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

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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.