Equivalence of Electric, Magnetic and Electromagnetic Chern Numbers for Topological Photonic Crystals

TL;DR

This paper proves that electric, magnetic, and electromagnetic Chern numbers are equivalent in topological photonic crystals, confirming that any of these definitions can be used in bulk-edge correspondence for such media.

Contribution

It provides a rigorous mathematical proof that the three different Chern number definitions in topological photonics are necessarily equal in relevant media.

Findings

All three Chern numbers agree in Haldane's media.

Any one of the three Chern numbers can be used in bulk-edge correspondence.

The proof uses vector bundle theoretic arguments.

Abstract

Haldane predicted an analog of the Integer Quantum Hall Effect in gyrotropic photonic crystals, where the net number of electromagnetic edge modes moving left-to-right is given by a bulk Chern number. His prediction --- topological effects are bona fide wave and not quantum phenomena --- has been confirmed in a number of experiments. However, theoretical physicists have tacitly used three different definitions for the bulk Chern numbers that enter the bulk-edge correspondence --- on the basis of electromagnetic Bloch functions, electric Bloch functions and magnetic Bloch functions. We use vector bundle theoretic arguments to prove that in media such as those considered by Haldane these three potentially different Chern numbers necessarily agree with one another, and consequently, any one of them can be used in Haldane's photonic bulk-edge correspondence.