  We study the dynamical behaviour of ultracold fermionic atoms loaded into an
optical lattice under the presence of an effective magnetic flux, induced by
spin-orbit coupled laser driving. At half filling, the resulting system can
emulate a variety of iconic spin-1/2 models such as an Ising model, an XY
model, a generic XXZ model with arbitrary anisotropy, or a collective one-axis
twisting model. The validity of these different spin models is examined across
the parameter space of flux and driving strength. In addition, there is a
parameter regime where the system exhibits chiral, persistent features in the
long-time dynamics. We explore these properties and discuss the role played by
the system's symmetries. We also discuss experimentally-viable implementations.
