Density Matrix and Tensor Network Renormalization

An introduction to tensor network renormalization, a powerful tool for the study of quantum many-body problems and complex systems.

Tao Xiang (Author)

9781009398701, Cambridge University Press

Hardback, published 31 July 2023

470 pages
28 x 19 x 2.9 cm, 0.883 kg

Renormalization group theory of tensor network states provides a powerful tool for studying quantum many-body problems and a new paradigm for understanding entangled structures of complex systems. In recent decades the theory has rapidly evolved into a universal framework and language employed by researchers in fields ranging from condensed matter theory to machine learning. This book presents a pedagogical and comprehensive introduction to this field for the first time. After an introductory survey on the major advances in tensor network algorithms and their applications, it introduces step-by-step the tensor network representations of quantum states and the tensor-network renormalization group methods developed over the past three decades. Basic statistical and condensed matter physics models are used to demonstrate how the tensor network renormalization works. An accessible primer for scientists and engineers, this book would also be ideal as a reference text for a graduate course in this area.

Preface
Abbreviations
Unit used
Notations and graphical representations
1. Introduction
2. Basic algebra of tensors
3. Tensor network representation of classical statistical methods
4. Tensor-network ansatz of wave functions
5. Criterion of truncation: symmetric systems
6. Real-space DMRG
7. Implementation of symmetries
8. DMRG with non-local basis states
9. Matrix Product States
10. Infinite Matrix Product States
11. Determination of MPS
12. Continuous Matrix Product States
13. Classical Transfer Matrix Renormalization
14. Criterion of truncation: non-symmetric systems
15. Renormalization of quantum transfer matrices
16. MPS solution of QTMRG
17. Dynamical Correlation Functions
18. Time-dependent methods
19. Tangent-space operations
20. Tangent-space approaches
21. Tree Tensor Network States
22. Two-dimensional tensor network states
23. Coarse graining tensor renormalization
Appendix A
References
Index.

Subject Areas: Condensed matter physics [liquid state & solid state physics PHFC]