Critical velocity of flowing supersolids of dipolar Bose gases in optical lattices

We study superfluidity of supersolid phases of dipolar Bose gases in two-dimensional optical lattices. We perform linear stability analyses for the corresponding dipolar Bose-Hubbard model in the hard-core boson limit to show that a supersolid can have stable superflow until the flow velocity reaches a certain critical value. The critical velocity for the supersolid is found to be significantly smaller than that for a conventional superfluid phase. We propose that the critical velocity can be used as a signature to identify the superfluidity of the supersolid phase in experiment.

Figure 1

(a) Ground-state phase diagram of the hard-core Bose-Hubbard model with the dipole-dipole interaction in the $(V/J,h/J)$ plane, where MI, SF, CSS, CS, and SS2 phases are present. The dashed red line represents the contour of $n=0.4$. (b) The filling factor $n$ as a function of $h/J$. The dotted red lines locate the boundaries between the different phases. The dipolar interaction is fixed to be $V=3.5J$, which is indicated by the dotted blue line in (a). (I) and (II) Schematic pictures of the CSS and SS2 phases.

Figure 2

Schematic pictures of the SF and CSS states with superflow in the spin representation. The arrows represent the local direction of the spins. The radius of the circles denotes the projection of the spins onto the $\mathit{xy}$ plane, which corresponds to the local condensate density $n_j^{\mathrm{con}}$. The red dashed lines denote the projection onto the $z$ axis corresponding to the local density $n_j$.

Figure 3

Stability phase diagrams of hard-core bosons flowing in a 2D lattice with quasimomentum $\mathbf{K}=(K,0)$. In (a) the regions of stable SF, stable CSS, stable SS2, Landau instability (LI), and dynamical instability (DI) are located when $V/J$ is varied for $n=0.4$, while in (b) those are located when the filling factor $n$ is varied for $V=3.5J$. The dotted blue, solid red, and dashed green lines represent the critical quasimomenta for LI, DI caused by phonons, and DI caused by roton-like excitations.

Figure 4

Excitation spectra $\omega (\mathbf{q})$ in the CSS phase for $V=3.5J$, $n=0.4$, and different values of $K$ in the $x$ direction, where $K=0$ (a), $K=K_{\mathrm{CSS}}^{\mathrm{L}}$ (b), and $K=0.83/d>K_{\mathrm{CSS}}^{\mathrm{D}}$ (c). (d) and (e) are magnifications of (b) and (c) focused on the region of $\left|q_x\right|\ll 1/d$ and $q_y=0$, where excitations with negative and complex energies arise. In (f), the imaginary part of $\omega (\mathbf{q})$ corresponding to (e) is shown.