Abstract
在晶格電子只能做最近鄰與次近鄰的跳躍下, 我們討論外加一均勻磁場對系統能量與霍爾電導的影響. 當每單位晶格的磁通量等於一有理數 p/q(p,q為互質的兩個整數), 則一初始能帶將會分裂成 q 個次能帶. 而電子填滿一個次能帶時, 由庫柏公式 (kubo formula) 來計算電子對霍爾電導的貢獻, 其結果為一整數. 這和 Laughlin 的理論相符合. 另一方面, 當次近鄰的跳躍強度在某一特定值時, 兩個相鄰的次能帶可能重疊在一起.而當跳躍強度再偏離此一特定值時, 電子對霍爾電導的貢獻將會改變e^2/h 的整數值 (0,1,2,...). 我們發現霍爾電導的改變量, 與在簡併附近次能帶的結構有密切的關係; 其改變可由一整數, 即貝利指數 (Berryindex) 來描述.We examine the energy and the Hall conductance of the tight-binding electronson a square lattice with the nearest-neighbor(NN) hopping as well as thenext-nearest-neighbor (NNN) hopping.In a uniform magnetic field perpendicularto the square lattice,the original band will split into q subbands when the flux perunit cell equal to p/q (p,q are relatively prime). From Kuboformula,we can determine the Hall conductance of a filledsubband to be an integer,which is consistent with the Laughlinargument. Two subbands may touch atcertain values of NNN hoppingstrength and the Hall conducatnce changes by aninteger multipleof e^2/h after the degeneracy is lifted. We give anexplanationof the change by the Berry index which is closelyrelated to the local structurearound the degenerate point wheresubbands touch.