Building blocks of topological band theory for photonic crystals

TL;DR

This paper develops a framework for classifying three-dimensional photonic bands using stable real-space invariants, addressing challenges posed by polarization singularities and providing a basis for identifying topological photonic phases.

Contribution

It introduces the use of stable real-space invariants to classify topological photonic bands, unifying band representation analysis with polarization singularity considerations.

Findings

Classified trivial photonic bands using SRSIs

Identified topological bands as those not constructible from trivial building blocks

Analyzed the influence of polarization singularities on Wilson loop behavior

Abstract

We derive a framework for classifying topological bands in three-dimensional photonic band structures, where the zero frequency polarization singularity implied by Maxwell's equations complicates the direct application of existing symmetry-based approaches. Building on recent advances in the regularization of photonic bands, we use the recently introduced concept of stable real-space invariants (SRSIs) to show how photonic band structures can be unambiguously characterized in terms of equivalence classes of band representations. We classify topologically trivial photonic bands using SRSIs, treating them as the fundamental building blocks of 3D photonic band structures. This means that if certain bands cannot be constructed from these building blocks, they are necessarily topological. Furthermore, we distinguish between photonic and electronic band structures by analyzing which SRSI values are allowed in systems with and without polarization singularity. We also explore the impact of the polarization singularity on the behavior of Wilson loops, providing new insights into the topological classification of 3D photonic systems.