Microscopic model for the boson integer quantum Hall effect

In two dimensions strongly interacting bosons in a magnetic field can form an integer quantum Hall state. This state has a bulk gap, no fractional charges or topological order in the bulk, but nevertheless quantized Hall transport and symmetry-protected edge excitations. Here we study a simple microscopic model for such a state in a system of two-component bosons in a strong orbital magnetic field. We show through exact-diagonalization calculations that the model supports the boson integer quantum Hall ground state in a range of parameters.

Figure 1

Phase diagram of the model Hamiltonian. The bIQH state is stable in a range of $\frac{g_d}{g_s}$ estimated to be between $\sim$0.8 and $\sim$1.3 from exact-diagonalization calculations with 12 and 16 particles. PS refers to the phase-separated state which appears for large $\frac{g_d}{g_s}$. For small $\frac{g_d}{g_s}$ the decoupled Pfaffian state $\mathrm{Pf}\times \mathrm{Pf}$ is stable.

Figure 2

Left panel: Low-energy spectrum for $N=16$ bosons and $N_{\Phi }=8$ flux quanta on the torus geometry. The ground state is a spin singlet ($S=0$) clearly separated from the higher energy excitations. Right panel: Low-energy spectrum for $N=15$ bosons and $N_{\Phi }=8$ flux quanta on the torus geometry. This situation corresponds to one less particle than the left panel. Since $N$ is odd, the spin is half integer. The exact degeneracy is a consequence of the magnetic translation symmetry when $N$ and $N_{\Phi }$ are relative prime numbers.

Figure 3

Neutral gap $\Delta$ as a function of $1/N$ at filling factor $\nu =2$. While a thermodynamical extrapolation is quite difficult, the trend strongly suggests that the gap will stay finite for large $N$.

Figure 4

Phase diagram for $N=12$ (upper panel) and $N=16$ (lower panel) on the torus geometry at filling $\nu =2$ when tuning the interaction parameter $g_d/g_s$. The red line indicates the gap ${\Delta }_{\mathrm{bIQH}}$ between the lowest energy state and the first-excited state. Each kink in ${\Delta }_{\mathrm{bIQH}}$ corresponds to a level crossing above the ground state. The blue line stands for the gap ${\Delta }_{\mathrm{Pf}\times \mathrm{Pf}}$ between the low-energy manifold of 9 states associated with the $\mathrm{Pf}\times \mathrm{Pf}$ phase and the first-excited state. In both cases, the gap is set to zero if the low-energy manifold that corresponds to each of these states does not have the proper quantum numbers. The shaded area indicates the region where the lowest energy state is fully polarized (i.e., a phase-separated state).