Exact solutions of non-Hermitian chains with asymmetric long-range hopping under specific boundary conditions

TL;DR

This paper analytically solves one-dimensional non-Hermitian chains with asymmetric long-range hopping under specific boundary conditions, revealing simple wave functions, complex spectra, and phenomena like the non-Hermitian skin effect.

Contribution

It provides exact solutions for non-Hermitian long-range hopping models under special boundary conditions, highlighting boundary-dependent spectral and wave function properties.

Findings

Wave functions are simple and independent of hopping range.

Eigenvalue spectra exhibit rich, model-dependent structures.

Non-Hermitian skin effect occurs under certain boundary conditions.

Abstract

We study one-dimensional general non-Hermitian models with asymmetric long-range hopping and explore to analytically solve the systems under some specific boundary conditions. Although the introduction of long-range hopping terms prevents us from finding analytical solutions for arbitrary boundary parameters, we identify the existence of exact solutions when the boundary parameters fulfill some constraint relations, which give the specific boundary conditions. Our analytical results show that the wave functions take simple forms and are independent of hopping range, while the eigenvalue spectra display rich model-dependent structures. Particularly, we find the existence of a special point coined as pseudo-periodic boundary condition, for which the eigenvalues are the same as the periodical system when the hopping parameters fulfill certain conditions, whereas eigenstates display non-Hermitian skin effect.