# An elementary proof and detailed investigation of the bulk-boundary correspondence in the generic two-band model of Chern insulators

## TL;DR

This paper provides an elementary, rigorous proof of the bulk-boundary correspondence in two-band Chern insulators, revealing new non-topological features and simplifying the analysis of edge states from bulk properties.

## Contribution

It introduces a simple, elementary framework for analyzing the bulk-boundary correspondence in two-band Chern insulators, applicable to arbitrary long-range hopping and spin-orbit couplings.

## Key findings

- Elementary proof of bulk-boundary correspondence robust to deformations
- Edge band loci and spin polarizations inferred from bulk Hamiltonian
- Reveals previously unnoticed non-topological features of Chern insulators

## Abstract

With the inclusion of arbitrary long-range hopping and (pseudo)spin-orbit coupling amplitudes, we formulate a generic model that can describe any two-dimensional two-band bulk insulators, thus providing a simple framework to investigate arbitrary adiabatic deformations upon the systems of any arbitrary Chern numbers. Without appealing to advanced techniques beyond the standard methods of solving linear difference equations and applying Cauchy's integral formula, we obtain a mathematically elementary yet rigorous proof of the bulk-boundary correspondence on a strip, which is robust against any adiabatic deformations upon the bulk Hamiltonian and any uniform edge perturbation along the edges. The elementary approach not only is more transparent about the underlying physics but also reveals various intriguing nontopological features of Chern insulators that have remained unnoticed or unclear so far. Particularly, if a certain condition is satisfied (as in most renowned models), the loci of edge bands in the energy spectrum and their (pseudo)spin polarizations can be largely inferred from the bulk Hamiltonian alone without invoking any numerical computation for the energy spectrum of a strip.