自從Andre Geim和Konstantin Novoselov成功作出單層石墨烯上後，科學家們對於這種特殊的二維結構便充滿了各種興趣，其異常量子霍爾效應更為其多種有趣特性之一，因為其霍爾平台並非出現在整數而只出現在奇數上。但在對於晶格之中的霍爾電導率仍有著許多問題存在於計算特殊晶格與減少計算時間之上，這也成為了我研究的動機。若能以簡單的方式找到晶格之中的霍爾電導率，我們也能推廣至不同晶格中探討其量子霍爾效應特性。而在初貝安弘發表了數篇利用拓樸的方式去探討量子霍爾效應的論文後，我決定嘗試以他的方式去計算各種晶格的霍爾電導率。而在籠目晶格與星型晶格之中，我發現了類似石墨烯的類迪拉克的霍爾電導率，其中星型晶格顯示出更為複雜的類電子－電洞性質區域結構。希望此星型晶格的量子霍爾特性能在未來被實驗所證實。

Since the single layer graphene are observed by Andre Geim and Konstantin Novoselov, the quantum phenomena emerging from graphene as attracted many physicists’s attentions in the two dimensional lattice structure. One of the important quantum phenomena is the unconventional quantum Hall effect which the Hall plateau (Chern number pattern) does not show successive integers but only odd integers. However, there are some other problems to calculate the quantum Hall conductivity, such as the determination of the Chern number pattern from a specific lattice structure and the reduction of the computational time. Motivated by these issues, I use Yasuhiro Hatsugai’s topological approaches to investigate the quantum Hall effect of other lattices. I use Hatsugai’s method to calculate the quantum Hall conductivity in two different lattices: Kagomé and star lattices. I find that the Dirac-like regions of quantum Hall conductivity also exists in both Kagomé and star lattices, which is similar to graphene. Unlike the graphene, the star lattice shows complicated Chern number structure in the electron-hole like regions. Hopefully, the Chern number pattern of the star lattice predicted by us can be determined by experiments in the near future.

[中文摘要+i]
[英文摘要+ii]
[圖次+iv]
[第一章 序論]
[1.1 簡述量子霍爾效應+1]
[1.2 量子霍爾效應之發展與回顧+3]
[第二章 理論與方法]
[2.1 磁場下正方晶格的緊束縛模型+4]
[2.2 磁場下蜂巢晶格的緊束縛模型+7]
[2.3 霍爾電導率與拓樸陳數的計算+10]
[第三章 結果與討論]
[3.1 簡述籠目晶格+14]
[3.2 籠目晶格之霍爾電導率+16]
[3.3 討論+22]
[3.4 星型晶格之霍爾電導率+23]
[參考文獻+33]

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