Surface charge theorem and topological constraints for edge states: An analytical study of one-dimensional nearest-neighbor tight-binding models

TL;DR

This paper analytically solves for all edge and scattering states in one-dimensional tight-binding models, establishing a bulk-boundary correspondence and a topological constraint linking boundary charge changes to edge state phase dependence.

Contribution

It provides an exact analytical solution for edge states in 1D tight-binding models with arbitrary unit cell size and modulations, and proves a surface charge theorem and a topological constraint.

Findings

Explicit proof of the surface charge theorem relating boundary charge to bulk polarization.

Derivation of a topological constraint on edge state energies phase dependence.

Establishment of a bulk-boundary correspondence for single-band boundary charge.

Abstract

For a wide class of noninteracting tight-binding models in one dimension we present an analytical solution for all scattering and edge states on a half-infinite system. Without assuming any symmetry constraints we consider models with nearest-neighbor hoppings and one orbital per site but arbitrary size of the unit cell and generic modulations of on-site potentials and hoppings. The solutions are parametrized by determinants which can be calculated from recursion relations. This representation allows for an elegant analytic continuation to complex quasimomentum consistent with previous treatments for continuum models. Two important analytical results are obtained: (1) An explicit proof of the surface charge theorem is presented including a unique relationship between the boundary charge $Q_B^{(\alpha)}$ of a single band $\alpha$ and the bulk polarization in terms of the Zak-Berry phase with no unknown integer left. This establishes a precise form of a bulk-boundary correspondence relating the {\it boundary charge of a single band} to bulk properties. (2) We derive a topological constraint for the phase-dependence of the edge state energies, where the phase variable describes a continuous shift of the lattice towards the boundary. The topological constraint is shown to be equivalent to the quantization of a topological index $I=\Delta Q_B-\bar{\rho}\in \{-1,0\}$ introduced in an accompanying letter [arXiv:1911.06890]. Here $\Delta Q_B$ is the change of the boundary charge $Q_B$ for a given chemical potential in the insulating regime when the lattice is shifted by one site towards the boundary, and $\bar{\rho}$ is the average charge per site (both in units of the elementary charge $e=1$). This establishes an interesting link between universal properties of the boundary charge and edge state physics discussed within the field of topological insulators.