  We study the interacting Fermi-Hubbard model in two spatial dimensions with
synthetic gauge coupling of the spin orbit Rashba type, at half-filling. Using
real space mean field theory, we numerically determine the phase as a function
of the interaction strength for different values of the gauge field parameters.
For a fixed value of the gauge field, we observe that when the strength of the
repulsive interaction is increased, the system enters into an antiferromagnetic
phase, then undergoes a first order phase transition to an non collinear
magnetic phase. Depending on the gauge field parameter, this phase further
evolves to the one predicted from the effective Heisenberg model obtained in
the limit of large interaction strength. We explain the presence of the
antiferromagnetic phase at small interaction from the computation of the
spin-spin susceptibility which displays a divergence at low temperatures for
the antiferromagnetic ordering. We discuss, how the divergence is related to
the nature of the underlying Fermi surfaces. Finally, the fact that the first
order phase transitions for different gauge field parameters occur at unrelated
critical interaction strengths arises from a Hofstadter-like situation, i.e.
for different magnetic phases, the mean-field Hamiltonians have different
translational symmetries.
