Asymmetric transfer matrix analysis of Lyapunov exponents in one-dimensional non-reciprocal quasicrystals

TL;DR

This paper introduces an asymmetric transfer matrix method to analyze localization and mobility edges in non-reciprocal quasicrystals, revealing distinct Lyapunov exponents on opposite sides of the localization center.

Contribution

The paper presents a novel asymmetric transfer matrix analysis for studying localization properties in non-reciprocal quasicrystals, expanding tools for non-Hermitian disordered systems.

Findings

Application to non-reciprocal Aubry-Andre9 model

Application to non-reciprocal off-diagonal Aubry-Andre9 model

Application to non-reciprocal mosaic quasicrystals

Abstract

The Lyapunov exponent, serving as an indicator of the localized state, is commonly utilized to identify localization transitions in disordered systems. In non-Hermitian quasicrystals, the non-Hermitian effect induced by non-reciprocal hopping can lead to the manifestation of two distinct Lyapunov exponents on opposite sides of the localization center. Building on this observation, we here introduce a comprehensive approach for examining the localization characteristics and mobility edges of non-reciprocal quasicrystals, referred to as asymmetric transfer matrix analysis. We demonstrate the application of this method to three specific scenarios: the non-reciprocal Aubry-Andr\'e model, the non-reciprocal off-diagonal Aubry-Andr\'e model, and the non-reciprocal mosaic quasicrystals. This work may contribute valuable insights to the investigation of non-Hermitian quasicrystal and disordered systems.