凝聚态场论常用公式（13）：Haper方程的有理磁通解

2023-04-01 17:10 作者:打电动的阿伟嘻嘻嘻 0人读过 | 我要投稿

考虑有理磁通 $%5Cphi%3D%5Cfrac%7Bp%7D%7Bq%7D%2C%5C%20k_ya%5Cequiv%5Cnu.$ 利用线性代数中解迭代关系式的技巧

$%5Cbegin%7Bpmatrix%7D%0Ag(m%2B1)%5C%5C%0Ag(m)%0A%5Cend%7Bpmatrix%7D%3D%5Cbegin%7Bpmatrix%7D%0AA_%7B11%7D(m)%26-1%5C%5C%0A1%260%0A%5Cend%7Bpmatrix%7D%5Cbegin%7Bpmatrix%7D%0Ag(m)%5C%5C%0Ag(m-1)%0A%5Cend%7Bpmatrix%7D%2C%5C%20A_%7B11%7D(m)%3D%5Cvarepsilon-2%5Ccos%5B2%5Cpi%20m%5Cfrac%7Bp%7D%7Bq%7D-%5Cnu%5D.$

其中迭代矩阵为 A(m).

利用有理磁通的周期性有 $A(m%2Bq)%3DA(m)%2C%5C%20%0A%5Cbegin%7Bpmatrix%7D%0Ag(m%2Bq)%5C%5C%0Ag(m%2Bq-1)%0A%5Cend%7Bpmatrix%7D%3DA(m%2Bq)%5Ccdots%20A(m)%0A%5Cbegin%7Bpmatrix%7D%0Ag(m)%5C%5C%0Ag(m-1)%0A%5Cend%7Bpmatrix%7D%2C$

$Q(%5Cvarepsilon%2C%5Cnu)%5Cequiv%20A(m%2Bq)%5Ccdots%20A(m).$

由本征方程 $g(m%2Bq)%3De%5E%7B-ik_xqa%7Dg(m)%2C%5C%20Q(%5Cvarepsilon%2C%5Cnu)%0A%5Cbegin%7Bpmatrix%7D%0Ag(m%2B1)%5C%5C%0Ag(m)%0A%5Cend%7Bpmatrix%7D%3D%0Ae%5E%7B-ik_xqa%7D%5Cboldsymbol%7B1%7D%5Cbegin%7Bpmatrix%7D%0Ag(m%2B1)%5C%5C%0Ag(m)%0A%5Cend%7Bpmatrix%7D.$

考虑行列式 $%5Cdet%5BQ(%5Cvarepsilon%2C%5Cnu)-e%5E%7B-ik_xqa%7D%5Cboldsymbol%7B1%7D%5D%3D0%2C%5C%20%5Cdet%7BQ(%5Cvarepsilon%2C%5Cnu)%7D%3D1.$

$%7B%5Crm%20Tr%7D%5BQ(%5Cvarepsilon%2C%5Cnu)%5D%3D2%5Ccos%5Bk_xqa%5D%5Cin%5B-2%2C2%5D.$

由此我们可以举例说明Haper方程有理磁通解的能带分裂.

以 $%5Cphi%3D%5Cfrac%7B1%7D%7B3%7D%2C%5Cfrac%7B1%7D%7B5%7D%2C%5Cnu%3D0%2C%5C%20%E4%B8%BA%E4%BE%8B%EF%BC%9A$

$%7B%5Crm%20Tr%7D%5BQ_%7B%5Cfrac%7B1%7D%7B3%7D%7D(%5Cvarepsilon%2C0)%5D%3D%5Cvarepsilon%5E3-6%5Cvarepsilon-2%5Cin%5B-2%2C2%5D%2C%5C%20%5Cvarepsilon%5Cin%5B-2.4495%2C-2%5D%2C%5B-0.7321%2C0%5D%2C%5B2.4495%2C2.7321%5D.$

$%7B%5Crm%20Tr%7D%5BQ_%7B%5Cfrac%7B1%7D%7B5%7D%7D(%5Cvarepsilon%2C0)%5D%3D%5Cvarepsilon%5E5-%5Cfrac%7B1%7D%7B2%7D(3%2B%5Csqrt%7B5%7D)%5Cvarepsilon%5E4%2B(-9%2B%5Csqrt%7B5%7D)%5Cvarepsilon%5E3%2B(7%2B3%5Csqrt%7B5%7D)%5Cvarepsilon%5E2-5(-4%2B%5Csqrt%7B5%7D)%5Cvarepsilon%2B%5Cfrac%7B1%7D%7B2%7D(9-5%5Csqrt%7B5%7D)%5Cin%5B-2%2C2%5D%2C$

$%5Cvarepsilon%5Cin%5B-2.2787%2C-2.2361%5D%2C%5B-0.7619%2C-0.4142%5D%2C%5B-0.1318%2C0.2600%5D%2C%5B2.2361%2C2.4142%5D%2C%5B3.0286%2C3.1199%5D.$

可以看出两种磁通分别把母带分裂为3，5条带.

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凝聚态场论常用公式（13）：Haper方程的有理磁通解

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