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研究生：	黃靜瑜 Ching-Yu Huang
論文名稱：	自旋系統的量子糾纏量子相變與拓樸序 Numerical study on quantum entanglement, quantum phase transition, and topological order in spin system
指導教授：	林豐利 Lin, Feng-Li
學位類別：	博士 Doctor
系所名稱：	物理學系 Department of Physics
論文出版年：	2012
畢業學年度：	100
語文別：	英文
論文頁數：	98
中文關鍵詞：	量子糾纏、量子相變、拓樸序、自旋系統、MPS/ TPS
英文關鍵詞：	quantum entanglement, quantum phase transition, topological order, spin system, MPS/ TPS
論文種類：	學術論文
相關次數：	點閱：172 下載：20
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這是一篇探討一維與二維量子自旋系統中基態波函數的性質.我們使用Matrix product state (MPS) 和Tensor product state (TPS) 描述基態波函數,並使用反覆投射(Iterative Projection)方法決定波函數的參數. 藉由計算物理量及基態的量子糾纏(entanglement)性質,發現量子糾纏可以用來描述量子相變的發生,並指出相變點. 接著我們討論臨界點附近entanglement的尺度性質(scaling behaviors). 一維系統中,許多物理量都遵守普遍(universal)性質.此外,我們也討論拓樸序(topological order)與奇異係數(singular values)簡併的關係.

This thesis concerns the ground state property and the quantum entanglement in the one- and two-dimensional quantum spin systems. We use the Iterative Projection method to find the ground states numerically in the form of the tensor product states, and then evaluate their expectation values and their entanglement measures such as the geometric entanglement by the method of tensor renormalization group. We find that these entanglement measures can characterize the quantum phase transitions by their derivative discontinuity right at the critical points. We also study the scaling behaviors of the entanglement measures near the quantum critical point by the ideas of quantum-state renormalization group transformations. We find some universal features for one-dimensional spin system. However, we fails to capture the area-law for two-dimensional spin system. We then study the connection between topological order and the degeneracy of the singular value spectrum by explicitly solving the two-dimensional dimerized quantum Heisenberg model in the form of tensor product state ansatz. By evaluating the topological entanglement entropy, we identify a new phase with topological order in this model, in which the singular value spectrum is doubly degenerate. Degeneracy of the singular value spectrum is robust against various types of perturbations, in accordance with our expectation for topological order.

Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv
Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v
Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi
List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix
Chapter
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1 Quantum phase transitions . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 The basic concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 The properties of quantum phase transition . . . . . . . . . . . . . 2
1.3 Topological order and spin liquid . . . . . . . . . . . . . . . . . . . 3
2 Entanglement measure and quantum phase transition . . . . . . . . . . . 7
2.1 Concurrence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.2 Entanglement entropy . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.3 Block entanglement entropy . . . . . . . . . . . . . . . . . . . . . . 10
2.4 Entanglement spectrum . . . . . . . . . . . . . . . . . . . . . . . . 12
2.5 Topological entanglement entropy . . . . . . . . . . . . . . . . . . . 13
2.6 Geometric entanglement . . . . . . . . . . . . . . . . . . . . . . . . 15
2.7 Scaling of the geometric entanglement in one-dimensional system . 16
3 Matrix product state . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
3.1 The basic concepts of the matrix product state . . . . . . . . . . . 17
3.2 Valence-bond picture . . . . . . . . . . . . . . . . . . . . . . . . . . 17
3.3 Vidal decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3.4 The canonical form . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
3.5 Simple examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3.6 Time-evolving block-decimation . . . . . . . . . . . . . . . . . . . . 23
3.7 Expectation value . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.8 Quantum state renormalization group in the matrix product state . 26
4 Tensor product state . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
4.1 The basic concepts of the tensor product state . . . . . . . . . . . . 28
4.2 The imaginary time evolution with the tensor product state . . . . 30
4.3 Tensor reorganization group . . . . . . . . . . . . . . . . . . . . . . 31
4.4 Quantum state renormalization group in the tensor product state . 34
5 Universal Quantum Resource via TPS . . . . . . . . . . . . . . . . . . . . 36
5.1 Teleportation quantum computation model . . . . . . . . . . . . . 37
5.2 Measurement based quantum computation . . . . . . . . . . . . . . 38
5.3 The cluster state . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
5.4 Quantum state as computation resource . . . . . . . . . . . . . . . 40
Multipartite Entanglement Measures and Quantum Criticality from MPS and
TPS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
2 Entanglement measure in one-dimensional spin system . . . . . . . . . . . 43
2.1 Spin 1/2 chain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
2.2 Spin 1 chain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
3 Entanglement measure in two-dimensional spin systems . . . . . . . . . . . 47
3.1 The two-dimensional transverse Ising model . . . . . . . . . . . . . 48
3.2 The two-dimensional XYX model . . . . . . . . . . . . . . . . . . . 49
3.3 The two-dimensional XXZ model . . . . . . . . . . . . . . . . . . . 51
4 Comment on scaling behavior of entanglement measure . . . . . . . . . . . 52
4.1 One-dimensional scaling via matrix product states . . . . . . . . . . 53
4.2 Two-dimensional scaling via tensor product states? . . . . . . . . . 55
Topological Order and Degenerate Singular Value Spectrum in two-Dimensional
Dimerized Quantum Heisenberg Model . . . . . . . . . . . . . . . . . . . . . . . 58
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
2 The Model and order parameters . . . . . . . . . . . . . . . . . . . . . . . 59
2.1 Order parameters for Neel and dimer phases . . . . . . . . . . . . . 61
2.2 Characterizing topological phase by degeneracy of singular value
spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
3 Numerical results for the phase diagram of J-J' model . . . . . . . . . . . . 62
4 Topological Entanglement Entropy . . . . . . . . . . . . . . . . . . . . . . 67
4.1 J-J' model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
4.2 Toric code like state . . . . . . . . . . . . . . . . . . . . . . . . . . 69
5 Degeneracy of Singular Value Spectrum and Topological Order . . . . . . . 70
5.1 Some examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
5.2 Topological order of J-J' model by singular value spectrum and its
robustness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
5.3 Remove short range entanglement . . . . . . . . . . . . . . . . . . . 75
The projection symmetry group analysis of J-J' model . . . . . . . . . . . . . . 78
1 General conditions on PSG on the square lattice . . . . . . . . . . . . . . . 78
2 Classication of Z2 PSG on the J-J' model . . . . . . . . . . . . . . . . . . 80
3 A summary of Z2 PSGs on the J-J' model . . . . . . . . . . . . . . . . . . 82
Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
A Schmidt decomposition and Singular value decomposition . . . . . . . . . . . . 88
B The properties of entanglement measure . . . . . . . . . . . . . . . . . . . . . . 90
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
Curriculum Vitae . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

                                

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