Competing ground states of strongly correlated bosons in the Harper-Hofstadter-Mott model

Using an efficient cluster approach, we study the physics of two-dimensional lattice bosons in a strong magnetic field in the regime where the tunneling is much weaker than the on-site interaction strength. We study both the dilute, hard-core bosons at filling factors much smaller than unity occupation per site and the physics in the vicinity of the superfluid-Mott lobes as the density is tuned away from unity. For hard-core bosons, we carry out extensive numerics for a fixed flux per plaquette $\varphi =1/5$ and $\varphi =1/3$. At large flux, the lowest-energy state is a strongly correlated superfluid, analogous to He-4, in which the order parameter is dramatically suppressed, but nonzero. At filling factors $\nu =1/2,1$, we find competing incompressible states which are metastable. These appear to be commensurate density wave states. For small flux, the situation is reversed and the ground state at $\nu =1/2$ is an incompressible density wave solid. Here, we find a metastable lattice supersolid phase, where superfluidity and density wave order coexist. We then perform careful numerical studies of the physics near the vicinity of the Mott lobes for $\varphi =1/2$ and $\varphi =1/4$. At $\varphi =1/2$, the superfluid ground state has commensurate density wave order. At $\varphi =1/4$, incompressible phases appear outside the Mott lobes at densities $n=1.125$ and $n=1.25$, corresponding to filling fractions $\nu =1/2$ and 1, respectively. These phases, which are absent in single-site mean-field theory, are metastable and have slightly higher energy than the superfluid, but the energy difference between them shrinks rapidly with increasing cluster size, suggestive of an incompressible ground state. We thus explore the interplay between Mott physics, magnetic Landau levels, and superfluidity, finding a rich phase diagram of competing compressible and incompressible states.

Figure 1

(a) Zero-temperature equation of state ($n_{\mathcal{C}}$ vs $\mu$) of a hard-core Bose gas in an artificial magnetic field. Solid and dashed curves indicate the results from different initial choices of the wave function. They converge to the same solution except at densities $n_{\mathcal{C}}=1/6$ and $n_{\mathcal{C}}=1/3$, which correspond to filling factors $\nu =1/2,1$ respectively. Incompressible plateaus, marked by the shaded gray regions, are observed which have higher energy than the compressible superfluid ground state. (b) Superfluid order parameter ${\rho }_{\mathcal{C}}$ as a function of $\tilde{\mu }$, showing that incompressible phases (dashed line) are uncondensed. When they differ, the energies of the states corresponding to the solid line have lower energy than the dashed.

Figure 2

Metastable checkerboard insulators at (a) $n=1/6\left(\nu =1/2\right)$ and (b) $n=1/3\left(\nu =1\right)$. These correspond to uncondensed phases which have higher energy than the corresponding superfluid ground state. The energy difference between the ground and metastable states is of the order of $0.1t$.

Figure 3

(a) Equation of state for $\varphi =1/5$ showing ground state (dashed line) and metastable (solid line). In this case, the incompressible plateau (dashed curve) at $\nu =1/2$ or $n=0.1$ corresponds to the ground state, while the superfluid has higher energy. (b) Density profile in the incompressible ground state at $n=0.1$ showing stripe order. (c) The metastable superfluid phase at $n\approx 0.1$ also supports density wave order and is analogous to a supersolid.

Figure 4

(a) Phase diagram of Harper-Hofstadter model at $\varphi =1/2$ using single-site (dashed line), $3\times 2$ (solid line), and $4\times 2$ (dots) clusters. Phase boundary separates $n=1$ Mott insulator and density wave (DW) ordered superfluid (SF). Vertical lines at $t/U\sim 0.06$ and $0.08$ mark the critical point $1\times 1$ and $3\times 2$ clusters, respectively. The results of the $4\times 2$ cluster are almost identical to the $3\times 2$ results, except at the critical point. (b) Superfluid order parameter at $\tilde{\mu }=0.4$ for a $4\times 2$ cluster. (c) Mean-field energy per site for the same parameters as in (b). At small $t$, the energy scales quadratically as $t^2/U$, as predicted by Freericks and Monien in the absence of a magnetic field [57]. The discontinuity in the slope of the energy vs $t$ marks the quantum phase transition to the superfluid. (d) Real-space structure in the superfluid showing stripes on the order of a few percent of the density. Much larger oscillations are found for the condensate order parameter (not shown).

Figure 5

Mean-field energy for two competing mean-field solutions at $\varphi =1/4$ and $\tilde{\mu }=0.9$ for $4\times 2$ (black) and $2\times 2$ (red) clusters. Dashed lines are noncondensed and correspond to states with integer total particle number, and may correspond to correlated states on top of the integer filling Mott insulator (see Fig. 6). Solid lines indicate superfluid solutions which have lower energy. Inset shows the energy difference between solid and dashed curves for $2\times 2$ (red) and $4\times 2$ (black) clusters for the same range of chemical potentials. The energy difference between superfluid and noncondensed states reduces rapidly with increasing cluster size.

Figure 6

Top: Density as a function of $\tilde{t}$ for a $4\times 2$ cluster at $\tilde{\mu }=0.9$. Solid curve shows the superfluid solution and dashed curve corresponds to noncondensed solutions with integer total particle number, or density $n=1,1+\epsilon ,1+2\epsilon$, where $\epsilon =1/8=0.125$. The superfluid always has lower energy than the noncondensed solutions, but the energy difference decreases rapidly with cluster size. Bottom: Condensate fraction $\rho /n$ as a function of $t/U$ for the superfluid solution.