Quadrupole Topological Photonic Crystals

TL;DR

This paper demonstrates the realization of quadrupole topological phases in photonic crystals, identifying quantized invariants through symmetry analysis and revealing boundary states and corner phenomena, advancing higher-order topological photonics.

Contribution

It introduces a method to realize quadrupole topological phases in Maxwell's equations for photonic crystals using crystalline symmetries, without threaded flux.

Findings

Quantized boundary polarizations and corner states confirmed

Three independent methods verify the quadrupole moment quantization

Boundary phenomena relate to fractional corner charges and filling anomalies

Abstract

Quadrupole topological phases, exhibiting protected boundary states that are themselves topological insulators of lower dimensions, have recently been of great interest. Extensions of these ideas from current tight binding models to continuum theories for realistic materials require the identification of quantized invariants describing the bulk quadrupole order. Here we identify the analog of quadrupole order in Maxwell's equations for a photonic crystal (PhC) and identify quadrupole topological photonic crystals formed through a band inversion process. Unlike prior studies relying on threaded flux, our quadrupole moment is quantized purely by crystalline symmetries, which we confirm using three independent methods: analysis of symmetry eigenvalues, numerical calculations of the nested Wannier bands, and the expectation value of the quadrupole operator. Furthermore, through the bulk-edge correspondence of Wannier bands, we reveal the boundary manifestations of nontrivial quadrupole phases as quantized polarizations at edges and bound states at corners. Finally, we relate the nontrivial corner states to the emergent phenomena of quantized fractional corner charges and a filling anomaly as first predicted in electronic systems. Our work paves the way to further explore higher-order topological phases in nanophotonic systems and our method of inducing quadrupole phase transitions is also applicable to other wave systems, such as electrons, phonons and polaritons.