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

Yongxu Fu, Yi Zhang

Published 2023-07-31Version 1

Surpassing the individual buildings of the non-Hermitian skin effect (NHSE) and the scale-free localization (SFL) observed lately, we systematically exploit the exponential decay behavior of bulk eigenstates via the transfer matrix approach in non-Hermitian systems. We concentrate on the one-dimensional (1D) finite-size non-Hermitian systems with 2*2 transfer matrices in the participation of boundary impurity. We analytically unveil that the unidirectional pure scale-free (UPSF) effect emerges with the singular transfer matrices, while the hybrid skin-scale-free (HSSF) effect emerges with the nonsingular transfer matrices even though impose open boundary condition (OBC). The UPSF effect exceeds the scope of the SFL in previous works, while the HSSF effect is a charming interplay between the finite-size NHSE and SFL. Our results reveal that the NHSE under OBC prevails in the blend with the SFL as the system tends to the thermodynamic limit. Our approach paves an avenue to rigorously explore the finite-size display of the NHSE and SFL in both Hermitian and non-Hermitian systems with generic boundary conditions.