Boundary Condition Independence of Non-Hermitian Hamiltonian Dynamics

TL;DR

This paper demonstrates that in the thermodynamic limit, the Green's function and density matrix evolution in non-Hermitian systems are independent of boundary conditions, despite the non-Hermitian skin effect.

Contribution

It provides a general proof that boundary condition independence holds for Green's functions in non-Hermitian systems of arbitrary dimension with finite hopping range.

Findings

Green's function is boundary condition independent in the thermodynamic limit.

Density matrix evolution in open quantum systems is boundary condition independent.

Explicit illustration in the non-Hermitian Su-Schrieffer-Heeger model.

Abstract

Non-Hermitian skin effect, namely that the eigenvalues and eigenstates of a non-Hermitian tight-binding Hamiltonian have significant differences under open or periodic boundary conditions, is a remarkable phenomenon of non-Hermitian systems. Inspired by the presence of the non-Hermitian skin effect, we study the evolution of wave-packets in non-Hermitian systems, which can be determined using the single-particle Green's function. Surprisingly, we find that in the thermodynamical limit, the Green's function does not depend on boundary conditions, despite the presence of skin effect. We proffer a general proof for this statement in arbitrary dimension with finite hopping range, with an explicit illustration in the non-Hermitian Su-Schrieffer-Heeger model. We also explore its applications in non-interacting open quantum systems described by the master equation, where we demonstrate that the evolution of the density matrix is independent of the boundary condition.