On the role of symmetries in the theory of photonic crystals |
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Authors: | Giuseppe De Nittis Max Lein |
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Institution: | 1. Department Mathematik, Universität Erlangen-Nürnberg, Cauerstrasse 11, D-91058 Erlangen, Germany;2. University of Toronto & Fields Institute, Bahen Centre, Department of Mathematics, 40 St. George Street, Toronto, ON M5S 2E4, Canada |
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Abstract: | We discuss the role of the symmetries in photonic crystals and classify them according to the Cartan–Altland–Zirnbauer scheme. Of particular importance are complex conjugation C and time-reversal T, but we identify also other significant symmetries. Borrowing the jargon of the classification theory of topological insulators, we show that C is a “particle–hole-type symmetry” rather than a “time-reversal symmetry” if one considers the Maxwell operator in the first-order formalism where the dynamical Maxwell equations can be rewritten as a Schrödinger equation; The symmetry which implements physical time-reversal is a “chiral-type symmetry”. We justify by an analysis of the band structure why the first-order formalism seems to be more advantageous than the second-order formalism. Moreover, based on the Schrödinger formalism, we introduce a class of effective (tight-binding) models called Maxwell–Harper operators. Some considerations about the breaking of the “particle–hole-type symmetry” in the case of gyrotropic crystals are added at the end of this paper. |
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Keywords: | Photonic crystal Gyrotropic effect Harper&ndash Maxwell operator Complex electromagnetic fields Photonic topological insulators Cartan&ndash Altland&ndash Zirnbauer classification |
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