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博碩士論文 etd-0808115-163844 詳細資訊
Title page for etd-0808115-163844
論文名稱
Title
在Haldane-Zeeman模型上拓樸相變之理論研究
Topological phase transition in Haldane-Zeeman model
系所名稱
Department
畢業學年期
Year, semester
語文別
Language
學位類別
Degree
頁數
Number of pages
43
研究生
Author
指導教授
Advisor
召集委員
Convenor
口試委員
Advisory Committee
口試日期
Date of Exam
2015-09-02
繳交日期
Date of Submission
2015-09-09
關鍵字
Keywords
量子霍爾效應、網路模型、轉移矩陣方法、李亞普諾夫指數、拓樸相變、緊束縛近似模型、量子反常霍爾效應
Chalker-Coddington network model, Topological phase transition, Tight-binding model, transfer matrix method, Lyapunov exponent, Qauntum Hall effect, Quantum anomalous Hall effect
統計
Statistics
本論文已被瀏覽 5656 次,被下載 363
The thesis/dissertation has been browsed 5656 times, has been downloaded 363 times.
中文摘要
在本研究中我們討論在二維六角晶格中Haldane-Zeeman模型所發生的拓樸相變與相變在臨界現象中所屬的普適類(universality class)。藉由調變Haldane-Zeeman模型的Haldane項(局部破壞時間反演對稱但總磁通量為零)強度或Zeeman效應強度,系統會發生拓樸相轉變;同樣地,調變粒子密度(費米能階)或外加磁場大小也能產生類似效果。我們計算系統基態之陳數(Chern number) 來有效區分各種量子態不同的拓樸性質,故而求得此系統之相轉變圖。同時我們發現,在系統半填滿狀態取低能量極限時,此模型可簡化為狄拉克方程式,恰如石墨烯的例子但此模型具有質量項。為研究具雜質的Haldane-Zeeman模型拓樸相變之臨界現象,我們採用了Chalker-Coddington網路模型(C-C network model)。我們以逆推的方式,利用低能量極限狄拉克方程式的相似性,將Haldane-Zeeman系統映射到其所對應的C-C network model。具體上,我們以Haldane-Zeeman系統中陳數為2的拓樸絕緣態為出發點(取Zeeman強度為零,具量子反常霍爾效應),考慮因雜質所造成的隨機散射與電子自旋在穿隧時可能發生的翻轉,接著運用網路模型調整穿隧系數計算出其中經finite-size scaling 後波函數的局域長度(localization length)來求得此拓樸相轉變的臨界特性。由我們所得到的臨界指數 ν≈2.7發現,此結果與過去研究整數量子霍爾效應平台間的相轉變相近。故可推知此量子反常霍爾效應與整數霍爾效應應屬於相同的普適類(universality class),亦即為unitary class。
Abstract
In this thesis, we study various topological phase transitions in the 2D Haldane-Zeeman model on the honeycomb lattice and unvail the universality class of the critical behavior for the transition we focus on. By tuning the strength of the Haldane term (locally breaking time reversal symmetry but keeping total magnetic flux zero) or Zeeman term in the model, a topological phase transition could occur. In addition, by tuning density of the particle (Fermi level) or adding external magnetic field, a phase transition could also occur. We classify different kinds of quantum ground states, either topologically trivial or nontrivial, by calculating the Chern number for each state in the system. As a result, we obtain the phase diagram of the model as a function of chosen parameters. Furthermore, in the low-energy limit we find that the system at half-filling can be approximately described by the Dirac equation, just like that in graphene while with a mass term here. By using this description, we map our model onto the so-called Chalker-Coddington network model in order to study the criticality of a certain topological phase transition in the model in the presence of disorder. Explicitly, we start by considering a phase with Chern number 2 (in the absence of the Zeeman term, and it exhibits quantum anomalous Hall effect) and take into account random scatterings due to disorder and possible spin-flip effect in the tunneling processes. We then employ the network model to obtain the localization length of the wave function after finite-size scaling as a function of the transmission parameter around the critical point. Eventually, the numerically extracted critical exponent ν≈2.7. This result is similar to what people find in the integer quantum Hall plateau transition. Therefore, we conclude that the critical behavior of this quantum anomalous Hall effect and integer quantum Hall effect should belong to the same universality class, i.e., a unitary class.
目次 Table of Contents
論文審定書+i
摘要+ii
Abstract+iii
目錄+v
圖次+vi
1.導論+1
2.理論分析與相圖+5
2.1 Haldane-Zeeman模型+5
2.2 陳數(Chern number)與相圖+7
2.3低能量近似描述: 狄拉克方程式+10
3.網路模型與量子霍爾相轉變+12
3.1 Chalker-Coddington網路模型的背景+12
3.2 映射Haldane-Zeeman系統至網路模型+17
3.3轉移矩陣方法(transfer matrix method)+22
4. 數值分析+27
4.1特定李亞普諾夫指數(Characteristic Lyapunov exponent)+27
4.2 Finite-size scaling +30
5.結論+34
6.參考文獻+35
參考文獻 References
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