BIRS Workshop Lecture Videos
Symmetry Classification of Topological Photonic Crystals Lein, Max
In 2005 Haldane conjectured that topological phenomena were not quantum but wave effects. He proposed [RH08] an electromagnetic analog of the Quantum Hall Effect, something that was confirmed in a number of spectacular experiments [Wan+08; Rec+13] a few years later. These and other, more recent works have naturally raised two questions: (1) How similar is the Quantum Hall Effect for light to the one from solid state physics? And (2) are there other, as-of-yet unknown topological effects in electromagnetic media? The crucial ingredient are symmetries, and when designing topological electromagnetic media, there are two axes to explore: One can choose the materials from which to build the photonic crystal (material symmetries) and then decide how to periodically arrange these materials (crys- tallographic symmetries). For material symmetries we answer both of these questions conclusively by first reformulating Maxwell’s equations in Schrödinger form [DL17], and then adapting the Cartan-Altland-Zirnbauer classification scheme for topological insulators [DL14; DL16]. With regards to question (1), gyrotropic media are in the same symmetry class (class A) as solids exhibiting the Quantum Hall Effect. This is a first step to proving photonic bulk-edge correspon- dences that would make Haldane’s conjecture precise: In a two-dimensional topological photonic crystal the Chern number quantifies the net number of edge modes traveling from left to right. Ques- tion (2) has a negative answer, in dimension d ≤ 3 there are no as-of-yet undiscovered topological effects due to material symmetries. In particular, despite some claims to the contrary, there is no electromagnetic analog of the Quantum Spin Hall Effect as that requires the presence of an odd time-reversal symmetry (a realization of class AII). Acknowledgements M. L. thanks JSPS for support of his research with a WAKATE B grant. G. D. research is supported by the grant Iniciación en Investigación 2015 - No 11150143 funded by FONDECYT.
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