Universal properties of boundary and interface charges in multichannel one-dimensional models without symmetry constraints

TL;DR

This paper extends the understanding of boundary and interface charges from single-channel to multichannel one-dimensional models, revealing topological invariants expressed as winding numbers of bulk Green's functions, and proves the nearsightedness principle for such systems.

Contribution

It generalizes the universal boundary charge properties to multichannel models using Green's functions and introduces topological indices for impurity-induced charge accumulation.

Findings

Boundary charge change relates to a winding number of Green's functions.

Topological invariant ranges from -N_c to 0, with N_c as the number of channels.

Proof of the nearsightedness principle for multichannel lattice models.

Abstract

The boundary charge that accumulates at the edge of a one-dimensional single-channel insulator is known to possess the universal property, that its change under a lattice shift towards the edge by one site is given by the sum of the average bulk electronic density and a topologically invariant contribution, restricted to the values $0$ and $-1$ [Phys. Rev. B 101, 165304 (2020)]. This quantized contribution is associated with particle-hole duality, ensures charge conservation and fixes the mod(1) ambiguity appearing in the Modern Theory of Polarization. In the present work we generalize the above-mentioned single-channel results to the multichannel case by employing the technique of boundary Green's functions. We show that the topological invariant associated with the change in boundary charge under a lattice shift in multichannel models can be expressed as a winding number of a certain combination of components of bulk Green's functions as function of the complex frequency, as it encircles the section of the energy axis that corresponds to the occupied part of the spectrum. We observe that this winding number is restricted to values ranging from $-N_c$ to $0$, where $N_c$ is the number of channels (orbitals) per site. Furthermore, we consider translationally invariant one-dimensional multichannel models with an impurity and introduce topological indices which correspond to the quantized charge that accumulates around said impurity. These invariants are again given in terms of winding numbers of combinations of components of bulk Green's functions. Through this construction we provide a rigorous mathematical proof of the so called nearsightedness principle formulated by W. Kohn [Phys. Rev. Lett. 76, 3168 (1996)] for noninteracting multichannel lattice models.