A mathematical study of periodic band inversion

TL;DR

This paper provides a rigorous mathematical analysis of periodic band inversion phenomena in a 2D electron system coupled to a photon cavity, including spectral gap behavior, Dirac cones, and topological invariants.

Contribution

It derives an effective Hamiltonian in the strong-coupling limit and proves properties of spectral gaps, Dirac cones, and Chern numbers, advancing understanding of topological band phenomena.

Findings

Periodic closing and reopening of the spectral gap explained.

Existence and persistence of Dirac cones at gap-closing points proved.

Computed Chern numbers for band clusters and analyzed their behavior.

Abstract

We give a mathematical analysis of the periodic band inversion phenomenon observed by Tan--Devakul for an electron in a two-dimensional periodic potential coupled to a circularly polarized photon cavity mode. In the strong-coupling limit, we derive an effective Bloch Hamiltonian and prove convergence of the low-lying bands. For a cosine potential, we explain the periodic closing and reopening of the first spectral gap, prove the existence and generic persistence of Dirac cones at the gap-closing points, and compute the Chern numbers associated to isolated band clusters. We also show that higher isolated band clusters cannot persist in the small-coupling regime. Finally, we resolve an apparent sign discrepancy between Berry curvature computations and Chern numbers by tracking the descent from the covering space to the Brillouin torus.