Entanglement and topology in Su-Schrieffer-Heeger cavity quantum electrodynamics

Cavity materials are a frontier to investigate the role of light-matter interactions on the properties of electronic phases of matter. In this work, we raise a fundamental question: can nonlocal interactions mediated by cavity photons destabilize a topological electronic phase? We investigate this question by characterizing entanglement, energy spectrum, and correlation functions of the topological Su-Schrieffer-Heeger chain interacting with an optical cavity mode. Employing density-matrix renormalization group and exact diagonalization, we demonstrate the stability of the edge state and establish an area law scaling for the ground state entanglement entropy, despite long-range correlations induced by light-matter interactions. These features are linked to gauge invariance and the scaling of virtual photon excitations entangled with matter, effectively computed in a low-dimensional Krylov subspace of the full Hilbert space. This work provides a framework for characterizing novel equilibrium phenomena in topological cavity materials.