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研究生： 張博勝
Chang, Po-Sheng
論文名稱： Numerical methods for 1D strongly correlated system
指導教授： 陳柏中
口試委員: 陳柏中
王道維
高英哲
學位類別： 碩士
Master
系所名稱： 理學院 - 物理學系
Department of Physics
論文出版年： 2014
畢業學年度： 102
語文別： 英文
論文頁數： 57
中文關鍵詞： 密度矩陣重整化群 、強關聯系統 、纏結熵
外文關鍵詞： density matrix renormalization group, strongly correlated system, entanglement entropy
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    • 多體系統的研究對於了解材料的性質很重要，但經常有希爾伯特空間(Hilbert space)隨著系統的大小呈指數的成長的困難，特別是那些展現奇特的行為的強關聯系統。然而，在過去的二十年間已經有許多解決多體問題的演算法被發展出來，且這些演算法並不會受限於系統大小的成長。由Steven R. White 所開發的密度矩陣重整化群(Density Matrix Renormalization Group)是最重要的一個，而這個方法也是我們在這個論文裡主要討論的演算法。除此之外，我們在這個論文裡還要討論有關一維多體系統的纏結(entanglement)性質，這在最近數年間吸引了許多的注意。

    The study of many-body systems is important for the properties of material and often has the difficulty in the exponential growth of Hilbert space with system size, especially for those strongly correlated systems which exhibit exotic behavior. However, in the past two decades have developed some useful algorithms of tackling many-body problems, which is not restricted by the growth of system size. The Density Matrix Renormalization Group (DMRG) invented by Steven R. White is the significant one and it is the major algorithm we discuss in this thesis. In addition, this thesis will show the entanglement properties of many-body systems in one dimension which attracts a lot of attention in recent years.

    1. Introduction 1 1.1. The simulation of strongly correlated systems . . . . . . . . . . . . . . . 1 1.2. Entanglement in quantum systems . . . . . . . . . . . . . . . . . . . . . 2 1.3. Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2. The entanglement measure 4 2.1. Entanglement of pure state . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.2. entanglement entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 3. The numerical methods 8 3.1. Matrix-product representation . . . . . . . . . . . . . . . . . . . . . . . . 8 3.1.1. Matrix-product diagram . . . . . . . . . . . . . . . . . . . . . . . 8 3.1.2. Matrix-product state . . . . . . . . . . . . . . . . . . . . . . . . . 10 3.1.3. Matrix-product operator . . . . . . . . . . . . . . . . . . . . . . . 16 3.1.4. The operations of MPS and MPO . . . . . . . . . . . . . . . . . . 19 3.2. Density Matrix Renormalization Group . . . . . . . . . . . . . . . . . . . 20 3.2.1. Infinite-size algorithm (iDMRG) . . . . . . . . . . . . . . . . . . . 22 3.2.2. Finite-size algorithm . . . . . . . . . . . . . . . . . . . . . . . . . 24 3.3. Infinite Time-Evolving Block Decimation . . . . . . . . . . . . . . . . . . 26 3.3.1. The power method in ground state search . . . . . . . . . . . . . 26 3.3.2. iTEBD algorithm in ground state search . . . . . . . . . . . . . . 27 4. Examples of numerical simulation 32 4.1. The Heisenberg spin chain . . . . . . . . . . . . . . . . . . . . . . . . . . 32 4.1.1. The simulation and its result . . . . . . . . . . . . . . . . . . . . 33 4.2. Ising model with transverse field . . . . . . . . . . . . . . . . . . . . . . . 35 4.2.1. The simulation and its result . . . . . . . . . . . . . . . . . . . . 37 5. The ferromagnetic entanglement 40 5.1. Entanglement entropy and conformal field theory . . . . . . . . . . . . . 40 5.2. Entanglemet scaling of ferromagnetic system . . . . . . . . . . . . . . . . 41 5.2.1. The XXZ spin chain . . . . . . . . . . . . . . . . . . . . . . . . . 41 5.2.2. The simulation results . . . . . . . . . . . . . . . . . . . . . . . . 43 5.2.3. Summary and discussion . . . . . . . . . . . . . . . . . . . . . . . 47 A. Useful Mathematical Tools 48 A.1. Singular Value Decomposition . . . . . . . . . . . . . . . . . . . . . . . . 48 A.2. Suzuki-Trotter expansion . . . . . . . . . . . . . . . . . . . . . . . . . . 49 B. Matrix Product Toolkit 51 B.1. Finite-size calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 B.2. Infinite-size calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 Bibliography 54

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