  We consider a non-equilibrium system of interacting emitters described by the
XXZ model, whose excitonic transitions are spatially and spectrally coupled to
a single mode cavity. We demonstrate that the output radiation field is
sensitive to an interplay between the hopping ($J$) and the interactions ($U$)
of the excitons. Moderate values of the short-ranged interaction are shown to
induce laser with maximal output at the Heisenberg point ($U=J$). In the laser
regime, charge-charge correlations emerge and they are shown to strongly depend
on the interaction-hopping ratio. In particular, the system shows
charge-density correlations below the Heisenberg point and ferromagnetic
correlations beyond the Heisenberg point. This contrast to the equilibrium
behavior of the XXZ chain occurs since the laser explores highly excited states
of the emitters.
