Electromagnetic coupling in tight-binding models for strongly correlated light and matter

The rapidly developing collaboration of condensed matter and quantum optical physics opens up exciting new possibilities for creating novel photon-matter states that allow manipulating material properties using optical cavities. This paper addresses the important challenge of formulating consistent, accurate, and convenient forms of the light-matter coupling for the low-energy tight-binding-type models useful for theoretical analysis. It achieves this by constructing precise mappings from the fundamental $a\phantom{\rule{0}{0ex}}b$ $i\phantom{\rule{0}{0ex}}n\phantom{\rule{0}{0ex}}i\phantom{\rule{0}{0ex}}t\phantom{\rule{0}{0ex}}i\phantom{\rule{0}{0ex}}o$ description of the cavity-matter system to low-energy theories, producing a truncated Hamiltonian coupling to a quantum Peierls phase. In a simple model system, the authors demonstrate the quick convergence of the truncated Hamiltonian with the number of bands, paving the way for rigorous theoretical studies in this emerging research field.