Proof of the bulk-edge correspondence through a link between topological photonics and fluctuation-electrodynamics

TL;DR

This paper rigorously proves the bulk-edge correspondence in topological photonics by linking topological invariants to fluctuation-induced light momentum, providing a solid mathematical foundation for a key principle in the field.

Contribution

It establishes a rigorous mathematical demonstration of the bulk-edge correspondence in dispersive photonic materials using fluctuation-electrodynamics.

Findings

The spectral density of fluctuation-induced light angular momentum relates to the Chern number.

The net number of unidirectional edge states determines fluctuation-induced light momentum.

A fundamental link between topological invariants and physical electromagnetic fluctuations is demonstrated.

Abstract

The bulk-edge correspondence links the Chern-topological numbers with the net number of unidirectional states supported at an interface of the relevant materials. This fundamental principle is perhaps the most consequential result of topological photonics, as it determines the precise physical manifestations of nontrivial topological features. Even though the bulk-edge correspondence has been extensively discussed and used in the literature, it seems that in the general photonic case with dispersive materials it has no solid mathematical foundation and is essentially a conjecture. Here, I present a rigorous demonstration of this fundamental principle by showing that the thermal fluctuation-induced light-angular momentum spectral density in a closed cavity can be expressed in terms of the photonic gap Chern number, as well as in terms of the net number of unidirectional edge states. In particular, I highlight the rather fundamental connections between topological numbers in Chern-type photonic insulators and the fluctuation-induced light-momentum