within33

α Pα2 Pα3 Pα4 Pα5 Pα6

3 2.171 2.230 1.019 2.163 2.1944 2.214 2.284 1.047 2.204 2.2455 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

model as an example. For a change δK(0)β in the original coupling constants, we

expand the corresponding change in the renormalized constants to quadratic order:

δK(n)α =

∑β

A(n,0)αβ δK

(0)β +

1

2

∑βη

B(n,0)αβη δK

(0)β δK(0)

η (3.12)

where A(n,0)αβ and B(n,0)

αβη can be determined by substituting Eq. 3.12 in Eq. 3.1 and

enforcing equality to second order in δK(0)α . A(n,0)

αβ is already given in Eq. 4.12. The

result for B is that for given β and η, for every γ, one requires

∑α

〈〈Sγ(µ), Sα(µ)〉〉V B(n,0)αβη = 〈〈Sγ(µ), Sβ(σ)Sη(σ)〉〉V

+∑αν

AαβAνη〈〈Sγ(µ), Sα(µ)Sν(µ)〉〉V

− 2∑α

Aαη〈〈Sγ(µ), Sβ(σ)Sα(µ)〉〉V

(3.13)

where the connected correlation functions are again sampled in the biased ensemble

〈·〉V . 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)α

∂K(0)β ∂K

(0)η

=1

2

(B(n,0)αβη + B(n,0)

αηβ

)(3.14)

In the coupling space of any pair β and η: K(0)β , K

(0)η , 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

K(n)α (K

(0)β , K(0)

η ) = 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

K(0)nn β

η nnn nnnn

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) 0.1480.37 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∂2K

(n)nn /∂K

(0)β ∂K

(0)η . The exact curvature for β = nn and η = nnn at the Onsager

point is also shown [28].

only quote the result calculated from the nearest neighbor renormalized constants K(n)α ,

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 〈〈Sγ, Sα〉〉 is of order N , where

N is the system size, but each of the terms on the right side is of order N2. Thus,

a delicate and exact cancellation of terms of order N2 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

an uncertainty typical for O(N3) 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(N2) 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 Nm+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 micro-

scopic 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:

H(σ) = −K∑〈x,y〉

σxσy (3.16)

where 〈x, y〉 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:

Z =∑ijk···

AaijklAbpqriA

cnojm · · · (3.17)

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

k l

i j m

n op q

r

e

s t u

v

w

Ab Ac

Aa

Ad Ae

i0

i2

i1

i3

A

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 indicesare labeled as i0i1i2i3. A grey circle represents a lattice site, or equivalently a tensorindex. A green box represents a tensor.

Aa by its relative position, x, to Aa with the notation σax. 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,

for example, both as σa0 and σb3 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:

Aai0i1i2i3 = eK(ηi0ηi1+ηi1ηi3+ηi3ηi2+ηi2ηi0 ) (3.18)

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

ηi ≡

−1, i = 0

1, i = 1

(3.19)

38

To perform MCRG, one needs to write the partition function as a configuration

sum in terms of a Hamiltonian H(σ):

Z =∑σ

e−H(σ) (3.20)

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

H(σ) =Nc∑β=1

KβSβ(σ) (3.21)

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:

H(σ) =

NA∑a=1

ln(Aai0i1i2i3) (3.22)

when σa0 = ηi0 , σa1 = ηi1 , σ

a2 = ηi2 , σ

a3 = 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:

H(σ) =

NA∑a

∑i0i1i2i3

lnAai0i1i2i3δσa0 ,ηi0δσa1 ,ηi1δσa2 ,ηi2δσa3 ,ηi3 (3.23)

In translationaly invariant systems, lnAai0i1i2i3 is independent from a, and

H(σ) =∑i0i1i2i3

lnAi0i1i2i3

NA∑a=1

δσa0 ,ηi0δσa1 ,ηi1δσa2 ,ηi2δσa3 ,ηi3

≡Nc∑β=1

KβSβ(σ)

(3.24)

39

where we have identified the logarithm of each tensor element, lnAi0i1i2i3 , as one

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

tuple i0i1i2i3 plays the role of β:

Kβ = Ki0i1i2i3 = lnAi0i1i2i3 (3.25)

and

Sβ(σ) = Si0i1i2i3(σ) =

NA∑a=1

δσa0 ,ηi0δσa1 ,ηi1δσa2 ,ηi2δσa3 ,ηi3 (3.26)

For a tensor network with n legs and bond dimension χ on each leg, there are

therefore Nc = χn 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 i0i1i2i31 0000, 11112 0100, 0010, 0001, 0111, 1011, 1101, 1110, 10003 0110, 10014 0101, 0011, 1010, 1100

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

and 1111 coupling terms:

S1(σ) =

NA∑a=1

δσa0 ,η0δσa1 ,η0δσa2 ,η0δσa3 ,η0

+

NA∑a=1

δσa0 ,η1δσa1 ,η1δσa2 ,η1δσa3 ,η1

(3.27)

and

K1 = lnA0000 = lnA1111 (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

Nc=4∑β=1

Sβ(µ) = NA (3.29)

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:

H(µ) =3∑

β=1

KβSβ(µ) +K4(NA − S1(µ)− S2(µ)− S3(µ))

=3∑

β=1

(Kβ −K4)Sβ(µ) + constant

=3∑

β=1

K ′βSβ(µ) + constant

(3.30)

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

K ′β:

A(n,0)αβ =

∂K′(n)α

∂K′(0)β

(3.31)

In Table 3.10, we report the matrix Pαβ =A(n,0)αβ

A(n,0)α1

, computed at the nearest-neighbor

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)

Table 3.10: The matrix Pαβ =A(n,0)αβ

A(n,0)α1

for the isotropic 2D square Ising model. A 2562

lattice was used with the renormalization level n = 5. The simulations were performedon 16 cores independently, each of which ran 3 × 106 Metropolis MC sweeps. Themean 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

uncertainty. In converting the computed δK ′β 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

so long as the resultant δK ′β = δKβ − δK4 conforms to the computed value. Here we

take δKβ = δK ′β 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

lnAi0i1,i2i3 = Kc

4 0 0 0

0 0 −4 0

0 −4 0 0

0 0 0 4

+ δK2

0.522(1) 1 1 0

1 0 0 1

1 0 0 1

0 1 1 0.522(1)

+ δK3

0.0184(3) 0 0 0

0 0 1 0

0 1 0 0

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:

H(σ) = −K∑〈x,y〉

δσxσy (3.33)

where 〈x, y〉 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

Nc = 34 = 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

-1.5-1

-0.50

0.51

1.52

2.5

0 200 400 600 800 1000

K α

VariationalStep

Figure 3.2: Optimization trajectory of the tensor network renormalized constants forthe three-state Potts model on a 162 lattice at K = 1.005053. All 81 renormalizedconstants are independently optimized and shown. Each curve represents one couplingterm.

classes and coupling terms, listed in Table 3.11.

β i0i1i2i31 00002 10003 11004 21005 01106 2110

Table 3.11: The tensor elements which belong to distinct symmetry classes. Only onerepresentative 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 for44

the 2D three-state Potts model is also one.) The tangent space to TNCM is again

reported as the matrix Pαβ =A(n,0)αβ

A(n,0)α1

in Table 3.12, from which its matrix form can be

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)

Table 3.12: The matrix Pαβ =A(n,0)αβ

A(n,0)α1

for the isotropic 2D square three-state Potts

model. A 2562 lattice was used with the renormalization level n = 5. The simulationswere performed on 16 cores independently, each of which ran 9× 105 Metropolis MCsweeps. The mean is cited as the result and the standard error as the statisticaluncertainty.

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

i0

i1i2

i3

i4

i5i6

i7

A

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 i0i1i2i3i4i5i6i71 00000000 8 111010002 10000000 9 100110003 11000000 10 110110004 01100000 11 011110005 11100000 12 001111006 11110000 13 100101107 01101000

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

terms are used to span the vector space of Hamiltonians representable by a 3D tensor

network, which admits a 11-dimensional critical manifold. The matrix Pαβ =A(n,0)αβ

A(n,0)α,1

is

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),

−0.218(2), −0.076(1), −0.0176(6)] (3.35)

46

Chapter 4


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:

HK(σ) = −∑α

N∑i=1

Ns(α)∑s=1

Ki,sα S

i,sα (σ), K ∼ Pv(K) (4.1)

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

terms at each lattice site, while the smallest plaquette has only one. Si,sα are products

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

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

vector K denotes the full set Ki,sα 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

K′ and Hamiltonian H ′K′ are related to their original counterparts by

H ′K′(µ) +Ng(K) = − ln∑σ

T (µ|σ)e−HK(σ) (4.2)

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

per site of a RG transformation [9] so that H ′K′ does not contain spin independent

terms. Let R be the RG map of the coupling constants implicitly defined by Eq. 4.2:

K′ = R(K) (4.3)

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

Pv′(K′) =

∫dKPv(K)δ(K′ −R(K)) (4.4)

Thus, upon coarse-graining of σ, the renormalization of the coupling constants, from

K to K′, induces a renormalization from v to v′.

49

The free energy of a quench-disordered system is

F (v) = −∫dKPv(K) ln

∑σ

e−HK(σ), (4.5)

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

F (v′) = −∫dKPv(K) ln

∑µ

e−H′R(K)

(µ)

= −∫dK′Pv′(K

′) ln∑µ

e−H′K′ (µ)

(4.6)

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):

fs(v) =1

bdfs(v

′), (4.7)

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 K′ = 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

the renormalized Hamiltonian H ′K′(µ) a bias potential V (µ) so that the distribution of

µ under the Hamiltonian H ′K′(µ) + V (µ) becomes equal to a preset target probability

pt(µ). By choosing the uniform distribution for the latter, i.e. pt(µ) = (12)N′ 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:

VJ(µ) =∑α

N∑i=1

Ns(α)∑s=1

J iαSiα(µ) (4.8)

The minimizing coefficients, Jmin = J iα,min, can be found by minimizing Ω with

a gradient descent procedure [6]. In disordered systems, the number of unknown

coefficients is large and we use only the diagonal part of the Hessian ∂2Ω

∂Ji,sα ∂Jj,tβin

addition to the gradient ∂Ω

∂Ji,sαin the minimization procedure. Convergence is facilitated

by the convexity of Ω[V ] [6]. Empirically, we find that the optimization cost increases

linearly with the number of coefficients J i,sα , making possible calculations on large

lattices. We have by virtue of Eq. 2.16:

K′ = −Jmin (4.9)

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

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

according to Pv′(K′) at each RG iteration. This distribution can be visualized with

histograms representing the marginal distribution of coupling type α:

Qv(Kα) =

ND∑iD=1

N∑i=1

Ns(α)∑s=1

δε(Kα − (Ki,sα )iD)

NDNαNs(α)(4.10)

Here δε is a delta-function approximant with support ε, 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

Jacobian ∂v′

∂vat 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]:

− lnPv(K) = C +∑β

vβUβ(K) (4.11)

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

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

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

the first four Uβ(K) could be U1 =∑

i,sKi,snn , U2 =

∑i,sK

i,snnn, U3 =

∑i,s(K

i,snn)2, and

U4 =∑

i,sKi,snnK

i,snnn. Taking the derivative ∂v′

∂vin the close proximity of v∗, we obtain:

Aβγ =∑α

∂v′α∂vβ·Bαγ, (4.12)

where

Aβγ = 〈UβU ′γ〉 − 〈Uβ〉〈U ′γ〉 (4.13)

Bαγ = 〈U ′αU ′γ〉 − 〈U ′α〉〈U ′γ〉 (4.14)

Here 〈·〉 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

HDI = −KDI

∑〈i,j〉

kijσiσj (4.15)

Here KDI > 0, 〈i, j〉 denotes nearest neighbors, and kij = 1 or 12with probability 1

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

on 1282 lattices for 4 RG iterations for three values of KDI, i.e. KDI = KDI,c, 0.60,

and 0.62. In addition, for KDI = KDI,c, we carry out a 5th iteration on a 2562 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

0

1

2

3

4

5

6

7

0.1 0.2 0.3 0.4 0.5 0.6 0.7

Q(K')

K'nn

n=1n=2n=3n=4

0.1 0.2 0.3 0.4 0.5 0.6 0.7K'nn

n=1n=2n=3n=4n=5

0.1 0.2 0.3 0.4 0.5 0.6 0.7K'nn

n=1n=2n=3n=4

Figure 4.1: Distribution of K ′nn 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:

HTFIM = −∑i

kiσzi σ

zi+1 −

∑i

hiσxi (4.16)

where σz and σx 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]:

HTrotter = −L∑i=1

βm∑j=1

kimσi,jσi+1,j

−L∑i=1

βm∑j=1

1

2ln

[coth

(him

)]σi,jσi,j+1

(4.17)

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.

0

2

4

6

8

10

0.20.30.40.50.60.70.80.9 1 1.1

Q(K')

Ky'nn

n=1n=2n=3n=4

0

2

4

6

8

10

0.20.30.40.50.60.70.80.9 1 1.1Ky'nn

n=1n=2n=3n=4

0

2

4

6

8

10

0.20.30.40.50.60.70.80.9 1 1.1Ky'nn

n=1n=2n=3n=4

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

4.2.3 2D spin glass

We now consider the spin glass Hamiltonian:

HK = −KSG

∑〈i,j〉

kijσiσj (4.18)

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]:

q ≡∫dKPv(K)

N∑i=1

〈σi〉2HK(4.19)

=

∫dKPv(K)

∑σσ′

∑i

σiσ′i

e−HK(σ)−HK(σ′)

Z2K

(4.20)

where ZK is the partition function for the Hamiltonian HK. By introducing a new

Ising variable ρi = σiσ′i, Eq. 4.20 defines an effective Hamiltonian for the ρ spins:

Heff(ρ) = − ln∑σ

e−KSG∑〈i,j〉 k

ij(1+ρiρj)σiσj (4.21)

so that q is equal to the quenched average of 〈∑

i ρi〉Heff(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 trans-

formations, 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

obtained for a Hamiltonian with KSG = 1.2 on a 1282 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

0123456789

10

0 0.2 0.4 0.6 0.8 1 1.2 1.4K'nn(spinglass)

n=0n=1n=2

Figure 4.3: Distribution of nearest neighbor couplings for a spin glass model withKSG = 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:

HRFIM = −KRFIM

∑〈i,j〉

σiσj − h0

∑i

hiσi (4.22)

with KRFIM positive, 〈i, j〉 nearest neighbor, and the his 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

SiM = 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.

0

12

3

4

56

7

8

0 0.2 0.4 0.6 0.8 1 1.2 1.4

Q(K')

K'nn

n=1n=2n=3n=4

81216202428

-0.05 0 0.05 0.1

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

-10 -5 0 5 10

Q(K')

K'M

n=1n=2n=3n=4

Figure 4.4: Distribution of the renormalized nearest neighbor (left) and local magneti-zation (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-disorder59

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

0

5

10

15

20

25

0 0.05 0.1 0.15 0.2 0.25 0.3

Q(K')

K'nn

n=1n=2n=3

02468

101214161820

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7K'nn

n=1n=2n=3

02468

1012141618

0 0.2 0.4 0.6 0.8 1 1.2 1.4K'nn

n=1n=2n=3

00.10.20.30.40.50.60.70.80.91

-5 -4 -3 -2 -1 0 1 2 3 4 5

P(K')

K'

n=1n=2n=3

00.10.20.30.40.50.60.70.80.91

-8 -6 -4 -2 0 2 4 6K'

n=1n=2n=3

00.10.20.30.40.50.60.70.80.9

-15 -10 -5 0 5 10 15K'

n=1n=2n=3

0

10

20

30

40

50

60

70

-0.02 0 0.02 0.04 0.06 0.08 0.1

P(K')

K'

n=1n=2n=3

0

10

20

30

40

50

60

-0.05 0 0.050.10.150.20.250.30.35K'

n=1n=2n=3

05

101520253035404550

-0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4K'

n=1n=2n=3

Figure 4.5: Distribution of the renormalized coupling constants for the 3D RFIMwith h0 = 0.35, and KRFIM = 0.25 (left), 0.264 (middle), and 0.28 (right). The toprow is for the nearnest neighbor couplings. The middle row is for the magnetizationcouplings. 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

0

5

10

15

20

25

0 0.050.10.150.20.250.30.350.4

P(K')

K'nn

n=1n=2n=3

0

5

10

15

20

25

0 0.1 0.2 0.3 0.4 0.5

P(K')

K'nn

n=1n=2n=3

0

5

10

15

20

25

0 0.1 0.2 0.3 0.4 0.5

P(K')

K'nn

n=1n=2n=3

02468

101214161820

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

P(K')

K'nn

n=1n=2n=3

02468

101214161820

0 0.10.20.30.40.50.60.70.80.9

P(K')

K'nn

n=1n=2n=3

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

We focus on the magnetization, i.e. M =∑N

i=1 σi. We study the time correlation

function CM (t) = 〈M(t)M(0)〉〈M2〉 . (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

-0.20

0.20.40.60.81

0 500 1000 1500 2000

C M(t)

t/MCsweeps(diluteIsing)

biased162biased322

unbiased162

unbiased322

-0.20

0.20.40.60.81

0 500 1000 1500 2000

C M(t)

t/MCsweeps(transversefieldIsing)

biased32x32biased64x64

unbiased32x32unbiased64x64

-0.20

0.20.40.60.81

0 1000 2000 3000 4000 5000

C M(t)

t/MCsweeps(spinglass)

biased322biased642

unbiased82

unbiased162

-0.20

0.20.40.60.81

0 500 1000 1500 2000

C M(t)

t/MCsweeps(2DrandomfieldIsing)

biased322biased642

unbiased322

unbiased642

Figure 4.7: Time correlation functions CM (t) = 〈M(t)M(0)〉〈M2〉 of the system magnetization

M = 1N

∑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 biasedensemble, the bias potential VJmin is obtained by VMCRG for b = 2 with 1 RGiteration. The data are collected, shown from top to bottom, for the dilute Ising modelwith KDI = KDI,c, the Trotter approximation of the transverse field Ising model withKTFIM = 1.0, the 2D spin glass with KSG = 1.2, and the 2D random field Ising modelwith 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

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 100 200 300 400 500

C M(t)

t/MCsweeps(3DrandomfieldIsing)

unbiased8x8unbiased16x16

biased8x8biased16x16

0.975

0.98

0.985

0.99

0.995

1

0 100 200 300 400 500

Figure 4.8: Time correlation functions CM (t) in the biased and unbiased ensemble forvarious 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 isplaced in the center of the figure to show the unbiased time correlation functions moreclearly. Also note that in the biased ensemble of the 3D RFIM, the time correlationfunction eventually converges to the average magnetization squared, 〈M〉2, which inthe 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

0.1

0.2

0.3

0.4

0.5

0.6

0 500 1000 1500 2000 2500 3000 3500 4000

Kinn

VariationalStep

Figure 4.9: Renormalized nearest neighbor constants at criticality during the opti-mization of Ω[V ] for one bond realization of a 128× 128 dilute Ising model, for thefirst 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 (σ′) 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: min1, exp(−∆V (σ′)

(the detailed balance condition can be easily proved). Here ∆V (σ′) 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

0.1

0.2

0.3

0.4

0.5

0.6

0 200 400 600 800 1000

Kinn

VariationalStep

Figure 4.10: Renormalized nearest neighbor constants at criticality during the opti-mization of Ω[V ] for one bond realization of a 256× 256 dilute Ising model for the5th 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

optimization for the renormalized constants for the local magnetic field, KiM , and the

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

66

-1.5-1

-0.50

0.51

1.52

2.5

0 500 1000 1500 2000 2500 3000

Kinn

VariationalStep

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

0

0.1

0.2

0.3

0.4

0.5

0.6

0 500 1000 1500 2000 2500 3000

Kinn

VariationalStep

Figure 4.12: Renormalized constants for the nearest neighbor product coupling duringthe 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


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:

HQ = −∑〈i,j〉

Q−1∑k=0

Ωki Ω

Q−kj − g

Ld∑i=1

Q−1∑k=1

Mki , (5.1)

68

where the system is in a d-dimensional hypercubic lattice with length L. i and j

label different lattice sites, and 〈i, j〉 denotes a nearest neighbor bond. The operators

Ωi and Mi act on the Q states of the local Hilbert space at site i, which we label

by |0〉i, ..., |s〉i, ...|Q − 1〉i. In this local basis, the Ωi is a diagonal matrix such that

Ωi|s〉i = ωs|s〉i, where ω = ei2π/Q and s = 0, · · · , Q − 1. Mi performs a cyclic

permutation: |0〉i → |Q − 1〉i, |1〉i → |0〉i, · · · , |Q − 1〉i → |Q − 2〉i, 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 =

Tr(e−βHQ), one takes

H0 = −∑〈i,j〉

Q−1∑k=0

Ωki Ω

Q−kj , H1 = −g

Ld∑i=1

Q−1∑k=1

Mki (5.2)

and considers the partition function in the interaction picture

Z = Tr[exp(−βH0)Texp(−∫ β

0

H1(τ)dτ)], (5.3)

where T is the time-ordering operator in imaginary time τ , and H1(τ) = eτH0H1e−τH0

is the H1, in the interaction picture. Eq. 5.3 can be written as a diagrammatic

69

expansion in the following way:

Z =∑σ

∞∑n=0

(−1)n

n!〈σ|e−βH0T

∫ β

0

H1(τn)dτn · · ·∫ β

0

H1(τ1)dτ1|σ〉

=∑σ

∞∑n=0

gn∫ β

0

dτn

∫ τn

0

dτn−1 · · ·∫ τ2

0

dτ1〈σ|e−(β−τn)H0

Ld∑i=1

Q−1∑k=1

Mki e−(τn−τn−1)H0 · · · e−τ1H0|σ〉

=∑σ

∞∑n=0

gn∑i1···in

∫ β

0

dτn

∫ τn

0

dτn−1 · · ·∫ τ2

0

dτ1〈σ|e−(β−τn)H0

Q−1∑k=1

Mkine−(τn−τn−1)H0 · · ·

Q−1∑k=1

Mki1e−τ1H0|σ〉

(5.4)

Here the states |σ〉 = ⊗i|σ〉i form a basis in the Hilbert space. Each index i1, · · · , in

runs from 1 to Ld. H0 is diagonal in the |σ〉 basis: H0|σ〉 = Q∑〈i,j〉 δσi,σj |σ〉 ≡

h0(σ)|σ〉. Eq. 5.4 suggests the following Monte Carlo (MC) scheme to sample the

partition function. For each i = 1, · · · , Ld 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 |σ〉 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 τ−l and τ+l respectively denote the time

immediately before and after the flip time τl. If a spin flip occurs at τl on site il,

σil(τ−l ) will be made to switch to any σil(τ

+l ) different from σil(τ

−l ) by the action of∑Q−1

k=1 Mkil. In Eq. 5.4, the earliest spin flip occurs at τ1 on site i1; the second one

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

factors equal to e−(τl+1−τl)h0(σ(τ+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.

That is, we partition the worldline into P intervals: [0, Pβ

], [Pβ, 2βP

], · · · , [ (P−1)βP

, 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:

P (σ) ∝ e−H(σ), where H(σ) = −∑α

KαSα(σ), (5.5)

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 invari-

ance 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 confor-

mal 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) = 〈σ(0, 0)σ(x, τ)〉, where

σ(x, τ) = eτH σ(x)e−τH , 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 between1There 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

vary Pβuntil the couplings along the space direction and those along the time direction

become equal for a large n. The Pβso determined is the sound velocity, vs.

5.3.1 Q = 2: The Ising model

When Q = 2, HQ=2 coincides up to an additive constant with the Hamiltonian of the

transverse-field Ising model (TFIM),

HIsing = −∑〈i,j〉

σzi σzj − g

Ld∑i=1

σxi , (5.7)

where σz,xi 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] :

E(k) = 2√

1− 2gc cos k + g2c = 2|k|+ o(|k|), (5.8)

where gc = 1. Note that sometimes the Ising Hamiltonian is written with spin

operators S = 12σ, in which case vs would be 1

2. The K(n)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

increasing n when Pβ

= vs. Note that the relative magnitude of K(n)1,x and K(n)

1,y switches

when Pβcrosses 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 term1, x nearest neighbor product along the time direction1, y nearest neighbor product along the space direction2 2nd neighbor product3 product of spins in the smallest plaquette4, x 3rd neighbor product along the time direction4, 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, thecouplings are themselves isotropic between space and time.

n K(n)1,x K

(n)1,y K

(n)4,x K

(n)4,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)

(a) Pβ = 2.0031299

n K(n)1,x K

(n)1,y K

(n)4,x K

(n)4,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)

(b) Pβ = 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)

(c) Pβ = 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 MCsampling is done on 16 cores in parallel, where each core does MC sampling of 20000Wolff steps. The optimization step is µ = 0.001. The number in the paranthesis is theuncertainty on the last digit.

75

Pβ

K(4)1,x K

(4)1,yz

3.42246 0.1603(3) 0.1597(2)3.40426 0.1594(3) 0.1602(2)

Table 5.3: The renormalized constants for the d = 2 TFIM. L = P = 128. K(4)1,x and

K(4)1,yz are respectively the renormalized nearest neighbor spin constants along the time

and the space direction at n = 4.

α Coupling term1, x δσiσj for 1st neighobor i, j along the time direction1, y δσiσj for 1st neighobor i, j along the space direction2 δσiσj for 2nd neighbor i, j3, x δσiσj for 3rd neighobor i, j along the time direction3, 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 whenα = 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 K(n)1,x K

(n)1,y K

(n)3,x K

(n)3,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)

(a) Pβ = 2.5995

n K(n)1,x K

(n)1,y K

(n)3,x K

(n)3,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)

(b) Pβ = 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 localupdates with 1000 variational steps. When n = 0, 1, 2, each variational step uses 100sweeps of MC averaging in parallel on 8 cores. When n = 3, each variational step uses500 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 K(n)1,x K

(n)1,y K

(n)3,x K

(n)3,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)

(a) Pβ = 3.146

n K(n)1,x K

(n)1,y K

(n)3,x K

(n)3,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)

(b) Pβ = 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

As one changes the parameter Pβin 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

reflecting the different extent of anisotropy in the system [11]. A change of Pβgenerates

a movement along this line of fixed-point Hamiltonians. In fact, fixing P , the change

β → β+ δβ induces a coordinate transformation: x0 → x′0 = (1− δββ

)x0, x1 → x′1 = x1,

where x0 and x1 are time and space coordinates, respectively. Here we have taken the

coordinate transformation to be passive, i.e. x = (x0, x1) and x′ = (x′0, x′1) 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 → x′µ = xµ + εµ(x), is described by the energy-stress

tensor, T µν , defined by

δH = − 1

(2π)D−1

∫∂εµ

∂xνTµνd

Dx (5.9)

where D is the space-time dimension of the system. As β is conjugate to H 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

HIsing(σ) = −K0

∑〈i,j〉0

σiσj −K1

∑〈i,j〉1

σiσj (5.10)

where 〈i, j〉0 and 〈i, j〉1 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,

δH(σ) =∑m,n

Am,n(σ)δJ

≡∑m,n

(σm,nσm+1,n − σm,nσm,n+1)δJ

(5.11)

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

where δλ = 1γδJ 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

the critical d = 1 TFIM with Pβ

= 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 βH → (β + δβ)H generates a change in the

fixed-point Hamiltonian δH(n)(µ) = −∑

α∂K

(n)α

∂βSα(µ)δβ with an anisotropy of the

extent δ(x1x0

) =x′1x′0− x1

x0= δβ

βx1x0. For the classical d = 2 Ising model, the change in the

unrenormalized Hamiltonian HIsing → HIsing +∑

m,nAm,nδJ generates a change in the

fixed-point Hamiltonian δH(n)(µ) = −∑

α∂K

(n)α

∂JSα(µ)δJ , with an anisotropy of the

extent δ(x1x0

) = 2δλx1x0. The δH(n)(µ) for the TFIM and for the d = 2 classical Ising

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

γ =δJ

δλ= 2

β ∂K(n)α /∂β

∂K(n)α /∂J

(5.12)

for all α. The Jacobians of the RG transformation, ∂K(n)α

∂Jand ∂K

(n)α

∂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

is analytically known and is√

22

= 0.7071 · · · [80]. A VMCRG calculation using Eq.

5.12 with n = 4 gives γ = 0.708± 0.001, so the ruler works.

With the coordinate transformation x0 → x′0 = (1−δλ)x0 and x1 → x′1 = (1+δλ)x1,

a part of the energy-stress tensor can now be read off from Eq. 5.9:

∑m,n

Am,n(σ)δJ = δH(σ) = − 1

2π

∫(−δλT00 + δλT11)d2x. (5.13)

80

We take the lattice spacing to be 1, and∑

m,n is equivalent to∫d2x. This gives

γA =1

π(T + T ), (5.14)

where, in 2D, T and T are respectively the holomorphic and the antiholomorphic

component of the energy-stress tensor, and are defined as T = 14(T00 − 2iT01 − T11)

and T = 14(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 〈A〉, 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

〈T 〉 = 〈T 〉 = −(2πL

)2 c24

[70, 71], and thus 〈A〉 = − 1γπ

(2πL

)2 c12, 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 H have trans-

lational symmetry, i.e. there is a translation operator T (x) that commutes with H.

Consider the finite-temperature correlator C(x, τ) at inverse temperature β:

C(x, τ) =∑k

〈k|ψ(x, τ)ψ(0, 0)e−βH |k〉 (5.15)

where ψ(x) is some generic local operator that is labeled by the position x. The |k〉s

are the common eigenstates of H and T (x), and ψ(x, τ) = eτHψ(x, 0)e−τH . Also,

81

ψ(x, 0) = T (x)ψ(0, 0)T (−x) and T (x)|k〉 = e−ikx|k〉. Then,

C(x, τ) =∑k

〈k|eτH T (x)ψ(0, 0)T (−x)e−τHψ(0, 0)e−βH |k〉

=∑k

eτEkeikx〈k|ψ(0, 0)T (−x)e−τHψ(0, 0)|k〉e−βEk

=∑k,k′

eτEkeikxeik′xe−τEk′ |〈k|ψ(0, 0)|k′〉|2e−βEk

(5.16)

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

distance τ , we keep only the low-lying states for k′, where Ek′ = vs|k′|. We also change

the summation variable k′ to k. Thus,

C(x, τ) ≈∫dkeikxe−|k|vsτ |〈0|ψ(0, 0)|k〉|2 (5.17)

The function g(k) ≡ |〈0|ψ(0, 0)|k〉|2 depends on the details of ψ and |k〉s, and is

generally difficult to compute. One should note that C(x, τ) is not generically isotropic

between x and vsτ for any ψ. For example, in the exactly solvable transverse-field

Ising model, if one takes ψ to be the Fermionic operator after the Jordan-Wigner

transformation, then g(k) does not depend on k [81], and one can evaluate

C(x, τ) ∼∫dkeikxe−|k|vsτ ∼ |vsτ |

x2 + (vsτ)2(5.18)

which is not isotropic between x and vsτ . In our VMCRG calculation, we are interested

in ψ = σz, 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.

84

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