Two-leg-ladder Bose-Hubbard models with staggered fluxes

We investigate the ground-state properties of ultracold atoms trapped in a two-leg ladder potential in the presence of an artificial magnetic field in a staggered configuration. We focus on the strongly interacting regime and use the Landau theory of phase transitions and a mean field Gutzwiller variational method to identify the stable superfluid phases and their boundaries with the Mott-insulator regime as a function of magnetic flux. In addition, we calculate the local and chiral currents of these superfluid phases, which show a staggered vortex-antivortex configuration. The analytical results are confirmed by numerical simulations using a cluster mean-field-theory approach.

Figure 1

Schematic of the two-leg ladder Bose Hubbard model with staggered flux $\alpha$ in neighboring plaquettes. The dashed box indicates the single unit cell used for the analytic and the cluster mean field calculations. The red dots represent the bosonic atoms on lattice sites.

Figure 2

Single-particle spectrum of the two-leg ladder system for different absolute magnetic flux strengths for $J=1$ and $K=1$.

Figure 3

Phase diagram for the two-leg ladder Bose-Hubbard model in the presence of a staggered flux of magnitude $\alpha$ for unit filling factor using Landau theory. The solid (red) curve marks the boundary between the Mott-insulator and the different superfluid phases for $K=J=1.0$. The region below the solid (red) curve comprises of two types of superfluids, SF-1 and SF-2 (see text for details), which are separated by green dashed lines. The dashed (blue) lines and dotted (black) lines mark the phase boundaries for $J=1$ and $K=0.5$ and 1.5, respectively.

Figure 4

Phase diagram of the Bose-Hubbard model for the two-leg ladder for different absolute values of staggered magnetic flux $\alpha$, for $K=1$ and $U=1$, calculated using a variational mean-field approach. The MI phases are indicated with their average occupancy per site, and SF indicated in the plot can be SF-1 for $-\pi <\alpha <\pi$ and SF-2 for the regime $-3\pi <\alpha <-\pi$ and $\pi <\alpha <3\pi$.

Figure 5

Same as Fig. 3, but the results are obtained by using the CMFT approach.

Figure 6

Variation of $j_c$ (top panel) and $|j_c|$ (bottom panel) with $n$ for $J=1$ and $K=0.25,1.0$, and 1.50 and ($U,\mu$) $=(8.0,11.5$).

Figure 7

Schematic of current patterns associated with the SF-1 and SF-2 phases. The red arrows denote the local currents given by equation (20). The blue circular arrows denote the local staggered vortices/antivortices deduced from the local current pattern. The local currents possess opposite rotational directions for the two superfluid phases.