Bosonic Integer Quantum Hall Effect in Optical Flux Lattices

In two dimensions strongly interacting bosons in a magnetic field can realize a bosonic integer quantum Hall state, the simplest two-dimensional example of a symmetry-protected topological phase. We propose a realistic implementation of this phase using an optical flux lattice. Through exact diagonalization calculations, we show that the system exhibits a clear bulk gap and the topological signature of the bosonic integer quantum Hall state. In particular, the calculation of the many-body Chern number leads to a quantized Hall conductance in agreement with the analytical predictions. We also study the stability of the phase with respect to some of the experimentally relevant parameters.

Figure 1

(a) Schematic description of the first Brillouin zone of the OFL model that we have considered for $N=5$. The blue, red, and black links represent the hopping amplitudes defined by the potential $\widehat{V}(\mathit{r})$. The phases are set to produce a lowest band carrying a Chern number $C=2$. (b) Density of states for $V/E_R=3$ and $N=6$. Note that we have rescaled the density of states of the lowest band by a factor 0.1. The band spread of the lowest band is ${\delta }_{1\mathrm{b}}=0.0126E_R$ and the band gap between the lowest band and the second band is ${\mathrm{\Delta }}_{1\mathrm{b}}=0.721E_R$. (c) Band spread and band gap of the lowest band as a function of the laser strength $V$ for $N=5$ and $N=6$.

Figure 2

Left Panel: Low energy spectrum of the two body interaction at $\nu =1$ for $N=6$, $N_B=14$, $N_x=7$, $N_y=2$, and $V=3E_R$. The energies are plotted as a function of the linearized momentum index where $(K_x,K_y)$ are the two integers defining the momentum sector. Note that the energies are shifted by the ground state energy $E_0$. Right Panel: Energy spectrum of the two body interaction at $\nu =1$ for $N=6$, $N_B=13$, $N_x=7$, $N_y=2$, and $V=3E_R$. Compared to the left panel, there is one particle less and thus one added quasihole. We observe an almost flat band of low energy states with one state per sector as expected for the IQH effect.

Figure 3

Gap as a function of the inverse of the particle number for $N=5$ (red dots) and $N=6$ (blue triangles) and $V=3E_R$ up to $N_B=17$. For each particle number, we choose $N_x$ and $N_y$ so that the aspect ratio is the closest possible to 1. However, there are still important aspect ratio variations between the different points which prevent us from doing a relevant extrapolation of the gap at the thermodynamical limit. Quite clearly, this gap does not close at the thermodynamical limit and will be of order $0.1V_{\mathrm{int}}$.

Figure 4

Many-body Chern number $C_{\mathrm{MB}}$ computed both using the surface integral and the path integral version, as a function of the discretization step $\delta$ of the $({\theta }_x,{\theta }_y)$ torus for $N=5$ and $N=6$. The system considered here has $N_B=12$, $N_x=6$, $N_y=2$, and $V=3E_R$. The dashed horizontal line at $C_{\mathrm{MB}}=2$ is a guide for the eye. In the inset, we show the result at $\nu =\frac{1}{2}$ when the lowest band has $C=1$ where the Laughlin state is realized for $N_B=6$, $N_x=4$, $N_y=3$, and $N=5$.

Figure 5

Influence of the laser strength $V$ on the many-body gap $\mathrm{\Delta }$ for $N_B=14$, $N_x=7$, and $N_y=2$. We have considered both $N=5$ (red dots) and $N=6$ (blue triangles). The behavior of $\mathrm{\Delta }$ exactly mimics the behavior of the one-body gap ${\mathrm{\Delta }}_{1\mathrm{b}}$ shown in Fig. 1.

Figure 6

Many-body gap as a function of the interaction strength $V_{\mathrm{int}}$ when the two lowest bands and their dispersion relations are taken into account, for $N=6$, $N_B=10$, $N_x=10$, $N_y=1$, and $V=3E_R$. For comparison, the one-body gap is ${\mathrm{\Delta }}_{1\mathrm{b}}=0.721E_R$. The inset gives the continuous evolution at $V_{\mathrm{int}}=0.175E_R$ from the two band model ($\lambda =0$) to the flat band limit ($\lambda =1$). Both evolutions of the gap (black line) and the overlap with the flat band ground state (green line) show the adiabatic connection between these two systems.