Devil’s staircases and supersolids in a one-dimensional dipolar Bose gas

We consider a single-component gas of dipolar bosons confined in a one-dimensional optical lattice, where the dipoles are aligned such that the long-ranged dipolar interactions are maximally repulsive. In the limit of zero intersite hopping and sufficiently large on-site interaction, the phase diagram is a complete devil’s staircase for filling fractions between 0 and 1, wherein every commensurate state at a rational filling is stable over a finite interval in chemical potential. We perturb away from this limit in two experimentally motivated directions involving the addition of hopping and a reduction in the on-site interaction. The addition of hopping alone yields a phase diagram, which we compute in perturbation theory in the hopping, where the commensurate Mott phases now compete with the superfluid. Further softening of the on-site interaction yields alternative commensurate states with double occupancies which can form a staircase of their own, as well as one-dimensional “supersolids” which simultaneously exhibit discrete broken symmetries and superfluidity.

Figure 1

(Color online) The devil’s staircase for $V\left(r\right)=1/r^3$, shown for fillings less than 1/2. Each filling fraction is stable over an interval $\Delta \mu$ given in Eq. (4). The resulting function is continuous, has 0 derivative almost everywhere, and possesses a fractal-like structure.

Figure 2

(Color online) Perturbative calculation of the Mott to SF phase boundary in the $\left(t,\mu \right)$ plane, shown here for $U=20$. Here the strength of all couplings is measured relative to that of the dipolar interaction strength $V_0$. Each lobe encloses a Mott insulating region in which the filling is fixed; the region outside the lobes is a superfluid of solitons. The 1/3-, 1/5-, 1/6-, 2/7-, 2/7-, and 3/7-filled lobes are shown here. Every commensurate state of the complete devil’s staircase has a Mott lobe, but the range of hoppings over which a state exists falls off sharply with its denominator. Using the values of $t/V_0$ from Table , this indicates that commensurate plateaux at 1/2 and 1/3 filling should be experimentally accessible to dipolar molecules.

Figure 3

(Color online) Numerically calculated values of $U_c^{\left(\text{CGS}\right)}$ (red) and $U_c^{\left(\text{qSS}\right)}$ (blue), for a selection of filling fractions $\nu$. Quoted values of $U$ are measured relative to $V_0$. A segment of the $y$ axis between $U_c^{\left(\text{CGS}\right)}\left(1/2\right)$ and $U_c^{\left(\text{qSS}\right)}\left(1/2\right)$ has been removed for better resolution of the rest of the phase diagram. The green and yellow boxes, bordered by black dots, indicate the approximate regions where the ground states consist of double occupancies in the 1/3- and 1/2-filled states, respectively. Between the shaded regions the ground states consist of double occupancies on other, higher-denominator states.

Figure 4

(Color online) Phase portrait in the vicinity of doubly commensurate states about 1/2 filling at $U=V_0$. Again, $\mu$, $t$, and $U$ in the figure are measured relative to $V_0$. The black lines indicate the boundaries of the Mott lobe at $U=20$. The red curve shows the chemical potential at which it becomes energetically favorable to add particles to the half-filled state as double occupancies for $U=V_0$. The blue curves show some Mott lobes of the doubly occupied staircase region; the region between these and the black lines (which delineate the region of stability of the 1/2-filled state at infinite $U$) is a supersolid state.