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