To infinity and back -- $1/N$ graph expansions of light-matter systems
We present a method for performing a full graph expansion for light-matter systems, utilizing the linked-cluster theorem. This method enables us to explore $1/N$ corrections to the thermodynamic limit $N\to \infty$ in the number of particles, giving us access to the mesoscopic regime. While this regime is yet largely unexplored due to the challenges of studying it with established approaches, it incorporates intriguing features, such as entanglement between light and matter that vanishes in the thermodynamic limit. As a representative application, we calculate physical quantities of the low-energy regime for the paradigmatic Dicke-Ising chain in the paramagnetic normal phase by accompanying the graph expansion with both exact diagonalization (NLCE) and perturbation theory (pcst++), benchmarking our approach against other techniques. We investigate the ground-state energy density and photon density, showing a smooth transition from the microscopic to the macroscopic regime up to the thermodynamic limit. Around the quantum critical point, we extract the $1/N$ corrections to the ground-state energy density to obtain the critical point and critical exponent using extrapolation techniques.