  We investigate classes of interacting quantum spin systems in a single-mode
cavity with a Dicke coupling, as a paradigmatic example of strongly correlated
light-matter systems. Coming from the limit of weak light-matter couplings and
large number of matter entities, we map the relevant low-energy sector of a
broad class of models in the non-superradiant phases onto the exactly solvable
Dicke model. We apply the outcomes to the Dicke-Ising model as a paradigmatic
example, in agreement with results obtained by mean-field theory. We further
accompany and verify our findings with finite-size calculations, using exact
diagonalization and the series expansion method pcst++.
