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Quantum matter at ultra-low temperatures offers a testbed for analyzing andcontrolling desired properties in strongly correlated systems. Under typicalconditions the nature of the atoms fixes the magnetic character of the system.Beyond classical light potentials leading to optical lattices and short rangeinteractions, high-Q cavities introduce novel dynamics into the system via thequantumness of light. Here we propose a theoretical model and we analyze itusing exact diagonalization and density matrix renormalization groupsimulations. We explore the effects of cavity mediated long range magneticinteractions and optical lattices in ultracold matter. We find that globalinteractions modify the underlying magnetic character of the system whileintroducing competition scenarios. Antiferromagnetic correlated bosonic matteremerges in conditions beyond to what nature typically provides. These allow newalternatives toward the design of robust mechanisms for quantum informationpurposes, exploiting the properties of magnetic phases of strongly correlatedquantum matter.

Spin Entanglement and Magnetic Competition via Long-range Interactions in Spinor Quantum Optical Lattices