## Spectra of a two-component Bose gas without the spin–orbit coupling (*q* = 0) at the filling factor ν = 4/3 for different spin polarizations *S*

**Figure 6.** Spectra of a two-component Bose gas without the spin–orbit coupling (*q* = 0) at the filling factor ν = 4/3 for different spin polarizations *S*. The results for a two-body contact interaction are given by the black crosses which correspond to the energy scale on the left. The red triangles with the energy scale on the right show the results for a system interacting via a three-body contact potential.

**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.