Sensitivity of non-Hermitian systems

TL;DR

This paper introduces a method to analyze the sensitivity of eigenvalues in non-Hermitian systems, revealing conditions for boundary condition insensitivity and predicting the skin effect in multi-dimensional models.

Contribution

It provides an analytical approach to determine eigenvalue sensitivity in non-Hermitian systems with arbitrary boundaries, extending to two-dimensional models.

Findings

Eigenvalues can be insensitive to boundary conditions under specific parameter regimes.

Conditions for the presence or absence of the skin effect are analytically derived.

The method extends to two-dimensional systems, predicting skin effect behavior.

Abstract

Understanding the extreme sensitivity of the eigenvalues of non-Hermitian Hamiltonians to the boundary conditions is of great importance when analyzing non-Hermitian systems, as it appears generically and is intimately connected to the skin effect and the breakdown of the conventional bulk boundary correspondence. Here we describe a method to find the eigenvalues of one-dimensional one-band models with arbitrary boundary conditions. We use this method on several systems to find analytical expressions for the eigenvalues, which give us conditions on the parameter values in the system for when we can expect the spectrum to be insensitive to a change in boundary conditions. By stacking one-dimensional chains, we use the derived results to find corresponding conditions for insensitivity for some two-dimensional systems with periodic boundary conditions in one direction. This would be hard by using other methods to detect skin effect, such as the winding of the determinant of the Bloch Hamiltonian. Finally, we use these results to make predictions about the (dis)appearance of the skin effect in purely two-dimensional systems with open boundary conditions in both directions.