摘 要
本論文研究q階耦合XY模型，並應用於探討液晶系統的相變化問題。Bruinsma與Aeppli對液晶Sm-A →Hex-B相變提出Hexatic與Herringbone兩種有序的耦合模型，我們一般化成q階耦合XY模型，以蒙地卡羅模擬方法對這個模型進行研究。對於二維q階耦合XY模型在三角晶格上模擬的結果發現，系統會出現q - state clock相變及XY相變。當交互作用係數在某些範圍時，會產生XY相變與q-state clock相變重合而變成一個新型態的相變。這個耦合的全新相變與一些液晶相變的臨界現象吻合，例如三階耦合XY模型衍生three-state Potts相變，當three-state Potts相變與XY相變結合可以模擬二維無襯液晶薄膜Sm-A →Hex-B相變，系統中耦合鍵向有序與鯡魚骨有序的相變，比熱臨界指數α≒0.3，模擬的結果能解釋二維無襯液晶薄膜Sm-A →Hex-B相變。另一方面關於耦合順磁有序與反磁有序的系統，我們模擬三階反磁耦合XY模型，我們發現系統會衍生反磁three-state Potts相變，當反磁three-state Potts相變與XY相變重合後是一階相變，這個相變與實驗量測出不具鲱魚骨相變的Sm-A →Hex-B相變有許多的相同點。

Abstract

In this study, we employed the Monte Carlo simulation method to investigate the q-state coupled XY model based on the Landau free energy of couple hexatic order and herringbone order proposed by Bruinsma and Aeppli. On two-dimensional triangular lattices simulation results reveal that the q-state coupled XY model will generate a q-state clock phase transition and a XY transition. The unique generated q-state clock phase transition and XY transition will couple in some coupling parameter domain. The novel coupled transitions behavior agree with the phase transition of some kinds of liquid crystal. For example, the three-state Potts phase transition generated by a 3-state coupled XY model and the Sm-A →Hex-B transition of free – standing two layers liquid crystal are matched. Their heat capacity anomaly is similar and the heat capacity exponent is both closed to α≒0.3.
We also investigated the system of coupled ferromagnetic order and antiferromagnetic order. Adapted the positive coupling parameter on the Hamiltonian of 3-state coupled XY model, the simulation results reveal that the system generate an antiferromagnetic three-state Potts transition. In some parameter domain the antiferromagnetic three-state Potts transition and XY transition are coupled, and become a novel transition. The novel transition may explain the Sm-A →Hex-B transition of some kinds of liquid crystal which lack herringbone order.

目錄
第一章 緒論 -------------------------------------------------------- 1
第二章 液晶SmA → HexB相變
2-1 : 液晶簡介 ------------------------------------------------- 5
2-2 : 液晶SmA → HexB相變之比熱異象 --------------- 9
2-2-1 : 二維熔化理論 ------------------------------------- 9
2-2-2 : 觀察六角相的實驗 ------------------------------ 12
2-2-3 : SmA → HexB 相變之比熱異象---------------- 13
第三章 XY模型
3-1 : KT相變 ---------------------------------------------------19
3-2 : 漩渦與漩渦對 ------------------------------------------20
3-3 : 二維XY模型 ------------------------------------------21
3-4 : 鍵向有序與XY模型 ----------------------------------27
第四章 耦合XY模型
4-1 : BA模型
4-1-1 : BA模型的建立 ----------------------------------- 31
4-1-2 : BA模型的分析 ----------------------------------- 33
4-1-3 : BA模型的修正 ----------------------------------- 36
4-2 : 耦合XY模型之總能函數 --------------------------- 39
第五章 蒙地卡羅模擬方法
5-1 : 蒙地卡羅理論 ------------------------------------------ 41
5–2 : 注意事項
5-2-1 : 新自旋的產生 ----------------------------------- 46
5-2-2 : 平均能量 ----------------------------------------- 46
5-2-3 : 熱力學物理量的計算 -------------------------- 47
5-3 : 臨界行為
5-3-1 : 臨界指數 ----------------------------------------- 49
5-3-2 : 普適律與尺度 ----------------------------------- 50
5-3-3 : 有限尺度分析 ----------------------------------- 51
5-3-4 : 相變階數的判斷 -------------------------------- 53
第六章 結果與討論
6-1 : Three-state Potts-like相變
6-1-1 : 緒論 ----------------------------------------------- 56
6-1-2 : 結果與分析 -------------------------------------- 57
6-1-3 : 結論 ----------------------------------------------- 62
6-2 : Six-state clock 相變
6-2-1 : 緒論 ----------------------------------------------- 75
6-2-2 : 結果與分析 -------------------------------------- 76
6-2-3 : 結論 ----------------------------------------------- 80
6-3 : q-state clock 相變
6-3-1 : 緒論 ----------------------------------------------- 97
6-3-2 : 結果與分析 -------------------------------------- 98
6-3-3 : 結論 ----------------------------------------------- 99
6-4 : Antiferromagnetic three-state Potts相變
6-4-1 : 緒論 ---------------------------------------------- 114
6-4-2 : 結果與分析 ------------------------------------- 115
6-4-3 : 結論 ---------------------------------------------- 119
6-5: 向列型液晶中非線性光學的研究
6-5-1 : 緒論---------------------------------------------- 138
6-5-2 : 實驗方法---------------------------------------- 138
6-5-3 : 結果與分析--------------------------------------- 140
第七章 總結 ----------------------------------------------------- 142
參考文獻 --------------------------------------------------------- 144

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