Deflation algorithm for the Green function of quasi-1D lattices

TL;DR

This paper introduces an optimized deflation algorithm for efficiently computing the Green function in quasi-1D lattices, improving numerical stability and reducing computational complexity in quantum transport calculations.

Contribution

The authors develop a deflation-based optimization of the transfer matrix method, including Jordan-block reconstruction, to enhance efficiency and stability in Green function computations.

Findings

Significant reduction in degrees of freedom for Schur factorization.

Minimal overhead from deflation and reconstruction compared to runtime savings.

Numerical stability maintained by avoiding ill-conditioned matrix inverses.

Abstract

We derive a method to efficiently compute the Green function of on arbitrary Hamiltonians defined on semi-infinite and periodic quasi-one-dimensional lattices. Computing the Green function is the backbone of quantum transport, electronic structure or linear response computations. Our method constitutes a "deflation optimization" of a well established algorithm often used in quantum transport that is based on generalized Schur factorizations of the linearized quadratic eigenvalue equation for the transfer matrix. Our deflation optimization may greatly reduce the number of degrees of freedom that must be processed in the Schur factorization. Deflation must be supplemented by a Jordan-block reconstruction of generalized eigenvectors, also developed here in detail. The overhead of deflation plus reconstruction is minimal as compared to the typical reduction in factorization runtime. Furthermore, by avoiding inverses of ill-conditioned matrices, the algorithm remains numerically stable.