Vacuum-field-mediated Landau-level reconstruction in a cavity-coupled two-dimensional electron gas

We study a two-dimensional electron gas in the integer quantum Hall regime coupled to a single spatially modulated cavity mode in the static off-resonant limit. Retaining both paramagnetic and diamagnetic contributions on equal footing, we derive a gauge complete single-particle Hamiltonian in which virtual photon exchange generates a periodic kinetic renormalization and a diamagnetic scalar potential. Both terms transfer momentum by $2k_0$ and displace guiding centers by $\mathrm{\Delta }X=2k_0{\mathrm{\ell }}_B^2$, forming a finite-$k_0$ momentum ladder. Diagonalizing the strip Hamiltonian, we find that the ladder reconstructs the Landau-level band structure and redistributes, rather than multiplies, the associated chiral edge states. Occupied-subspace Chern numbers remain $C_{\nu }=\nu$ for $\nu =1,2,3$ throughout $0\le {\lambda }_{\mathrm{c}\mathrm{a}\mathrm{v}}\le 0.5$, while the bulk gaps decrease without closing. Strip Kubo responses show cavity-induced displacements of the Hall-step energies; at finite width the plateau heights are noninteger and approach the integers as $L_x$ increases. The ladder is spectrally resolved at $k_0{\mathrm{\ell }}_B=0.5$ and collapses at $k_0{\mathrm{\ell }}_B=0.1$. These results isolate a spatially local, finite-$k_0$ momentum ladder as a clean-system mechanism by which a structured vacuum field reconstructs quantum Hall spectra and strip transport without changing the integer topology, complementary to cavity-mediated all-to-all hopping.