**Figure 2.** Properties of the system as a function of the strength of the spin–orbit interaction, *q*^{2}/*B*. (a) Low-lying excitation energies. (b) Left scale: interaction energy of the ground state. Right scale: occupation of the *n* = 0 level in the ground state. (c) Overlap of ground state with different test states as further explained in the text: Laughlin projections, and ground states of two-body contact interaction in the first and second excited LL. The system size is *N* = 6 at half filling, with spin-independent interactions g_{s_1s_2}=0.2. Energies are in units of *gB*.

**Abstract**

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 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 fillings, ν = 1/2 and ν = 1/4. While for these filling factors incompressible phases disappear at the degeneracy point, we find gaps in the spectra of denser systems at ν = 3/2 and ν = 2. For filling factors ν = 2/3 and ν = 4/3, we discuss the connection of the exact ground states to the non-Abelian spin singlet states, obtained as the ground states of (*k* + 1)-body contact interactions.