Accurate self-energy algorithm for quasi-1D systems

TL;DR

This paper introduces a robust numerical algorithm for calculating surface Green's functions and self-energies in quasi-one-dimensional systems, improving accuracy and efficiency, especially for large systems, while addressing issues with surface states.

Contribution

The authors develop a new self-energy calculation method that avoids ill-conditioned matrix inversions and enhances computational speed and accuracy for quasi-1D systems.

Findings

High accuracy for most energies

Efficient handling of large Hamiltonian matrices

Effective detection and management of surface states

Abstract

We present a complete prescription for the numerical calculation of surface Green's functions and self-energies of semi-infinite quasi-onedimensional systems. Our work extends the results of Sanvito et al. [1] generating a robust algorithm to be used in conjunction with ab initio electronic structure methods. We perform a detailed error analysis of the scheme and find that the highest accuracy is found if no inversion of the usually ill conditioned hopping matrix is involved. Even in this case however a transformation of the hopping matrix that decreases its condition number is needed in order to limit the size of the imaginary part of the wave-vectors. This is done in two different ways, either by applying a singular value decomposition and setting a lowest bound for the smallest singular value, or by adding a random matrix of small amplitude. By using the first scheme the size of the Hamiltonian matrix is reduced, making the computation considerably faster for large systems. For most energies the method gives high accuracy, however in the presence of surface states the error diverges due to the singularity in the self-energy. A surface state is found at a particular energy if the set of solution eigenvectors of the infinite system is linearly dependent. This is then used as a criterion to detect surface states, and the error is limited by adding a small imaginary part to the energy. [1] S. Sanvito, C. J. Lambert, J. H. Jefferson, and A. M. Bratkovsky, Phys. Rev. B 59, 11936 (1999).