Advances in Theoretical and Mathematical Physics

Volume 23 (2019)

Number 6

Symmetry classification of topological photonic crystals

Pages: 1467 – 1531



Giuseppe De Nittis (Facultad de Mateméticas & Instituto de Física, Pontificia Universidad Católica de Chile, Santiago, Chile)

Max Lein (Advanced Institute of Materials Research, Tohoku University, Sendai, Japan)


In a seminal paper Haldane conjectured that topological phenomena are not particular to quantum systems, and indeed, later 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 due to our choice of materials; 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 are in the same topological class, class A. Our ideas are directly applicable to other classical waves.

G. D.’s research is supported by FONDECYT regular No. 1190204. M. L. acknowledges support from JSPS through a WAKATE B grant (grant number 16K17761) and the WPI-AIMR through a Fusion grant.

Published 20 March 2020