We investigate the anomalous behavior of localization length of a non-interacting one-dimensional Anderson model at zero temperature. We report numerical calculations of the Thouless expression of localization length, based on the Kernel polynomial method (KPM), which has an computational complexity, where N is the system size. The KPM results show excellent agreement with perturbative results in a large system size limit, confirming the validity of the Thouless formula. In the perturbative regime, we show that the KPM approximation of the Thouless expression produces the correct localization length at the band center in the thermodynamic limit. The Thouless expression relates localization length in terms of density of states in a one-dimensional disordered system. By calculating the KPM estimates of the density of states, we find a cusp-like behavior around the band center in the perturbative regime. This cusp-like singularity can not be obtained by approximate analytical calculations within the second-order approximations, reflects the band-center anomaly.

我们研究零温度下非相互作用的一维安德森模型的定位长度的异常行为。我们基于内核多项式方法（KPM）报告了局部化长度Thouless表达式的数值计算，该方法具有计算复杂性，其中N是系统大小。KPM结果显示，在较大的系统规模限制下，其与摄动结果具有极好的一致性，从而证实了Thouless公式的有效性。在微扰状态下，我们表明，Thouless表达式的KPM近似在热力学极限的带中心产生正确的定位长度。Thouless表达式根据一维无序系统中的状态密度来关联定位长度。通过计算状态密度的KPM估计，我们在扰动状态下发现了能带中心周围的尖峰状行为。这种尖峰状的奇点不能通过二阶近似内的近似分析计算获得，反映了带中心异常。