A Mathematical Theory of Integer Quantum Hall Effect in Photonics

TL;DR

This paper provides a mathematical foundation for realizing the integer quantum Hall effect in photonic systems by analyzing interface modes in gyromagnetic photonic crystals with specific symmetry properties.

Contribution

It introduces a theoretical framework explaining how interface modes in gyromagnetic photonic crystals can support the quantum Hall effect, highlighting the role of quadratic degeneracy at the M-point.

Findings

Quadratic degeneracy occurs at the M-point in the Brillouin zone.

Opposite magnetic fields on either side of an interface induce bifurcation of interface modes.

The results underpin the first experimental realization of the integer quantum Hall effect in photonics.

Abstract

This paper investigates interface modes in a square lattice of photonic crystal composed of gyromagnetic particles with $C_{4v}$ point group symmetry. The study shows that Dirac or linear degenerate points cannot occur at the three high-symmetry points in the Brillouin zone where two Bloch bands touch. Instead, a touch point at the M-point has a quadratic degeneracy in the generic case. It is further proved that when a magnetic field is applied to the two sides of an interface in opposite directions, two interface modes supported along that interface can bifurcate from the quadratic degenerate point. These results provide a mathematical foundation for the first experimental realization of the integer quantum Hall effect in the context of photonics.