Bosonic quantum Hall states in single-layer two-dimensional optical lattices

Quantum Hall (QH) states of two-dimensional (2D) single-layer optical lattices are examined using the Bose-Hubbard model (BHM) in the presence of an artificial gauge field. We study the QH states of both the homogeneous and inhomogeneous systems. For the homogeneous case, we use cluster Gutzwiller mean-field (CGMF) theory with cluster sizes ranging from $2\times 2$ to $5\times 5$. We then consider the inhomogeneous case, which is relevant to experimental realization. In this case, we use CGMF and exact diagonalization (ED). The ED studies are using lattice sizes ranging from $3\times 3$ to $4\times 12$. Our results show that the geometries of the QH states are sensitive to the magnetic flux $\alpha$ and cluster sizes. For homogeneous systems, among various combinations of $1/5\le \alpha \le 1/2$ and filling factor $\nu$, only the QH state of $\alpha =1/4$ with $\nu =1/2,1,3/2$, and 2 occur as ground states. For other combinations, the competing superfluid (SF) state is the ground state and QH state is metastable. For BHM with envelope potential, all the QH states observed in homogeneous systems exist for box potentials, but none exist for the harmonic potential. The QH states also persist for very shallow Gaussian envelope potential. As a possible experimental signature, we study the two-point correlations of the QH and SF states.

Figure 1

The solid blue lines between the lattice sites represent the intersite bonds. The gray dashed lines demarcate cells around each lattice site, which are used in representing cluster or attributing properties to each of the lattice sites. For illustration, one of the cells is highlighted in yellow and as an example a $2\times 2$ cluster is identified with orange color.

Figure 2

A $2\times 2$ cluster within the lattice. The light and bold dashed lines marked boundaries of cells and cluster, respectively. The solid (dashed) green (light gray) arrows represent the exact hopping term (Hermitian conjugate) within the cluster. Similarly, the solid (dashed) red (gray) arrows represent approximate hopping term (Hermitian conjugate) across clusters with one order of $\varphi$ and operator.

Figure 3

The $M\times 1$ row of a cluster with occupation number $n_0,n_1,\cdots ,n_{M-1}$. Each square box represents a lattice site and each of $n_i$ corresponds to $i\mathrm{th}$ lattice site in that row. Here, $n_i$ runs from 0 to $N_b-1$ for each lattice site.

Figure 4

The $M\times N$ cluster with occupation number $n_0^j,n_1^j,\cdots ,n_{M-1}^j$ for $j\mathrm{th}$ row of the cluster. Each square box represents a lattice site and each of $n_i^j$ corresponds to each $j\mathrm{th}$ row of cluster and $i\mathrm{th}$ lattice site in that row. Here, $n_i^j$ runs from 0 to $N_b-1$ for each lattice site.

Figure 5

MI-SF phase boundary around $\rho$ = 1 Mott lobe with $\alpha =0$ from SGMF theory (blue dashed line). For $\alpha =1/3$, from SGMF (black dot-dashed line) and from $3\times 2$ CGMF theory (brown solid line). From the CGMF calculation enhancement in the phase boundary is obtained.

Figure 6

The number density $\rho$ with synthetic magnetic field $\alpha >0$. The SF states are compressible; as a result, $\rho$ varies linearly with $\mu$, which correspond to the green curves. The incompressible QH states correspond to constant $\rho$ (blue lines) or plateaus for specific values of filling factor $\nu$. (a) $\alpha =1/5$ and the plateaus correspond to $\nu =n/2,n=1,2,\cdots ,9$. (b) $\alpha =1/2$ and the plateaus correspond to $\nu =1/2,1$, and $3/2$.

Figure 7

(a) Hall state with stripe phase for $\alpha =1/5,\nu =1/2$ with average number $\rho =0.1$. (b) Zero SF order parameter $\varphi$ for the same.

Figure 8

The variation in the lattice occupancy $\rho$ of the FQH states with stripe and checkerboard geometry for high flux $\alpha =1/2$ obtained using CGMF for the filling factor $\nu =1/2$. This is a metastable state, and the ground state is in SF phase. (a) The FQH state has average number density $\rho =0.25$ with stripe pattern and it is obtained from the $2\times 4$ cluster. (b) The checkerboard FQH state with the same number density obtained from CGMF theory with the $4\times 4$ cluster. In both the cases, the ground states, SF phase, like the FQH state, have stripe and checkerboard geometries with $2\times 4$ and $4\times 4$ clusters, respectively.

Figure 9

Density distribution of the IQH state for $\alpha =1/5$ and $\nu =1$ with hard-wall boundary. The average density of atoms in this state is $\rho =0.2$. (a) The IQH state has stripe geometry in the CGMF results with $2\times 5$ clusters. (b) It is, however, transformed to checkerboard geometry when $3\times 5$ clusters are considered in the CGMF computations.

Figure 10

The variations in $\rho$ for IQH state of $\alpha =1/5$ and $\nu =1$ for a single cluster of different sizes. (a) The result from $3\times 5$ cluster has checkerboard pattern, and is the unit cell of the large lattice shown in Fig. 9. (b) $4\times 5$ cluster has less variations in $\rho$ compared to $3\times 5$. (c) $5\times 5$ cluster shows a rich variation in $\rho$ and unlike in panels (a) and (b) the central lattice site has maxima in density.

Figure 11

Two-point correlation function for low flux $\alpha =1/5$ with the $5\times 5$ and $5\times 3$ clusters for the QH and SF states, respectively. The correlation is calculated along the $x$ direction for the single cluster. Here $y=0$ and 1 represent the edge and bulk, respectively. (a) As a characteristic feature of QH state, the correlation function of the $\nu =1$ IQH state decays nonmonotonically in the bulk, and there is no difference between the hard-wall and periodic boundary conditions. (b) For the corresponding SF state, there is no trend in the bulk correlation function with hard-wall boundary (solid green line with down triangle symbol), but it decays monotonically at the edge (solid brown line with circle symbol). With periodic boundary condition (dashed lines), the range of values change, and both the bulk and edge exhibit monotonic decay in correlation.

Figure 12

A $3\times 3$ cluster and form of the hopping terms between the lattice sites. For clarity, each lattice site is represented in terms of cells. The light and bold dashed lines mark boundaries of cells and cluster, respectively. The solid (dashed) light gray arrows represent the exact hopping term (Hermitian conjugate) within the cluster. Similarly, the solid (dashed) gray arrows represent approximate hopping term (Hermitian conjugate) across clusters with one order of $\varphi$ and operator. The hopping terms involving the central lattice site, represented in green color, are all exact.

Figure 13

The $M\times N$ cluster with occupation number $n_0,n_1,\cdots ,n_{m^{\prime }}$ at each lattice site for CGMF. Each square box represents a lattice site and each $n_i$ corresponds to each $i$ lattice site. Here, $n_i$ runs from 0 to $N_b-1$ for each lattice site.