Symmetry Classification of Topological Photonic Crystals

TL;DR

This paper classifies topological photonic crystals based on material symmetries, revealing only gyrotropic media are topologically non-trivial and clarifying the analogy between photonic and electronic quantum Hall effects.

Contribution

It provides a comprehensive symmetry-based classification of topological photonic media, identifying only gyrotropic media as topologically non-trivial in 2D and 3D.

Findings

Only gyrotropic media are topologically non-trivial in 2D and 3D.

No undiscovered topological effects analogous to Quantum Spin Hall Effect.

Photonic Quantum Hall Effect is topologically equivalent to electronic Quantum Hall Effect.

Abstract

In a seminal paper Haldane conjectured that topological phenomena are not particular to quantum systems, and indeed experiments realized unidirectional, backscattering-free edge modes with electromagnetic waves. This raises two immediate questions: (1) Are there other topological effects in electromagnetic media? And (2) is Haldane's Quantum Hall Effect for light really analogous to the Quantum Hall Effect? We conclusively answer both of these questions by classifying topological photonic crystals according to material (as opposed to crystallographic) symmetries. It turns out there are four topologically distinct types of media, of which only one, gyrotropic media, is topologically non-trivial in $d = 2 , 3$. That means there are no as-of-yet undiscovered topological effects; in particular, there is no analog of the Quantum Spin Hall Effect in classical electromagnetism. Moreover, at least qualitatively, Haldane's Quantum Hall Effect for light is analogous to the Quantum Hall Effect from condensed matter physics as both systems as in the same topological class, class A. Our ideas are directly applicable to other classical waves.