Tunable defect interactions and supersolidity in dipolar quantum gases on a lattice potential

Point defects in self-assembled crystals, such as vacancies and interstitials, attract each other and form stable clusters. This leads to a phase separation between perfect crystalline structures and defect conglomerates at low temperatures. We propose a method that allows one to tune the effective interactions between point defects from attractive to repulsive by means of external periodic fields. In the quantum regime, this allows one to engineer strongly correlated many-body phases. We exemplify the microscopic mechanism by considering dipolar quantum gases of ground-state polar molecules and weakly bound molecules of strongly magnetic atoms trapped in a weak optical lattice in a two-dimensional configuration. By tuning the lattice depth, defect interactions turn repulsive, which allows us to deterministically design a novel supersolid phase in the continuum limit.

Figure 1

(a) Sketch of setup: Particles are prepared in a two-dimensional self-assembled crystal with an additional quasicommensurate optical lattice with varying strength $\lambda$. Particles are colored by their numbers of nearest neighbors $N_b$. Interstitials: (b1) Classical steady state of two interstitials for $\lambda =0$. In the absence of an optical lattice, defects collapse to dislocation pairs which are attractive (arrows) and form a stable pair in a defect string [70]. Dots indicate the position of the undistorted lattice. (b2) For a deep lattice $\lambda =6$ with spacing $a_0$ the interaction between interstitials is purely repulsive. (b3) Interstitials for lattice spacing $a_0/2$ and $\lambda =6$ can form a bound pair with triangular order. Vacancies: (c1) Vacancies in a self-assembled crystal and $\lambda =0$ form defect strings, similar to interstitials [70]. Vacancies in a deep lattice with $\lambda =6$ and spacing (c2) $a_0$ and (c3) $a_0/2$ are purely repulsive.

Figure 2

Effective interaction $V_{\mathrm{eff}}$ between interstitials and vacancies in a dipolar crystal as a function of the distance $r$ for various $\lambda$ and $\mathrm{\Gamma }=15$ from a classical Monte Carlo simulation. (a) For $\lambda =0,V_{\mathrm{eff}}$ for interstitials is purely attractive (black). With increasing $\lambda$ and lattice filling $n=1$, the interaction can be tuned from attractive to purely repulsive. The dashed line is the analytical prediction $\mathrm{\Delta }F/k_{B}T=\mathrm{\Gamma }/r^3$ for a lattice model. (b) $V_{\mathrm{eff}}$ between interstitials can also be tuned with $\lambda$ in a lattice with filling $n=1/4$ (which corresponds to a lattice spacing $a_0/2)$. For $\lambda \lesssim 6$ the shape of the interaction is similar to the filling $n=1$. For $\lambda \simeq 6$, however, a bound defect configuration becomes metastable [compare with Fig. 1(b3)]. (c) Tunability of vacancy interactions for lattice filling $n=1$. $V_{\mathrm{eff}}$ changes sign with increasing $\lambda$. (d) Tunability of vacancy interactions for lattice filling close to $n=1/4$. Note that for vacancies and $\lambda >6$, hopping is basically suppressed. For interstitials, due to the pair interaction a sampling up to $\lambda =10$ is possible. Note that the roughness of the interaction is an artifact of the projection of a two-dimensional interaction to 1D. The two-dimensional potential surface for $\lambda =0$ is shown in Ref. [70].

Figure 3

(top) Superfluid fraction and (bottom) structure factor as a function of the lattice depth $\lambda$ with $r_d\approx 7.5$ for filling $n=1/4$, i.e., without defects $(N_d=0$, circle), and for $n\gtrsim 1/4$, i.e., with defects: $N_d=4$ (solid squares, interstitial density 0.02) and $N_d=8$ (open squares, interstitial density 0.04).

Figure 4

Phase diagram $r_d$ vs $\lambda$ for Eq. (1). $f_s$ and $S^{\max }$ for $r_d=15$ (dashed line) are reported in Fig. 5 (see text). The phase diagram displays the emergence of superfluid (blue), crystal (gray), and supersolid (red) phases for varying parameters $r_d$ and $\lambda$. Indeed, the white area is not easy to understand entirely, as it is probably located around a complex critical region. The study of this region goes well beyond the scope of the present article. Error bars on $r_d$ have been estimated using a finite-size scaling analysis.

Figure 5

$f_s$ and $S^{max}$ vs $\lambda$ for $N=26$ (circles) and $N=52$ (squares) considering $r_d=15$ (see dashed line in Fig. 4).

Figure 6

Quantum Monte Carlo snapshots with (a) $N=80$ and (b) $N=88$ particles with $r_d\approx 7.5$ (320 sites, squares) and $\lambda =10$. In the presence of interstitials in (b) delocalized paths emerge (thick red line) and result in a finite $f_s$. (c) Spherical averaged one-body density matrix [with the same set of parameters as in (a) and (b)] for $n=1/4$ (crystal, open squares) and for a supersolid phase on lattice (solid squares). The error bars lie within point size.

Figure 7

(top) Superfluid fraction and (bottom) structure factor of the reduced temperature $t=k_{B}T/({\hslash }^{2}n/m)$ with $\lambda =10$ and $N_d=4$ (solid squares, interstitial density 0.02) and $N_d=8$ (open squares, interstitial density 0.04).