Dipolar ordering transitions in many-body quantum optics: Analytical diagrammatic approach to equilibrium quantum spins

Quantum spin models with a large number of interaction partners per spin are frequently used to describe modern many-body quantum optical systems like arrays of Rydberg atoms, atom-cavity systems or trapped ion crystals. For theoretical analysis the mean-field (MF) ansatz is routinely applied. However, besides special cases of all-to-all or strong long range interactions, the MF ansatz provides only approximate results. Here we present a systematic correction to MF theory based on diagrammatic perturbation theory for quantum spin correlators in thermal equilibrium. Our analytic results are universally applicable for any lattice geometry and spin-length S. We provide pre-computed and easy-to-use building blocks for Ising, Heisenberg and transverse field Ising models in the symmetry-unbroken regime. We showcase the quality and simplicity of the method by computing magnetic phase boundaries and excitations gaps. We also treat the Dicke-Ising model of ground-state superradiance where we show that corrections to the MF phase boundary vanish.