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Fractional quantum Hall states of a Bose gas with spin-orbit coupling

We study the fractional quantum Hall phases of a pseudospin-1/2 Bose gas in an artificial gauge field. In addition to an external magnetic field, the gauge field also mimics an intrinsic spin-orbit coupling of the Rashba type. While the spin degeneracy of the Landau levels is lifted by the spin-orbit coupling, the crossing of two Landau levels at certain coupling strengths gives rise to a new degeneracy. We therefore take into account two Landau levels, and perform exact diagonalization of the many-body Hamiltonian. We study and characterize the quantum Hall phases which occur in the vicinity of the degeneracy point. Notably, we describe the different states appearing at the Laughlin filling, ν=1/2. While for this filling incompressible phases disappear at the degeneracy point, denser systems at ν=3/2 and ν=2 are found to be clearly gapped. For filling factors ν=2/3 and ν=4/3, we discuss the connection of the exact ground state to the non-Abelian spin singlet states, obtained as the ground state of k+1 body contact interactions.