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

Long-range-ordered phase in a quantum Heisenberg chain with interactions beyond nearest neighbors