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

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 the arbitrary size of the unit cell and generic modulations of on-site potentials and hoppings. The solutions are parametrized by determinants that can be straightforwardly calculated from recursion relations. We show that this representation allows for an elegant analytic continuation to complex quasimomentum consistent with previous treatments for continuum models. Two important analytical results are obtained based on the explicit knowledge of all eigenstates. (1) An explicit proof of the surface charge theorem is presented including a unique relationship between the boundary charge QB(α) of a single band α and the bulk polarization in terms of the Zak-Berry phase. In particular, the Zak-Berry phase is determined within a special gauge of the Bloch states such that no unknown integer is left. This establishes a precise form of a bulk-boundary correspondence relating the 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=ΔQB-ρ{-1,0} introduced in an accompanying paper [M. Pletyukhov, Phys. Rev. B 101, 161106 (2020)10.1103/PhysRevB.101.161106]. Here, ΔQB is the change of the boundary charge QB for a given chemical potential in the insulating regime when the lattice is shifted by one site towards the boundary, and ρ 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. In accordance with previous results for continuum systems, we also establish the localization of the boundary charge and determine the explicit form of the density given by an exponential decay and a pre-exponential function following a power law with generic exponent -1/2 at large distances.