Hybrid scale-free skin effect in non-Hermitian systems: A transfer matrix approach

TL;DR

This paper introduces a transfer matrix approach to analyze the hybrid scale-free skin effect in non-Hermitian 1D systems, revealing new phenomena and unifying the understanding of skin and scale-free effects under various boundary conditions.

Contribution

It analytically uncovers the emergence of unidirectional scale-free effects and hybrid skin-scale-free effects in non-Hermitian systems using transfer matrices, extending previous knowledge.

Findings

Unidirectional SF effect occurs with singular transfer matrices.

Hybrid SFS effect appears with nonsingular transfer matrices.

Skin effect dominates in the thermodynamic limit with hybrid SFS.

Abstract

Surpassing the individual characteristics of the non-Hermitian skin effect (NHSE) and the scale-free (SF) effect observed recently, we systematically exploit the exponential decay behavior of bulk eigenstates via the transfer matrix approach in non-Hermitian systems. We concentrate on one-dimensional (1D) finite-size non-Hermitian systems with 2*2 transfer matrices in either the absence or presence of the boundary impurity. We analytically unveil that the unidirectional SF effect emerges with the singular transfer matrices, while the hybrid scale-free skin (SFS) effect appears with the nonsingular transfer matrices even when an open boundary condition (OBC) is imposed. The unidirectional SF effect exceeds the scope of the SF effect in previous works, while the hybrid SFS effect is an interesting interplay between the skin effect and the SF effect in finite-size systems. Our results reveal that the skin effect under the OBC prevails when it coexists with the SF effect as the system approaches the thermodynamic limit in the presence of the hybrid SFS effect. Our approach paves the way for rigorous and unified explorations of the skin and SF effects in both Hermitian and non-Hermitian systems with generic boundary conditions.