# Left: first eigenvalues of the single-particle Hamiltonian, E_n^pm as a function of *q*^{2}/*B*, see equation (13)

**Figure 1.** Left: first eigenvalues of the single-particle Hamiltonian, E_n^\pm as a function of *q*^{2}/*B*, see equation (13). Right: weights of the two components of each eigenvector of the single-particle Hamiltonian for the '−' states, \alpha _n^-, and \beta _n^- as a function of *q*^{2}/*B*, see equation (16).

**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.