Integer Quantum Hall State in Two-Component Bose Gases in a Synthetic Magnetic Field

We study two-component (or pseudospin-$1/2$) Bose gases in a strong synthetic magnetic field. Using exact diagonalization, we show that a bosonic analog of an integer quantum Hall state with no intrinsic topological order appears at the total filling factor $\nu =1+1$ when the strengths of intracomponent and intercomponent interactions are comparable with each other. This provides a prime example of a symmetry-protected topological phase in a controlled setting of quantum gases. The real-space entanglement spectrum of this state is found to be comprised of counterpropagating chiral modes consistent with the edge theory derived from an effective Chern-Simons theory.

Figure 1

Candidates for incompressible GSs in the ($N_{\varphi }$, $N$) plane, calculated on a sphere geometry in the SU(2)-symmetric case $g_{\uparrow \downarrow }=g$. Filled circles indicate GSs with the total angular momentum $L=0$, where incompressible states can appear; the area of each symbol is proportional to the neutral gap ${\Delta }_n$. Empty circles indicate the GSs with $L>0$. Solid and broken lines show the relation $N=\nu (N_{\varphi }+\delta )$ for $\delta =0$ and 2, respectively. Data points are missing for large $N_{\varphi }$ and $N$ due to an exponentially long computation time. Below the dotted line with $\nu =2/3$, GSs are highly degenerate [21].

Figure 2

(a) Charge and neutral energy gaps, ${\Delta }_c$ and ${\Delta }_n$, calculated on a sphere with $N_{\varphi }=7$. (b),(c) Size dependences of the energy gaps at $\nu =2$ and 4 on the sphere and torus geometries. For a torus, we set $L_x=L_y$ (periodic square). For a sphere, we set $\delta =0$; namely, calculations are performed along the solid lines in Fig. 1. The dotted lines indicate linear extrapolation of the charge gap to the thermodynamic limit, using the data with $N\ge 10$.

Figure 3

Real-space entanglement spectrum of the $\nu =2$ IQH state with $(N_{\varphi },N)=(8,16)$ on a sphere. We plot the entanglement energies $\{-\log p_i\}$ defined from the eigenvalues $\{p_i\}$ of ${\rho }_A$, where $A$ is the northern hemisphere; the lower the entanglement energy is, the larger weight it has in ${\rho }_A$. The entanglement energies are classified by the number of $\uparrow$ and $\downarrow$ atoms, $N_{\uparrow }^A$ and $N_{\downarrow }^A$, and by the $z$ component of the total angular momentum, $L_z^A$, on $A$.