Collective Modes of the Two-Component ν=1 Quantum Hall System in a Cavity

We analyze the collective mode spectrum of a two-component quantum Hall at filling factor ν=1 in the strong magnetic field limit, when the system is immersed in a quantum cavity hosting a quantized electromagnetic mode of frequency ω₀, assumed to be much smaller than the cyclotron frequency ω_c. The simplest realization of this system involves a cavity mode with a uniform electric field. We consider different ansatz’s for the collective mode wavefunctions, noting that the state may take a different form when the system is represented in a different gauge. When the cavity mode field is uniform and in the plane of the electron gas, our analysis shows that for momenta q ≠ 0, for electrons with an active spin degree of freedom, the spin wave mode spectrum is essentially identical to what is found in the absence of the cavity. For bilayer systems with balanced fillings in each layer, the situation turns out to be more subtle, due to the breaking of SU(2) symmetry in the Coulomb interactions and the resulting particle-hole fluctuations in the ground and excited states. We discuss results from different approaches, including density matrix renormalization group theory in the thin cylinder regime, as well as the expected impacts of non-uniformity in the cavity electric field, and electric potentials that break translational symmetry in the absence of the cavity.