Chebyshev-polynomial expansion of the localization length of Hermitian and non-Hermitian random chains

TL;DR

This paper introduces a Chebyshev-polynomial expansion method to efficiently compute the energy-dependent localization length and density of states in Hermitian and non-Hermitian random chains, outperforming traditional methods especially for non-Hermitian models.

Contribution

The authors develop a novel Chebyshev-polynomial expansion approach for calculating localization length and density of states, providing a more efficient alternative to existing methods for both Hermitian and non-Hermitian systems.

Findings

Efficient computation of energy-dependent localization length for Hermitian models.

Derivation of a Chebyshev-polynomial expansion formula for the density of states in non-Hermitian models.

Potential for the algorithm to be the only efficient method for density of states in interacting non-Hermitian models.

Abstract

We carry Chebyshev-polynomial expansion of the inverse localization length of Hermitian and non-Hermitian random chains as function of energy. For Hermitian models, the expansion produces numerically this energy-dependent function in one run of the algorithm. This is in strong contrast to the standard transfer-matrix method, which produces the inverse localization length for a fixed energy in each run. For non-Hermitian models, as in the transfer-matrix method, our algorithm computes the inverse localization length for a fixed (complex) energy. We also find a formula of the Chebyshev-polynomial expansion of the density of states of non-Hermitian models. As explained in more detail in the Introduction, our algorithm for non-Hermitian models may be the only available efficient algorithm for finding the density of states of models with interactions.