  Low-dimensional systems of interacting fermions in a synthetic gauge field
have been experimentally realized using two-component ultra-cold Fermi gases in
optical lattices. Using a two-leg ladder model that is relevant to these
experiments, we have studied the signatures of topological Lifshitz transitions
and the effects of the inter-species interaction $U$ on the gauge-invariant
orbital current in the regime of large intra-leg hopping $\Omega$. Focusing on
non-insulating regimes, we have carried out numerically exact density-matrix
renormalization-group (DMRG) calculations to compute the orbital current at
fixed particle number as a function of the interaction strength and the
synthetic gauge flux per plaquette. Signatures of topological Lifshitz
transitions where the number Fermi points changes are found to persist even in
the presence of very strong repulsive interactions. This numerical observation
suggests that the orbital current can be captured by an appropriately
renormalized mean-field band structure, which is also described here.
Quantitative agreement between the mean-field and the DMRG results in the
intermediate interaction regime where $U \lesssim \Omega$ is demonstrated. We
also have observed that interactions can change the sign of the current
susceptibility at zero field and induce Lifshitz transitions between two
metallic phases, which is also captured by the mean-field theory. Correlation
effects beyond mean-field theory in the oscillations of the local inter-leg
current are also reported. We argue that the observed robustness against
interactions makes the orbital current a good indicator of the topological
Lifshitz transitions.
