Proof of bulk-edge correspondence for band topology by Toeplitz algebra

TL;DR

This paper rigorously proves the bulk-edge correspondence in topological insulators across various dimensions using Toeplitz algebra, linking bulk topological invariants to edge mode indices with accessible methods for physicists.

Contribution

It introduces a formula connecting Toeplitz algebra to the bulk Fourier series and applies differential calculus to establish the bulk-edge correspondence without K-theory.

Findings

Bulk topological number equals edge-mode index in general dD topological insulators.

Winding number of bulk equals Fredholm index of edge Hamiltonian.

Chern number matches spectral flow of 2D edge Hamiltonian.

Abstract

We rigorously yet concisely prove the bulk-edge correspondence for general $d$-dimensional ($d$D) topological insulators in complex Altland-Zirnbauer classes, which states that the bulk topological number equals to the edge-mode index. Specifically, an essential formula is discovered that links the quantity expressed by Toeplitz algebra, i.e., hopping terms on the lattice with an edge, to the Fourier series on the bulk Brillouin zone. We then apply it to chiral models and utilize exterior differential calculations, instead of the sophisticated \emph{K}-theory, to show that the winding number of bulk system equals to the Fredholm index of 1D edge Hamiltonian, or to the sum of edge winding numbers for higher odd dimensions. Moreover, this result is inherited to the even-dimensional Chern insulators as each of them can be mapped to an odd-dimensional chiral model. It is revealed that the Chern number of bulk system is identical to the spectral flow of 2D edge Hamiltonian, or to the negative sum of edge Chern numbers for higher even dimensions. Our methods and conclusions are friendly to physicists and could be easily extended to other physical scenarios.