Strongly correlated systems : numerical methods

Vlll Contents 2.2 Truncated Diagonalization: The Numerical Renormalization Group Idea 32 2.2.1 Two-Spin Problem 32 2.2.2 ManySpins 33 2.2.3 ASimple Ge...

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Adolfo Avella



Ferdinando Mancini

Editors

Strongly

Correlated

Numerical Methods

With 106

Figures

Springer

Systems

Contents

Foreword Elbio

1

xvii

Dagotto

Ground State and Finite Temperature Lanczos Methods

I

P. Prelovsek and J. Bonca 1.1

Introduction

1.2

Exact

1

Diagonalization

and Lanczos Method

3

1.2.1

Models, Geometries and System Sizes

3

1.2.2

Lanczos

5

Diagonalization Technique Properties and Dynamics Properties and Dynamics at T > 0

1.3

Ground State

1.4

Static 1.4.1

Finite

Static 1.4.2

Finite

Temperature Quantities Temperature

9 Lanczos Method: 12

1.4.3

Finite

1.4.4

Implementation Low Temperature

Lanczos Method:

1.4.5

Microcanonical Lanczos Method

1.4.6

Statical and

14 Lanczos Method

Dynamical Quantities

16 18 at T > 0:

19

Applications

2

8

Lanczos Method:

Dynamical Response Temperature

7

1.5

Reduced Basis Lanczos Methods

22

1.6

Real Time

27

1.7

Discussion

Dynamics Using

Lanczos Method

28

References

29

The Density Matrix Renormalization Group

31

Adrian E. 2.1

Feiguin

Introduction

31

vii

Contents

Vlll

2.2

Truncated

Group 2.2.1

2.3

32

Idea

32

2.2.2 2.2.3

A

2.2.4

The Case of

2.2.5

The Block Decimation Idea

The

Simple

Density

Geometrical

33 35

Analogy

37

Spins

38

Matrix Truncation: The Kernel of the DMRG

The Reduced

2.3.2

The

2.3.3

The Schmidt

41 42

Decomposition

43

the Truncated Wave-Function

Optimizing

2.4.3

2.4.4

Adding

2.4.5

Single Super-Hamiltonian Building Obtaining the Ground-State: Lanczos Diagonalization Density Matrix Truncation and the a

46 47

the

Storing

47 Rotation 49

Basis

50

Matrices and States

50

The Finite-Size DMRG 2.5.1

Obtaining Quasi-Exact

Results with the Finite-Size

52

DMRG 2.5.2 2.5.3

53

Observables

Measuring

States

Targeting Calculating

Excited States

55

2.5.5

Wave-Function Prediction

56

2.5.6

Generalization

Dimensions

Higher Complex Geometries Why Does the DMRG Work? to

When and

57

58

2.6.1

Entanglement

58

2.6.2

59

2.6.3

Entanglement and the Schmidt Decomposition Quantifying Entanglement

2.6.4

The Area Law

61

2.6.5

Entanglement and the DMRG

63

Outlook: DMRG and Tensor Network Methods

63

60

64

References

3

55

2.5.4



and

2.7

44 44

Site to the Block

to the New

40

Value

Infinite-Size DMRG

2.4.2

2.6

...

Density Matrix Decomposition Singular

2.3.1

2.4.1

2.5

The Numerical Renormalization

Two-Spin Problem Many Spins

2.3.4

2.4

Diagonalization:

Matrix Product State

Algorithms: DMRG,

TEBD and Relatives

67

Ulrich Schollwock 3.1

Introduction

67

3.2

Ground State Calculations in One Dimension

68

3.2.1

Matrix Product States

68

Contents

jx

3.2.2

Matrix Product

3.2.3

The Variational MPS

Operators or

75

Finite-System

DMRG Algorithm

3.3

79

3.2.4

iDMRG—Infinite-System

3.2.5

Why

Does it Work and

DMRG Revisited

81

Does it Fail?

Why

86

Time Evolution of Matrix Product States 3.3.1

87

3.3.2

Conventional Time Evolution: Pure States Conventional Time Evolution: Mixed States

91

3.3.3

tMPS, tDMRG, TEBD: Variations

95

3.3.4

How Far Can We Go?

on

a

88

Theme

96

References

4

97

Quantum Criticality

with the Multi-scale

Entanglement

Renormalization Ansatz

Glen

Evenbly

99

and Guifre Vidal

4.1

Introduction

4.2

Entanglement Renormalization

4.3

Foundations of

4.2.2

Foundations of the MERA

105

4.2.3

Choice of MERA Scheme

108

Symmetries in Tensor 4.3.2

4.5

4.6

102

4.2.1

4.3.1

4.4

99 and the MERA

Entanglement

Renormalization

102

Network States

108 110

Spatial Symmetries Global Internal Symmetries

112

Scale-Invariant MERA

113

4.4.1

Basic

114

4.4.2

Transitional

4.4.3

Local

4.4.4

Scale-Invariant

Properties Layers

Density

114

Matrix

114 116

Objects

Benchmark Results

119

4.5.1

with MPS

Comparison

4.5.2

Evaluation of Conformal Data: The

121

Ising

Model

....

'28

Conclusions

'29

References

5

The Time-Dependent Adrian E.

Density Matrix Renormalization Group

131

Feiguin

5.1

Introduction

5.2

The

5.3

125

' 3'

Adaptive Time-Dependent DMRG (tDMRG)

5.2.1

The Suzuki-Trotter Approach

5.2.2

Evolution

The tDMRG

Using Suzuki-Trotter Expansions

Algorithm

133 133 135

'35

5.4

Time-Step Targeting Method

'37

5.5

Sources of Error

'39

Contents

x

5.6

Comparing Suzuki-Trotter

5.7

Evolution in

5.8

Applications 5.8.1 5.8.2

5.9

5.10

6

Imaginary

and

141

Time-Step Targeting

141

Time

143 144

Transport

Time-Dependent

146

Correlation Functions

148

Growth

The

Enemy: Entanglement Global Quench: Qualitative 5.9.1

149

Picture

5.9.2

Local Quench: Qualitative Picture

150

5.9.3

Cost

150

Computational

151

Discussion

References

151

Loop Algorithm

153

Synge Todo 6.1

Introduction

153

6.2

Path

Integral Representation Mapping to Classical System 6.2.2 Single Spin-1/2 in a Magnetic Field 6.2.3 Continuous Time Path Integral Representation 6.2.4 World-Line Representation of XXZ Spin Model

154

6.2.1

154

6.2.5 6.3

Negative Sign Loop Algorithm

162 162

6.3.2

162

6.3.3

Swendsen-Wang Algorithm Kandel-Domany Framework

6.3.4

Continuous

Imaginary

Rejection

163

Time Limit

165

Free Scheme

Loop Algorithm

6.3.6

Implementation

6.3.7

Generalizations

167

for XXZ Model and Technical

170

Aspects

171 173

Measurements 6.4.1

6.4.2 6.4.3 6.4.4 6.4.5

6.6

159 160

Problem

6.3.1

6.3.5

6.5

156 ....

Markov Chain Monte Carlo

and

6.4

155

173

and Correlation Functions

Diagonal Operators Susceptibilities and Dynamical Structure Correlation Length and Excitation Gap Energy and Specific Spin Stiffness

Heat

Factors

....

174 175 176 177

Loop Representation 6.5.1 Improved Estimators 6.5.2 Pure Loop Representation Pure Loop Algorithm 6.5.3

178

Conclusion

182

References

178 180 181

183

Contents

7

xi

Stochastic Series Expansion Quantum Monte Carlo

185

Roger

G. Melko

7.1

Introduction

185

7.2

Quantum

187

7.2.1

7.3

Monte Carlo Formalism

188

7.2.2

Finite-Temperature Representation Zero-Temperature Projector Representation

7.2.3

Local and Non-Local

191

190

Schemes

Updating Spin-1/2 Heisenberg Model 7.3.1 Finite-Temperature SSE in the S: 7.3.2 Zero-Temperature Projector

193 Basis

194

in the Valence Bond Basis

7.4

Transverse Field

Ising Finite-Temperature SSE in the 5" Basis Zero-Temperature Projector in the S: Basis

7.4.1 7.4.2

7.5

197

Model

199

200 203

Discussion

204

References

8

205

Variational Monte Carlo and Markov Chains

for

Computational Physics

207

Sandro Sorella 8.1

Introduction

8.2

Quantum Monte Carlo: The Variational Approach 8.2.1

207

Introduction:

Importance

208

of Correlated

Wave Functions

8.3

8.2.2

Expectation

8.2.3

Finite

208

Value of the Energy

Variance

213

Property

Markov Chains: Stochastic Walks in 8.3.1

211

Configuration Space.

...

Detailed Balance and Effective Hamiltonian

8.4

The

8.5

Stochastic Minimization of the

8.6

Conclusion

Coupled

225

Energy

230

235

Cluster Theories for

Molecular

218

223

Metropolis Algorithm

References 9

216

Strongly Correlated 237

Systems

Karol Kowalski, Kiran Bhaskaran-Nair, Jiff Brabec and Jin Pittner 9.1

9.2

Single 9.2.1

239

Reference CC (SRCC) Methods Standard

241

Approximations Equations Higher-Order

243

9.2.2

Solvability

9.2.3

Perturbative Inclusion of

Clusters

244

Implementations

of the SRCC Methods

248

9.2.4 9.3

237

Introduction

Parallel

of the CC

251

Multireference CC Theories 9.3.1

Wave

Operator

Formalism and Bloch

Equation

252

Contents

xii

9.3.2

State-Universal MRCC Formulations

253

9.3.3

Intruder State Problem

255

9.3.4

Incomplete/General Model Spaces State-Specific Methods Inclusion of High-Order Clusters

257

in MRCC Formalisms

259

9.3.5 9.3.6

9.3.7

256

Parallel Calculations with the MRCC Methods: 264

Reference-Level Parallelism 9.4

266

Conclusions

267

References

10

Diagrammatic Monte Carlo Nikolay 10.1 10.2

.

.

273

273

Diagrammatic 10.2.2 Worm 10.3.1 10.3.2

10.3.3

11

Algorithm Techniques.

Introduction

10.2.1

10.3

and Worm

Prokof ev 274

Monte Carlo

Updates Advantages Algorithm

276 278

and Potential Problems

280

Ising and XY Models Path-Integral Representation Wandering Amongst the Feynman Diagrams

281 285 289

References

291

Fermionic and Continuous Time Quantum Monte Carlo

293

Emanuel Gull and Matthias Troyer 11.1 11.2

Diagrammatic 11.21

11.3

11.4

11.5

293

Introduction

296

Monte Carlo

Monte Carlo Basics

11.2.2

Diagrammatic

11.2.3

The

Interaction

296

Monte Carlo

Negative Sign

298 300

Problem

302

Expansion

302

11.3.1

Partition Function

11.3.2

Updates

304

11.3.3

Measurements

304

11.3.4

Generalizations

305

Expansion

306

Hybridization Expansion 11.4.1

Partition Function

11.4.2

306

Updates

308

11.4.3

Measurements

308

11.4.4

Generalizations

310

Expansion

310

Applications 11.5.1

Nanoscience

11.5.2

Single

11.5.3

Cluster DMFT

Site

Dynamical

310 Mean Field

Theory

311

312

Contents

xiii

11.5.4

Diagrammatics Beyond

11.5.5

Lattice and Large Cluster DMFT Calculations and

11.6

313

Extrapolations to the Infinite System Applications to Real Materials

314

11.5.6 11.5.7

Real-Time

316

Outlook

References

Index

DMFT

Dynamics

315

317 317

321