Scheme to evaluate the Green function by recursive polynomial expansion, and its applications

本文提出了一种基于递归多项式展开的计算大哈密顿矩阵系统的绿色函数的新方法。它具有以下特点：（1）不需要对角化哈密顿量，（2）可以计算绿色函数的非对角元素和对角元素，（3）可以计算绿色函数与其它量子算符的乘积。（4）该方法适用于具有离散性质的系统，（5）在同一算法上还可以计算特征值和特征向量，（6）计算所需的CPU时间和内存大小与系统大小N成正比（即有序N方法）。将该方法与第一性原理计算相结合，在大的无序系统中，我们可以从第一原理分析几个物理性质，例如电子电导率。在实际应用中，我们将该方法应用于体心立方和非晶Fe系统，并与含时方法如粒子源方法和受迫振子方法进行了比较，证明了该方法的时间演化算符可以由用该方法得到的绿色函数的傅里叶变换导出，此外，我们证明了该方法可以推广到计算热绿色函数的虚时间解析延拓，在实际计算中，我们必须在有限阶终止展开。我们估计的数值误差，由于终止，并表明，它减少成比例的扩展阶N。

We developed a new method to evaluate the Green function for the system described by a huge Hamiltonian matrix on the basis of recursive polynomial expansion. It has the following features ; (1) it is not necessary to diagonalize the Hamiltonian, (2) we can evaluate off-diagonal elements of the Green function as well as diagonal elements, (3) it is possible to evaluate products of the Green functions and other quantum operators. Then, several physical properties can be evaluated, (4) the method is applicable to the system with discrete nature, (5) we can also evaluate eigenvalues and eigenvectors on the same algorithm, (6) CPU time and memory size needed in the calculation is proportional to the system size N (i.e. ordered N method).Combining this method with the first principles calculations, we can analyze several physical properties such as electronic conductivities from first principles in the large disordered systems. In practice, we applied this method to bcc and amorphous Fe system, and shows the usefulness of the method.We also compared the method with time-dependent method such as particle source method and forced oscillator method, and showed that the time evolution operator can be derived from the Fourier transformation of the Green function obtained by our method.Furthermore, we showed that the method can be extended to the evaluation of thermal Green function by analytical continuation to the imaginary time.In the practical calculations, we have to terminate the expansion at finite order. We estimate the numerical error due to the termination, and showed that it decreases as proportional to the expansion order N.