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We consider the quantum XX-model in the presence of competingnearest-neighbour and global-range interactions, which is equivalent to aBose-Hubbard model with cavity mediated global range interactions in the hardcore boson limit. Using fermionic techniques the problem is solved exactly inone dimension in the thermodynamic limit. The ground state phase diagramconsists of two ordered phases: ferromagnetic (F) and antiferromagnetic (AF),as well as an XY-phase having quasi-long-range order. We have also studiedquantum relaxation after sudden quenches. Quenching from the AF phase to the XYregion remanent AF order is observed below a dynamical transition line. In theopposite quench, from the XY region to the AF-phase beyond a staticmetastability line AF order arises on top of remanent XY quasi-long-rangeorder, which corresponds to dynamically generated supersolid state in theequivalent Bose-Hubbard model with hard-core bosons.

Quantum XX-model with competing short- and long-range interactions: Phases and phase transitions in and out of equilibrium