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研究生: 廖育捷
Liao, Yu-Jie
論文名稱: 高階張量重整化群在二維橫場易辛模型的應用
Applications of higher order tensor renormalization group in 2D quantum Ising model
指導教授: 江府峻
Jiang, Fu-Jiun
口試委員: 黃靜瑜
Huang, Ching-Yu
陸健榮
Lu, Chien-Rong
江府峻
Jiang, Fu-Jiun
口試日期: 2024/01/17
學位類別: 碩士
Master
系所名稱: 物理學系
Department of Physics
論文出版年: 2024
畢業學年度: 112
語文別: 英文
論文頁數: 30
中文關鍵詞: 量子易辛模型相變張量網路方法
英文關鍵詞: quantum Ising model, phase transition, tensor network method
DOI URL: http://doi.org/10.6345/NTNU202400429
論文種類: 學術論文
相關次數: 點閱:106下載:16
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  • 本研究深入探討了二維橫場易辛模型(2D Transverse Field Ising Model, 2D TFIM)在正方晶格上的相變行為,透過張量網路方法來求得其磁化率的高階動差。易辛模型作為凝態物理中的一個基本模型,已被廣泛用於研究鐵磁性和反鐵磁性等物質的相變。本次研究透過二維的無限時間演化區塊分解(2D Infinite Time-Evolving Block Decimation, 2D ITEBD )[1]來得到二維橫場易辛模型的基態,再透過高階張量重整化群(Higher-Order Tensor Renormalization Group ,HOTRG) [2]來計算磁化率的高階矩並且透過磁化率的高階矩來計算Binder ratio來決定系統的相變點。在未來可以計算其他熱力學性質如相關長度和比熱,更進一步完整這份研究。

    This study delves into the phase transition behavior of the two-dimensional Transverse Field Ising Model (2D TFIM) on a square lattice. The higher-order moments of the magnetization are obtained through the tensor network method. The Ising model, serving as a fundamental model in condensed matter physics, has been widely used to investigate the phase transitions of materials such as ferromagnets and antiferromagnets. In this research, the ground state of the 2D TFIM is obtained through two-dimensional infinite-time evolution block decimation(2D ITEBD) [1]. Subsequently, the higher-order moments of the magnetization are calculated using the Higher-Order Tensor Renormalization Group (HOTRG). [2] The Binder ratio is determined through these higher-order moments of the magnetization to identify the system's phase transition point. In the future, other thermodynamic properties such as the correlation length and the specific heat can be computed to further enhance the comprehensiveness of this study.

    Chapter 1 Introduction1 Chapter 2 Model:2D Transverse Filed Ising model on the Square Lattice 3 2.1 Hamiltonian 3 2.2 Phase Transition 3 Chapter 3 Tensor Network Method 5 3.1 Tensor 5 3.2 Singular Value decomposition (SVD) 6 3.3 Matrix Product State (MPS) &Projected-Entangled-Pair-State (PEPS) 7 3.4 Infinite Time-Evolving Block Decimation (ITEBD) 12 3.5 Higher-Order Tensor Renormalization Group (HOTRG) 14 3.6 Calculation of higher-order moments 17 Chapter 4 Result 20 Chapter 5 Conclusions 28 Reference 29

    [1] J. Jordan, R. Oru´s, G. Vidal,1 F. Verstraete, and J. I. Cirac(2008). Classical Simulation of Infinite-Size Quantum Lattice Systems in Two Spatial Dimensions. PRL 101, 250602

    [2] Z. Y. Xie, J. Chen, M. P. Qin, J. W. Zhu, L. P. Yang, and T. Xiang(2012). Coarse-graining renormalization by higher-order singular value decomposition. PHYSICAL REVIEW B 86, 045139

    [3] Lars Onsager(1944). Crystal Statistics. I. A Two-Dimensional Model with an Order-Disorder Transition. Phys. Rev. 65, 117

    [4] Rom´an Or´us(2014). A Practical Introduction to Tensor Networks: Matrix Product States and Projected Entangled Pair States. Annals of Physics 349 (2014) 117-158

    [5] Chase Roberts, Ashley Milsted, Martin Ganahl, Adam Zalcman, Bruce Fontaine, Yijian Zou, Jack Hidary, Guifre Vidal, and Stefan Leichenauer(2019) .TensorNetwork: A Library for Physics and Machine Learning. arXiv:1905.01330

    [6] Ashley Milsted, Martin Ganahl, Stefan Leichenauer, Jack Hidary, Guifre Vidal(2019). TensorNetwork on TensorFlow: A Spin Chain Application Using Tree Tensor Networks. arXiv:1905.01331

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    [8] Satoshi Morita , Naoki Kawashima(2019). Calculation of higher-order moments by higher-order tensor renormalization group. Computer Physics Communications 236 (2019) 65–71

    [9] Charles R. Harris, K. Jarrod Millman, Stéfan J. van der Walt, Ralf Gommers, Pauli Virtanen, David Cournapeau, Eric Wieser, Julian Taylor, Sebastian Berg, Nathaniel J. Smith, Robert Kern, Matti Picus, Stephan Hoyer, Marten H. van Kerkwijk, Matthew Brett, Allan Haldane, Jaime Fernández del Río, Mark Wiebe, Pearu Peterson, Pierre Gérard-Marchant, Kevin Sheppard, Tyler Reddy, Warren Weckesser, Hameer Abbasi, Christoph Gohlke & Travis E. Oliphant(2020). Array programming with NumPy. Nature volume 585, pages357–362(2020)

    [10] H. W. J. Blote and Y. Deng, Phys. Rev. E 66, 066110(2002).

    [11] Tensor Networks 2021 – Lecture notes from Ludwig-Maximilians-Universität München FAKULTÄT FÜR PHYSIK
    https://www2.physik.uni-muenchen.de/lehre/vorlesungen/sose_21/tensor_networks_21/skript/index.html

    [12] R R dos Santos, L Sneddon and R B Stinchcombe (1981) The 2D transverse Ising model at T=0: a finite-size rescaling transformation approach. J. Phys. A: Math. Gen. 14 (1981) 3329-3339.

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