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研究生:吳柏寬
研究生(外文):Po-Kuan Wu
論文名稱:古典蒙地卡羅對三維XY模型非線性與異向性效應之研究
論文名稱(外文):Classical Monte Carlo Studies on 3D XY Models with Effects of Nonlinearity and Anisotropy
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:林瑜琤陳柏中
口試委員(外文):Yu-Cheng LinPo-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:63
中文關鍵詞:蒙地卡羅模擬XY模型圖型處理器易辛模型有限尺度標度變換異向性
外文關鍵詞:Monte CarloXY modelGPUCUDAIsing ModelFinite Size ScalingAnisotropy
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在統計與凝態物理中,相變與臨界現象是相當重要的主題。而在研究這類問題上,晶格模型扮演了非常重要的角色。基於普適性與臨界現象的理論,理論模型的臨界行為經常能對應到真實物理系統。因此,對於理論模型的研究是了解臨界現象的關鍵之一。
蒙地卡羅方法被廣泛用於了解晶格模型的相變上。利用隨機過程,可以利用在態空間中隨機取樣得到各種熱力學性質的近似值。而基於有限尺度標度變換,從有限尺度的結果可以推估臨界指數的值,也可以幫助了解相變的性質。然而,在有限大小的結果中,不相關的場會造成一些修正項,在計算臨界指數時,這會造成系統誤差。因此,必須模擬更大的晶格。在本文中,為了在模擬大晶格模型時有更高的效率,我們利用了圖形處理器(GPU) 將程序平行化處理。
在本論文中,利用圖形處理器上的蒙地卡羅模擬,我們研究了簡單XY模型,以及將交互作用項推廣為非線性的XY 模型。簡單XY模型的臨界現象與Helium-4的相變屬於相同的普適類。而在q 大於4時,Zq異向性都是危險不相關的。推廣的XY模型的情形則是不同,當自旋間的交互作用越來越接近delta 函數,模型的行為會越來越接近Potts模型,相變變為一階。在參數位於某些區域時,可以明顯觀察到異向性是相干的,甚至可能因為異向性強度的增加,相變轉變為一階相變。

In statistical and condensed matter physics, the phase transition and the critical behavior are very important topics. To study on them, the lattice models play important roles. For the theory of universality, the behaviors of models correspond to realistic physical systems. Therefore, the studies of theoretic models are keys to understand the critical behaviors.
The Monte Carlo simulation is a widely-used way to study the phase transitions of lattice models. With the stochastic process, we can sample in the space of states and get the approximations of the thermal observables. According to the finite-size scaling, from the finite-size results, the critical exponents can be extracted and the properties of the transition can be studied.
However, in finite-size cases, there are correction terms cause by the irrelevant fields, so them cause the systematic errors of exponents. To simulate the system with larger sizes more quickly, in this thesis, we parallelize the procedures of Monte Carlo simulations on GPUs.
With the GPU Monte Carlo simulations, we study the simple XY model and the generalized cases with nonlinear interactions between spins. The simple XY model is in the same universality class with the lambda-transition of Helium-4, and the Zq anisotropy is dangerously irrelevant for q>=4. In the generalized XY models, the behavior approaches the Potts models as the potential of interactions approaching the delta function. In some region of the parameters, the anisotropy is significantly relevant. The transitions may even become first-order as anisotropy increases.

致謝 ii
摘要 iii
Abstract iv
1 Introduction 1
1.1 Phase transitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Critical behavior and scaling law . . . . . . . . . . . . . . . . . . . . . . 4
1.2.1 Critical exponent . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.2.2 Universality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.2.3 Scaling laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.3 Ising Model, XY model and Potts model . . . . . . . . . . . . . . . . . . 8
2 Classical Monte Carlo Method with Graphics Processing Unit 12
2.1 Classical Monte Carlo . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.1.1 Importance sampling and Markov chain Monte Carlo . . . . . . . 13
2.1.2 Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.1.3 Parallel tempering . . . . . . . . . . . . . . . . . . . . . . . . . 17
v
2.2 GPU architecture and CUDA framework . . . . . . . . . . . . . . . . . . 18
2.3 Implementation of classical Monte Carlo simulation on GPU . . . . . . . 21
2.4 Multispin coding on GPU . . . . . . . . . . . . . . . . . . . . . . . . . . 24
3 Data Analysis and Finite-Size Scaling 27
3.1 Data analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
3.2 Finite-size scaling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
4 Three-dimensional XY model with Zq Anisotropy 33
4.1 3D XY model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
4.2 3D XY Model with Zq anisotropy and emergent U(1) symmetry . . . . . 39
5 Nonlinear Three-dimensional XY Model with Zq Anisotropy 45
5.1 Nonlinear 3D XY model . . . . . . . . . . . . . . . . . . . . . . . . . . 46
5.2 Nonlinear 3D XY model with Zq anisotropy . . . . . . . . . . . . . . . . 51
5.2.1 From continuous to first-order . . . . . . . . . . . . . . . . . . . 51
5.2.2 From irrelevance to relevance . . . . . . . . . . . . . . . . . . . 53
6 Summary 59
Bibliography 60

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