$\mathbb{Z}_{2}$ Topological Index for Continuous Photonic Materials

TL;DR

This paper introduces a $Z_2$ topological index for continuous photonic materials, establishing a photonic analogue of electronic topological insulators and linking bulk properties to edge modes.

Contribution

It develops a gauge-invariant $Z_2$ index for photonic band structures, demonstrating topological inequivalence of different plasma types and proposing a bulk-edge correspondence.

Findings

The $Z_2$ index is robust under continuous parameter variations.

Electric and magnetic plasmas are topologically inequivalent.

Numerical examples illustrate the application of the topological index.

Abstract

Electronic topological insulators are one of the breakthroughs of the 21st century condensed matter physics. So far, the search for a light counterpart of an electronic topological insulator has remained elusive. This is due to the fundamentally different natures of light and matter and the different spins of photons and electrons. Here, it is shown that the theory of electronic topological insulators has a genuine analogue in the context of light wave propagation in time-reversal invariant continuous materials. We introduce a Gauge invariant $Z_2$ index that depends on the global properties of the photonic band structure and is robust to any continuous weak variation of the material parameters that preserves the time-reversal invariance. A nontrivial $Z_2$ index has two possible causes: (i) the lack of smoothness of the pseudo-Hamiltonian in the ${\bf{k}} \to \infty$ limit, and (ii) the entanglement between positive and negative frequency eigenmode branches. In particular, it is proven that electric-type plasmas and magnetic-type plasmas are topologically inequivalent for a fixed wave polarization. We propose a bulk-edge correspondence that links the number of edge modes with the topological invariants of two continuous bulk materials, and present detailed numerical examples that illustrate the application of the theory.