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We show that ultra-cold atoms with synthetic spin-orbit coupling in Ramanlattices can be used as versatile quantum simulators to explore the connectionsbetween correlated Chern insulators and strongly-coupled four-Fermi fieldtheories related to the Gross-Neveu model in (2+1) dimensions. Exploiting thismultidisciplinary perspective, we identify a large-$N$ quantum anomalous Hall(QAH) effect in absence of any external magnetic field, and use it to delimitregions in parameter space where these correlated topological phases appear,the boundaries of which are controlled by strongly-coupled fixed points of thefour-Fermi relativistic field theory. We further show how, for stronginteractions, the QAH effect gives way to magnetic phases described by atwo-dimensional quantum compass model in a transverse field. We present adetailed description of the phase diagram using the large-$N$ effectivepotential, and variational techniques such as projected entangled pairs.

Correlated Chern insulators in two-dimensional Raman lattices: a cold-atom regularization of strongly-coupled four-Fermi field theories