Book contents
- Frontmatter
- Contents
- Preface
- Unit Used
- Notations and Graphical Representations
- Abbreviations
- 1 Introduction
- 2 Basic Algebra of Tensors
- 3 Tensor Network Representation of Classical Statistical Models
- 4 Tensor Network Representation of Operators
- 5 Tensor Network Ansatz of Wave Functions
- 6 Criterion of Truncation: Symmetric Systems
- 7 Real-Space DMRG
- 8 Implementation of Symmetries
- 9 DMRG with Nonlocal Basis States
- 10 Matrix Product States
- 11 Infinite Matrix Product States
- 12 Determination of MPS
- 13 Continuous Matrix Product States
- 14 Classical Transfer Matrix Renormalization
- 15 Criterion of Truncation: Nonsymmetric Systems
- 16 Renormalization of Quantum Transfer Matrices
- 17 MPS Solution of QTMRG
- 18 Dynamical Correlation Functions
- 19 Time-Dependent Methods
- 20 Tangent-Space Approaches
- 21 Tree Tensor Network States
- 22 Two-Dimensional Tensor Network States
- 23 Coarse-Graining Tensor Renormalization
- Appendix Other Numerical Methods
- References
- Index
22 - Two-Dimensional Tensor Network States
Published online by Cambridge University Press: 18 January 2024
- Frontmatter
- Contents
- Preface
- Unit Used
- Notations and Graphical Representations
- Abbreviations
- 1 Introduction
- 2 Basic Algebra of Tensors
- 3 Tensor Network Representation of Classical Statistical Models
- 4 Tensor Network Representation of Operators
- 5 Tensor Network Ansatz of Wave Functions
- 6 Criterion of Truncation: Symmetric Systems
- 7 Real-Space DMRG
- 8 Implementation of Symmetries
- 9 DMRG with Nonlocal Basis States
- 10 Matrix Product States
- 11 Infinite Matrix Product States
- 12 Determination of MPS
- 13 Continuous Matrix Product States
- 14 Classical Transfer Matrix Renormalization
- 15 Criterion of Truncation: Nonsymmetric Systems
- 16 Renormalization of Quantum Transfer Matrices
- 17 MPS Solution of QTMRG
- 18 Dynamical Correlation Functions
- 19 Time-Dependent Methods
- 20 Tangent-Space Approaches
- 21 Tree Tensor Network States
- 22 Two-Dimensional Tensor Network States
- 23 Coarse-Graining Tensor Renormalization
- Appendix Other Numerical Methods
- References
- Index
Summary
This chapter discusses the methods of solving PEPS or other two-dimensional tensor network states, including variational optimization and the annealing simulation. The variation optimization determines the local terms by minimizing the ground-state energy. The annealing simulation takes the full or simple update strategy to filter out the ground state through the imaginary time evolution. The nonlinear effect arises in evaluating the derivative of uniform PEPS and is avoided by utilizing automatic differentiation. Both variational optimization and the annealing simulation involve a contraction of double-layer tensor network states. This contraction is the primary technical barrier in the study of PEPS. A nested tensor network approach is introduced to combat this difficulty.
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- Density Matrix and Tensor Network Renormalization , pp. 351 - 363Publisher: Cambridge University PressPrint publication year: 2023