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研究生:郭子傑
研究生(外文):Tzu-Chieh Kuo
論文名稱:二維易辛模型考慮次近鄰交互作用其相變化及淬火動力學
論文名稱(外文):The Quench Dynamics and the Critical Behavior of the J1-J2 Ising Model
指導教授:高英哲高英哲引用關係
指導教授(外文):Ying-Jer Kao
口試委員:陳柏中林瑜琤
口試委員(外文):Po-Chung Chen
口試日期:2014-07-25
學位類別:碩士
校院名稱:國立臺灣大學
系所名稱:物理研究所
學門:自然科學學門
學類:物理學類
論文種類:學術論文
論文出版年:2014
畢業學年度:102
語文別:英文
論文頁數:47
中文關鍵詞:古典蒙地卡羅演算法有限尺度效應圖形處理單元二維方格易辛模型考慮次近鄰之交互作用淬火動力學Kibble-Zurek機制
外文關鍵詞:Classical Monte Carlofinite-size scalingGPUJ1&;#8722;J2 Ising modelquench dynamicsKibble-Zurek mechanism
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於圖形處理單元(GPU) 環境中使用平行演算法及蒙地卡羅演算法模擬了二維方格易辛模型並考慮次近鄰之交互作用,其中最近鄰(J1) 與次近鄰(J2) 之交互作用皆為反鐵磁性且互為競爭關係,本篇展現了如何計算出臨界指數與交互作用比例(J2/J1) 之關係,及利用Metropolis演算法模擬非平衡淬火至臨界溫度並計算出動力學指數。

We perform the Monte Carlo simulations of the J1 &;#8722;J2 (frustrated) Ising model on a square lattice with competing coupling J1 > 0 (nearest-neighbor, anti-ferromagnetic) and J2 > 0 (next-nearest neighbor, also anti-ferromagnetic) using the graphic processing unit (GPU). In this thesis, we present the critical exponents evolution as one tunes J2/J1 and the extraction of the dynamical exponent using non-equilibrium quenching with Metropolis algorithm to the critical point.

口試委員會審定書 i
致謝iii
中文摘要v
Abstract vii
Contents ix
List of Figures xi
List of Tables xiii
1 Introduction 1
2 Theory 3
2.1 Ising model on a square lattice . . . . . . . . . . . . . . . . . . . . . . . 3
2.2 J1 &;#8722; J2 Ising model on a square lattice . . . . . . . . . . . . . . . . . . . 3
2.3 Monte Carlo simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.3.1 Classical Monte Carlo method . . . . . . . . . . . . . . . . . . . 5
2.3.2 Metropolis algorithm . . . . . . . . . . . . . . . . . . . . . . . . 7
2.3.3 Parallel tempering Monte Carlo method . . . . . . . . . . . . . . 7
2.3.4 Calculated observables . . . . . . . . . . . . . . . . . . . . . . . 8
2.4 Finite-size scaling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.5 Non-equilibrium quenching . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.5.1 Kibble-Zurek Mechanism . . . . . . . . . . . . . . . . . . . . . 12
2.5.2 Complete finite-size scaling form with linear quench and nonlinear
quench . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.6 Statistics and data analysis . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.7 GPU . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.7.1 GPU architecture . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.7.2 Algorithm of J1 &;#8722; J2 Ising model on GPU . . . . . . . . . . . . 25
3 Results 27
3.1 Critical temperatures and critical exponents . . . . . . . . . . . . . . . . 27
3.2 Extraction of the dynamic exponent . . . . . . . . . . . . . . . . . . . . 37
4 Summary and Discussion 41
Bibliography 43

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