Topological markers for a one-dimensional fermionic chain coupled to a single-mode cavity

We study a Su-Schrieffer-Heeger chain coupled to a single mode photonic cavity. Considering an off-resonant regime we use the high-frequency expansion in order to obtain an effective fermionic Hamiltonian with cavity-mediated interactions. We characterize the effects of the cavity on topology in a finite size chain by studying three different markers adapted for interacting systems: correlation functions between edges in a chain with open boundary conditions, and a winding number based on the single-particle Green's function and bulk electric polarization via the many-body formula by Resta for a chain with periodic boundary conditions. There is excellent agreement between the winding number and polarization approaches to compute the phase diagram, with the presence of the edge states being confirmed through the calculations of the two-point correlation function. Our approach provides an alternative perspective on cavity-modified topological phases through a study of an effective interacting electronic Hamiltonian and complements methods that treat the full light-matter Hamiltonian directly.