Non-Bloch band theory for non-Hermitian continuum systems

TL;DR

This paper extends non-Bloch band theory to non-Hermitian continuum systems, emphasizing the role of boundary conditions and transfer matrices in understanding the non-Hermitian skin effect beyond lattice models.

Contribution

It introduces a framework for analyzing non-Hermitian continuum systems, incorporating boundary conditions and discretization, which was lacking in prior lattice-based theories.

Findings

Boundary conditions are essential for continuum non-Hermitian systems.

Discretization must match hopping range with boundary conditions.

Transfer matrix helps determine the generalized Brillouin zone.

Abstract

One of the most pronounced non-Hermitian phenomena is the non-Hermitian skin effect, which refers to the exponential localization of bulk eigenstates near the boundaries of non-Hermitian systems. Whereas non-Bloch band theory has been developed to describe the non-Hermitian skin effect in lattice systems, its counterpart in continuum systems still lacks a quantitative characterization. Here, we generalize the non-Bloch band theory to non-Hermitian continuum systems. In contrast to lattice systems for which the bulk Hamiltonian alone determines the non-Hermitian skin effect and energy spectrum, we find for continuum systems that the number of boundary conditions, i.e., the number of independent differential equations satisfied by wavefunctions at two boundaries, must also be included as essential information. We show that the appropriate discretization of continuum systems into lattice models requires matching the hopping range of the latter with the number of boundary conditions in the former. Furthermore, in periodic non-Hermitian continuum systems, we highlight the application of the transfer matrix in determining the generalized Brillouin zone. Our theory serves as a useful toolbox for investigating the rich non-Bloch physics in non-Hermitian continuum systems, such as photonic crystals, elastic media, and certain cold-atom systems.