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

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 $x_{\phi }\in [0,L]$ is given by a perfectly linear function of $x_{\phi }$ 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.