Tensor Network Contractions Methods and Applications to Quantum Many-Body Systems
Tensor network is a fundamental mathematical tool with a huge range of applications in physics, such as condensed matter physics, statistic physics, high energy physics, and quantum information sciences. This open access book aims to explain the tensor network contraction approaches in a systematic...
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| Hlavní autoři: | , , , , , , |
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| Médium: | E-kniha Kniha |
| Jazyk: | angličtina |
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Cham
Springer Nature
2020
Springer Springer International Publishing AG |
| Vydání: | 1 |
| Edice: | Lecture Notes in Physics |
| Témata: | |
| ISBN: | 9783030344894, 3030344894, 3030344886, 9783030344887 |
| On-line přístup: | Získat plný text |
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- 3.7 A Shot Summary -- References -- 4 Tensor Network Approaches for Higher-Dimensional Quantum Lattice Models -- 4.1 Variational Approaches of Projected-Entangled Pair State -- 4.2 Imaginary-Time Evolution Methods -- 4.3 Full, Simple, and Cluster Update Schemes -- 4.4 Summary of the Tensor Network Algorithms in HigherDimensions -- References -- 5 Tensor Network Contraction and Multi-Linear Algebra -- 5.1 A Simple Example of Solving Tensor Network Contraction by Eigenvalue Decomposition -- 5.1.1 Canonicalization of Matrix Product State -- 5.1.2 Canonical Form and Globally Optimal Truncations ofMPS -- 5.1.3 Canonicalization Algorithm and Some Related Topics -- 5.2 Super-Orthogonalization and Tucker Decomposition -- 5.2.1 Super-Orthogonalization -- 5.2.2 Super-Orthogonalization Algorithm -- 5.2.3 Super-Orthogonalization and Dimension Reduction by Tucker Decomposition -- 5.3 Zero-Loop Approximation on Regular Lattices and Rank-1 Decomposition -- 5.3.1 Super-Orthogonalization Works Well for Truncating the PEPS on Regular Lattice: Some Intuitive Discussions -- 5.3.2 Rank-1 Decomposition and Algorithm -- 5.3.3 Rank-1 Decomposition, Super-Orthogonalization, and Zero-Loop Approximation -- 5.3.4 Error of Zero-Loop Approximation and Tree-Expansion Theory Based on Rank-Decomposition -- 5.4 iDMRG, iTEBD, and CTMRG Revisited by Tensor Ring Decomposition -- 5.4.1 Revisiting iDMRG, iTEBD, and CTMRG: A Unified Description with Tensor Ring Decomposition -- 5.4.2 Extracting the Information of Tensor Networks From Eigenvalue Equations: Two Examples -- References -- 6 Quantum Entanglement Simulation Inspired by Tensor Network -- 6.1 Motivation and General Ideas -- 6.2 Simulating One-Dimensional Quantum Lattice Models -- 6.3 Simulating Higher-Dimensional Quantum Systems -- 6.4 Quantum Entanglement Simulation by Tensor Network:Summary -- References -- 7 Summary -- Index
- Intro -- Preface -- Acknowledgements -- Contents -- Acronyms -- 1 Introduction -- 1.1 Numeric Renormalization Group in One Dimension -- 1.2 Tensor Network States in Two Dimensions -- 1.3 Tensor Renormalization Group and Tensor Network Algorithms -- 1.4 Organization of Lecture Notes -- References -- 2 Tensor Network: Basic Definitions and Properties -- 2.1 Scalar, Vector, Matrix, and Tensor -- 2.2 Tensor Network and Tensor Network States -- 2.2.1 A Simple Example of Two Spins and Schmidt Decomposition -- 2.2.2 Matrix Product State -- 2.2.3 Affleck-Kennedy-Lieb-Tasaki State -- 2.2.4 Tree Tensor Network State (TTNS) and Projected Entangled Pair State (PEPS) -- 2.2.5 PEPS Can Represent Non-trivial Many-Body States: Examples -- 2.2.6 Tensor Network Operators -- 2.2.7 Tensor Network for Quantum Circuits -- 2.3 Tensor Networks that Can Be Contracted Exactly -- 2.3.1 Definition of Exactly Contractible Tensor Network States -- 2.3.2 MPS Wave-Functions -- 2.3.3 Tree Tensor Network Wave-Functions -- 2.3.4 MERA Wave-Functions -- 2.3.5 Sequentially Generated PEPS Wave-Functions -- 2.3.6 Exactly Contractible Tensor Networks -- 2.4 Some Discussions -- 2.4.1 General Form of Tensor Network -- 2.4.2 Gauge Degrees of Freedom -- 2.4.3 Tensor Network and Quantum Entanglement -- References -- 3 Two-Dimensional Tensor Networks and Contraction Algorithms -- 3.1 From Physical Problems to Two-Dimensional Tensor Networks -- 3.1.1 Classical Partition Functions -- 3.1.2 Quantum Observables -- 3.1.3 Ground-State and Finite-Temperature Simulations -- 3.2 Tensor Renormalization Group -- 3.3 Corner Transfer Matrix Renormalization Group -- 3.4 Time-Evolving Block Decimation: Linearized Contraction and Boundary-State Methods -- 3.5 Transverse Contraction and Folding Trick -- 3.6 Relations to Exactly Contractible Tensor Networks and Entanglement Renormalization