On the Role of Symmetries in the Theory of Photonic Crystals

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 consider the Maxwell operator in the first-order formalism where the dynamical Maxwell equations can be rewritten as a Schr\"odinger 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 Schr\"odinger 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.