  We show that ultra-cold atoms with synthetic spin-orbit coupling in Raman
lattices can be used as versatile quantum simulators to explore the connections
between correlated Chern insulators and strongly-coupled four-Fermi field
theories related to the Gross-Neveu model in (2+1) dimensions. Exploiting this
multidisciplinary perspective, we identify a large-$N$ quantum anomalous Hall
(QAH) effect in absence of any external magnetic field, and use it to delimit
regions in parameter space where these correlated topological phases appear,
the boundaries of which are controlled by strongly-coupled fixed points of the
four-Fermi relativistic field theory. We further show how, for strong
interactions, the QAH effect gives way to magnetic phases described by a
two-dimensional quantum compass model in a transverse field. We present a
detailed description of the phase diagram using the large-$N$ effective
potential, and variational techniques such as projected entangled pairs.
