Universal properties of boundary and interface charges in multichannel one-dimensional continuum models

TL;DR

This paper extends the understanding of boundary and interface charges to multichannel one-dimensional continuum models, proving their linear dependence on boundary shift and introducing a topological invariant, with analytical tools for weak potentials.

Contribution

It generalizes previous single-channel and tight-binding results to multichannel continuum systems, establishing a rigorous proof of boundary charge behavior and developing a low-energy analytical framework.

Findings

Boundary charge change is linear in boundary shift plus a topological invariant.

Introduces a boundary invariant I characterizing boundary charge.

Provides a low-energy Green's function-based theory for weak potentials.

Abstract

We generalize our recent results for the boundary and interface charges in one-dimensional single-channel continuum [Phys. Rev. B 104, 155409 (2021)] and multichannel tight-binding [Phys. Rev. B 104, 125447 (2021)] models to the realm of the multichannel continuum systems. Using the technique of boundary Green's functions, we give a rigorous proof that the change in boundary charge upon the shift of the system towards the boundary by the distance $x_{\varphi}\in[0, L]$ is given by a perfectly linear function of $x_{\varphi}$ plus an integer-valued topological invariant $I$ -- the so called boundary invariant. For systems with weak potential amplitudes, we additionally develop Green's function-based low-energy theory, allowing one to analytically access the physics of multichannel continuum systems in the low-energy approximation.