Strongly Correlated Systems: High Temperature Superconductors Heavy Fermion Compounds Organic materials M.N.Kiselev. 2 OAK RIDGE NATIONAL LABORATORY ... Crystal Structure and Fermi Surface. 15 OAK RIDGE NATIONAL LABORATORY U. S. DEPARTMENT OF ENERGY
Remarkably, no signature of a magnetic transition can be seen in the pressure window pc*< p
The application of these new techniques to correlated materials is particularly worth-while. The properties of these systems are determined by the presence of valence electrons in dor fshells. Because of their more localized nature, these electrons e
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
