# Collective modes for helical edge state interacting with quantum light

## TL;DR

This paper explores the interaction between a 2D topological insulator's edge state and quantum light, revealing a superradiant ground state with broken time reversal symmetry, collective excitations, and modified optical properties.

## Contribution

It introduces a novel analysis of light-matter interaction in topological insulator edge states, highlighting superradiance and collective modes induced by quantum light.

## Key findings

- Emergence of a superradiant ground state with broken time reversal symmetry.
- Identification of polariton continuum excitations and a Higgs mode.
- Modification of optical conductivity with renormalized effective mass and finite frequency shifts.

## Abstract

We investigate the light-matter interaction between the edge state of a 2D topological insulator and quantum electromagnetic field. The interaction originates from the Zeeman term between the spin of the edge electrons and the magnetic field, and also through the Peierls substition. The continuous U(1) symmetry of the system in the absence of the vector potential reduces into discrete time reversal symmetry in the presence of the vector potential. Due to light-matter interaction, a superradiant ground state emerges with spontaneously broken time reversal symmetry, accompanied by a net photocurrent along the edge, generated by the vector potential of the quantum light. The spectral function of the photon field reveals polariton continuum excitations above a threshold energy, corresponding to a Higgs mode and another low energy collective mode due to the phase fluctuations of the ground state. This collective mode is a zero energy Goldstone mode that arises from the broken continuous U(1) symmetry in the absence of the vector potential, and acquires finite a gap in the presence of the vector potential. The optical conductivity of the edge electrons is calculated using the random phase approximation by taking the fluctuation of the order parameter into account. It contains the collective modes as a Drude peak with renormalized effective mass, which moves to finite frequencies as the symmetry of the system is lowered by the inclusion of the vector potential.