Contents
Preface
Unit Used
Notations and Graphical Representations
List of Abbreviations
Introduction
1.1 Quantum Many-Body Problems
1.2 From NRG to DMRG
1.3 From DMRG to Tensor Network Algorithms
1.4 Applications
2 Basic Algebra of Tensors
2.1 Diagrammatic Representation of Tensors
2.2 QR and LQ Decompositions
2.3 LU Decomposition with Partial Pivoting
2.4 Singular Value Decomposition
2.5 Polar Decomposition
2.6 Higher-Order Singular Value Decomposition
2.7 Low-Rank Approximation of Tensors
2.8 Automatic Differentiation
2.9 Trotter-Suzuki Decomposition
3 Tensor Network Representation of Classical Statistical Models
3.1 Tensor Network Models
3.2 Matrix-Network Models
3.3 Tensor Network Representation in the Original Lattice
3.4 Tensor Network Representation in the Dual Space
3.5 Vertex-Sharing Lattice Models
3.6 Duality Properties of Tensor Network Models
4 Tensor Network Representation of Operators
2 Notations and Graphical Representations
4.1 Matrix Product Operators (MPO)
4.2 Imaginary Time Evolution Operato
4.3 Quantum Transfer Matrix
4.4 MPO Representation of Quantum Transfer Matrix
5 Tensor Network Ansatz ofWave Functions
5.1 Area Law of Entanglement Entropy
5.2 Matrix Product States (MPS)
5.3 One-Dimensional AKLT States
5.4 Multiscale Entanglement Renormalization Ansatz (MERA)
5.5 Projected Entangled Pair State (PEPS)
5.6 Projected Entangled Simplex State (PESS)
6 Criterion of Truncation: Symmetric Systems
6.1 Density Matrix
6.2 Reduced Density Matrix
6.3 Schmidt Decomposition
6.4 Variational Approach
6.5 Edge and Bond Density Matrices
7 Real-Space DMRG
7.1 Two Kinds of Algorithms
7.2 DMRG in the MPO Language
7.3 Error Analysis
7.4 Heisenberg Spin Chains
7.5 Periodic System
7.6 Multiple Target States
7.7 Two-Dimensional Systems
8 Implementation of Symmetries
8.1 Symmetry Consideration
8.2 Continuous Abelian Symmetries
8.3 Spin Reflection Symmetry
8.4 Spatial Reflection Symmetry
8.5 Non-Abelian Symmetries
9 DMRG with Nonlocal Basis States
9.1 General Consideration
9.2 Momentum-Space DMRG
9.3 DMRG in a General Basis Space
9.4 Optimization of Single-Particle Basis States
9.5 Optimizing Active Basis Space
Notations and Graphical Representations 3
10 Matrix Product States
10.1 The DMRG Wave Function
10.2 Canonical Representations
10.3 Canonical Transformation
10.4 Implementation of Symmetries
11 Infinite Matrix Product States
11.1 Translation Invariant MPS
11.2 Transfer Matrix and Canonical Transformation
11.3 Expectation Values of Physical Observables
11.4 String Order Parameter
11.5 MPS with a Finite Unit Cell
12 Determination of MPS
12.1 Variational Optimization
12.2 Excited states
12.3 Imaginary Time Evolution
12.4 Purification
13 Continuous Matrix Product States
13.1 Lattice Discretization of Continuous Quantum Field Theory
13.2 Continuum limit of MPS
13.3 Expectation Values
13.4 Canonicalization
13.5 Determination of Continuous MPS
14 Classical Transfer Matrix Renormalization
14.1 Classical Transfer Matrix
14.2 TMRG
14.3 Fixed-Point MPS: One-site Approach
14.4 Fixed-Point MPS: Two-Site Approach
14.5 Corner Transfer Matrix Renormalization
15 Criterion of Truncation: Nonsymmetric Systems
15.1 Nonsymmetric Density Matrix
15.2 Transformation Matrices
15.3 Canonicalization of the Transformation Matrices
15.4 Biorthonormalization
15.5 Low-Rank Approximation to the Environment Density Matrix
16 Renormalization of Quantum Transfer Matrices
16.1 Quantum Transfer Matrix and Thermodynamics
16.2 Correlation Functions
4 Notations and Graphical Representations
16.3 QTMRG
16.4 Thermodynamics of the Heisenberg Spin Chain
17 MPS Solution of QTMRG
17.1 Biorthonormal MPS
17.2 Biorthonormalization
17.3 Fixed-Point Equations
17.4 Translation Invariant System with a Finite Unit Cell
18 Dynamical Correlation Functions
18.1 Spectral Functions
18.2 Continued-Fraction Expansion
18.3 Dynamical Moments
18.4 Lanczos-DMRG Method
18.5 Dynamical Calculations with MPS
18.6 Correction-Vector Method
18.7 Spin Structure Factor of the Heisenberg Model
19 Time-Dependent Methods
19.1 Pace-Keeping DMRG
19.2 Time-Evolving Block Decimation
19.3 Adaptive Time-Dependent DMRG
19.4 Folded Transfer Matrix Method
20 Tangent-Space Approaches
20.1 Tangent Vectors of Uniform MPS
20.2 Time-Dependent Variational Principle
20.3 Single-mode excitations
20.4 Excitations Represented with PEPS
21 Tree Tensor Network States
21.1 Canonical Representation
21.2 Canonicalization
21.3 Husimi lattice
21.4 Determination of Tree Tensor Network State
21.5 Upper Bound of the Correlation Length
21.6 Thermodynamics
22 Two-Dimensional Tensor Network States
22.1 PEPS
22.2 Variational Optimization
22.3 Imaginary Time Evolution
22.4 Tensor Derivatives by Automatic Differentiation
Notations and Graphical Representations 5
22.5 Contraction of Double-Layer Tensor Networks
23 Coarse-Graining Tensor Renormalization
23.1 Coarse-Graining Approaches
23.2 TRG
23.3 Second Renormalized TRG
23.4 Determination of the Environment Tensor
23.5 Tensor Network Renormalization (TNR)
23.6 Loop Tensor Network Renormalization (Loop-TNR)
23.7 HOTRG
23.8 Second Renormalized HOTRG
23.9 Comparison of Different Methods
23.10 Three-Dimensional Classical Models
23.11 Two-Dimensional Quantum Lattice Models
Appendix A Other Numerical Methods
A.1 Power Method
A.2 Lanczos Method
A.3 Conjugate Gradient Method
A.4 Arnoldi Method
A.5 Quantum Monte Carlo Simulation
References
Index