Bogoliubov-de Gennes对角化与Schur分解方法的等价性

本文以Kitaev的一维量子线模型为例，分别利用传统的Bogoliubov de Gennes对角化方法和Schur分解方法求解该模型的本征能量以及本征波函数，从理论分析和数值计算两个方面对两种方法进行对比。结果表明，BdG对角化方法得到的准粒子能量是能量本征值的两倍，而Schur分解方法可以直接得到准粒子能量。数值计算表明，两者是完全一样的。另外，在确定的参数下，两种方法得到的准粒子算符对初始的费米子算符的展开系数只相差一个常数相因子。所以，最后的结论是BdG对角化跟Schur分解两种方法是等价的。

The equivalence between the Bogoliubov de Gennes diagonalization and the Schur decomposition

The equivalence between the Bogoliubov de Gennes (BdG) diagonalization method and the Schur decomposition has been verified through numerical computations to the Kitaev model of a one-dimensional quantum wire. The quasiparticle energies obtained from the BdG method are twice the eigenenergies but the Schur decomposition gives the quasiparticle energies directly. The numerical results show that quasiparticle energies from the two methods are consistent with each other perfectly. In addition the expansion coefficients of the quasiparticle operators from the two methods have only a constant phase difference.