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
19 - Time-Dependent Methods
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 introduces the method for solving time-dependent problems of quantum many-body systems. It includes the pace-keeping DMRG, time-evolving block decimation (TEBD), adaptive time-dependent DMRG, and folded transfer matrix methods. The pace-keeping DMRG, which solves the time-dependent Schrodinger equation, works independently of the dimensionality, nor the model Hamiltonian, with or without impurities. The time-evolving block decimation (TEBD) is more efficient than the pace-keeping DMRG if a one-dimensional Hamiltonian with the nearest-neighboring interactions is studied. The adaptive time-dependent DMRG provides an efficient scheme to implement TEBD with the skill of DMRG. On the other hand, the folded transfer matrix method handles the transfer matrix like TMRG by folding the transfer matrix so that the entanglement entropy along the positive and negative time evolution directions can partially cancel each other. This folding scheme significantly extends the time scale within which a time-dependent problem can be reliably investigated.
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- Density Matrix and Tensor Network Renormalization , pp. 293 - 309Publisher: Cambridge University PressPrint publication year: 2023