  Quantum matter at ultra-low temperatures offers a testbed for analyzing and
controlling desired properties in strongly correlated systems. Under typical
conditions the nature of the atoms fixes the magnetic character of the system.
Beyond classical light potentials leading to optical lattices and short range
interactions, high-Q cavities introduce novel dynamics into the system via the
quantumness of light. Here we propose a theoretical model and we analyze it
using exact diagonalization and density matrix renormalization group
simulations. We explore the effects of cavity mediated long range magnetic
interactions and optical lattices in ultracold matter. We find that global
interactions modify the underlying magnetic character of the system while
introducing competition scenarios. Antiferromagnetic correlated bosonic matter
emerges in conditions beyond to what nature typically provides. These allow new
alternatives toward the design of robust mechanisms for quantum information
purposes, exploiting the properties of magnetic phases of strongly correlated
quantum matter.
