On the Role of Symmetries in the Theory of Photonic Crystals

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 consider 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.

arXiv comment34 pages, 2 figures

References in corpus (20)

• Classification of topological insulators and superconductors in three spatial dimensions
• Photonic Floquet Topological Insulators
• Novel Symmetry Classes in Mesoscopic Normal-Superconducting Hybrid Structures
• Photonic Analogue of Two-dimensional Topological Insulators and Helical One-Way Edge Transport in Bi-Anisotropic Metamaterials
• Periodic table for topological insulators and superconductors
• Reflection-Free One-Way Edge Modes in a Gyromagnetic Photonic Crystal
• Analogs of quantum Hall effect edge states in photonic crystals
• Weyl points and line nodes in gapless gyroid photonic crystals
• Topological creation and destruction of edge states in photonic graphene
• Photonic analog of graphene model and its extension -- Dirac cone, symmetry, and edge states --
• Effective dynamics for Bloch electrons: Peierls substitution and beyond
• Geometrical Aspects in Optical Wavepacket Dynamics
• Quantum Hall effect and the topological number in graphene
• Classification of "Real" Bloch-bundles: Topological quantum systems of type AI
• Topological nature of polarization and charge pumping in ferroelectrics
• Applications of Magnetic PsiDO Techniques to Space-adiabatic Perturbation Theory
• Exponentially Localized Wannier Functions in Periodic Zero Flux Magnetic Fields
• Effective Light Dynamics in Perturbed Photonic Crystals
• Topological Polarization in Graphene-like Systems
• Topological aspects in the photonic crystal analog of single-particle transport in quantum Hall systems

Cited by in corpus (11)

• Topological Photonics
• Nonlinear one-way edge-mode interactions for frequency mixing in topological photonic crystals
• Simultaneous Multi-frequency Topological Edge Modes between One-dimensional Photonic Crystals
• Elliptic operators with honeycomb symmetry: Dirac points, Edge States and Applications to Photonic Graphene
• Soft self-assembly of Weyl materials for light and sound
• Non-Commutative Chern Numbers for Generic Aperiodic Discrete Systems
• Topological edge states for disordered bosonic systems
• The K-Theoretic Bulk-Boundary Principle for Dynamically Patterned Resonators
• The Schrödinger Formalism of Electromagnetism and Other Classical Waves --- How to Make Quantum-Wave Analogies Rigorous
• Effective Light Dynamics in Perturbed Photonic Crystals
• Linear and nonlinear electromagnetic waves in modulated honeycomb media