  Spin ensembles coupled to optical cavities provide a powerful platform for
engineering synthetic quantum matter. Recently, we demonstrated that cavity
mediated infinite range interactions can induce fast scrambling in a Heisenberg
$XXZ$ spin chain (Phys. Rev. Research {\bf 2}, 043399 (2020)). In this work, we
analyze the kaleidoscope of quantum phases that emerge in this system from the
interplay of these interactions. Employing both analytical spin-wave theory as
well as numerical DMRG calculations, we find that there is a large parameter
regime where the continuous $U(1)$ symmetry of this model is spontaneously
broken and the ground state of the system exhibits $XY$ order. This kind of
symmetry breaking and the consequent long range order is forbidden for short
range interacting systems by the Mermin-Wagner theorem. Intriguingly, we find
that the $XY$ order can be induced by even an infinitesimally weak infinite
range interaction. Furthermore, we demonstrate that in the $U(1)$ symmetry
broken phase, the half chain entanglement entropy violates the area law
logarithmically. Finally, we discuss a proposal to verify our predictions in
state-of-the-art quantum emulators.
