  We analyze the strongly correlated regime of a two-component trapped
ultracold fermionic gas in a synthetic non-Abelian U(2) gauge potential, that
consists of both a magnetic field and a homogeneous spin-orbit coupling. This
gauge potential deforms the Landau levels (LLs) with respect to the Abelian
case and exchanges their ordering as a function of the spin-orbit coupling. In
view of experimental realizations, we show that a harmonic potential combined
with a Zeeman term, gives rise to an angular momentum term, which can be used
to test the stability of the correlated states obtained through interactions.
We derive the Haldane pseudopotentials (HPs) describing the interspecies
contact interaction within a lowest LL approximation. Unlike ordinary
fractional quantum Hall systems and ultracold bosons with short-range
interactions in the same gauge potential, the HPs for sufficiently strong
non-Abelian fields show an unconventional non-monotonic behaviour in the
relative angular momentum. Exploiting this property, we study the occurrence of
new incompressible ground states as a function of the total angular momentum.
In the first deformed Landau level (DLL) we obtain Laughlin and Jain states.
Instead, in the second DLL three classes of stabilized states appear: Laughlin
states, a series of intermediate strongly correlated states and finally
vortices of the integer quantum Hall state. Remarkably, in the intermediate
regime, the non-monotonic HPs of the second DLL induce two-particle
correlations which are reminiscent of paired states such as the Haffnian state.
Via exact diagonalization in the disk geometry, we compute experimentally
relevant observables such as density profiles and correlations, and we study
the entanglement spectra as a further tool to characterize the obtained
strongly correlated states.
