Optical $N$-insulators: topological obstructions in the atomistic susceptibility tensor

TL;DR

This paper introduces optical N-insulators, a new class of 2D topological materials characterized by an invariant related to the susceptibility tensor, revealing obstructions to localized polarization bases akin to electronic topological phases.

Contribution

It defines the optical N-invariant, connects it to the susceptibility tensor's singularities, and classifies topological electromagnetic phases considering crystalline symmetries.

Findings

Optical N-insulators possess a winding number N that counts singularities in the susceptibility tensor.

Nontrivial optical phases (N≠0) cannot have localized polarization bases.

The classification incorporates crystalline symmetries using K-theory.

Abstract

A powerful result of topological band theory is that nontrivial phases manifest obstructions to constructing localized Wannier functions. In Chern insulators, it is impossible to construct Wannier functions that respect translational symmetry in both directions. Similarly, Wannier functions that respect time-reversal symmetry cannot be formed in quantum spin Hall insulators. This molecular orbital interpretation of topology has been enlightening and was recently extended to topological crystalline insulators which include obstructions tied to space group symmetries. In this article, we introduce a new class of two-dimensional topological materials known as optical $N$-insulators that possess obstructions to constructing localized molecular polarizabilities. The optical $N$-invariant $N\in\mathbb{Z}$ is the winding number of the atomistic susceptibility tensor $\chi$ and counts the number of singularities in the electromagnetic linear response theory. We decipher these singularities by analyzing the optical band structure of the material -- the eigenvectors of the susceptibility tensor -- which constitutes the collection of optical Bloch functions. The localized basis of these eigenvectors are optical Wannier functions which represent the molecular polarizabilities at different lattice sites. We prove that in a nontrivial optical phase $N\neq 0$, such a localized polarization basis is impossible to construct. Utilizing the mathematical machinery of $K$-theory, these optical $N$-phases are refined further to account for the underlying crystalline symmetries of the material, generating a classification of the topological electromagnetic phase of matter.