  We investigate the emergence of long-range electron hopping mediated by
cavity vacuum fields in disordered quantum Hall systems. We show that the
counter-rotating (anti-resonant) light-matter interaction produces an effective
hopping between disordered eigenstates within the last occupied Landau band.
The process involves a number of intermediate states equal to the Landau
degeneracy: each of these states consists of a virtual cavity photon and an
electron excited in the next Landau band with the same spin. We study such a
cavity-mediated hopping mechanism in the dual presence of a random disordered
potential and a wall potential near the edges, accounting for both paramagnetic
coupling and diamagnetic renormalization. We determine the cavity-mediated
scattering rates, showing the impact on both bulk and edge states. The effect
for edge states is shown to increase when their energy approaches the
disordered bulk band, while for higher energy the edge states become
asymptotically free. We determine the scaling properties while increasing the
Landau band degeneracy. Consequences on the quantum Hall physics and future
perspectives are discussed.
