Frustration-induced supersolids in the absence of intersite interactions

We discuss a mechanism for the realization of supersolids in lattices in the absence of intersite interactions that surprisingly works as well at unit filling. This mechanism, that we study for the case of the sawtooth lattice, is based on the existence of frustrated and unfrustrated plaquettes. For sufficiently large interactions and frustration the particles gather preferentially at unfrustrated plaquettes breaking spontaneously translational invariance, resulting in a supersolid. We show that for the sawtooth lattice the supersolid exists for a large region of parameters for densities above half filling. Our results open a feasible path for realizing supersolids in existing ultracold atomic gases in optical lattices without the need for long-range interactions.

Figure 1

Sawtooth lattice discussed in the paper. The V-shaped plaquettes are unfrustrated, whereas the $▵$-shaped plaquettes are frustrated, since we assume $t>0$ and $t^{\prime }<0$. The presence of frustrated and unfrustrated plaquettes is crucial for the supersolid mechanism discussed in the paper.

Figure 2

Single particle dispersion for Model (1) with $U=0$ for different values of $|t^{\prime }|/t$.

Figure 3

Phase diagram of model (1) for $\rho =1$ as a function of $|t^{\prime }|/t$ and $U/t$ (see text).

Figure 4

Extrapolated gap $E_G$ as a function of $|t^{\prime }|/t$. The inset shows the exponential to power-law decay of the single particle correlation function ${\mathrm{\Gamma }}_r$ at the MI-SS boundary. Only odd $r$ is shown to avoid oscillations.

Figure 5

(a) Extrapolated values of $K$ (black squares) with respect to $|t^{\prime }|/t$ for $U/t=3$. The $K=2$ line (red dashed) is drawn to find the critical point for the ${\mathrm{SF}}_0$-MI transition. (b) Scaled momentum distribution $N\left(k\right)L^{-3/4}$ for different lengths as a function of $|t^{\prime }|/t$ for $U/t=3$.

Figure 6

Number fluctuation between odd and even sites of the $A$ and $B$ legs; the inset shows ${\mathrm{\Delta }}_n$ and ${\mathrm{\Delta }}_D$.

Figure 7

(Top) Scaled structure factor $S(k=\pi /2)$ with respect to $|t^{\prime }|/t$ for $U/t=3.0$. (Bottom) Finite-size scaling of $S(k=\pi /2)$ for $U/t=3.0$ and different $|t^{\prime }|/t$ values.

Figure 8

Phase diagram of the sawtooth chain as a function of $\rho$ and $|t^{\prime }|/t$ for $U/t=2.0$. Black lines with squares show the gapped regions. The SS phase is bounded by the red circles. The ${\mathrm{SF}}_{\pi /2}$ phase occupies a large portion of the phase diagram below the SS region and for $|t^{\prime }|/t>1/\sqrt{2}$, also around $\rho =1$. The region bounded by magenta triangles shows a macroscopic jump in the density. The brown dashed lines are two representative cuts at $|t^{\prime }|/t=1/\sqrt{2}$ and $1.5$ (see text). The red dot corresponds to the flat-band point where $|t^{\prime }|/t=1/\sqrt{2}$.

Figure 9

$\rho \left(\mu \right)$ for $L=40, U/t=2$, and $|t^{\prime }|/t=1/\sqrt{2}$ (blue curve) and $1.5$ (red curve).

Figure 10

Structure factor $S(k=\pi /2)$ for different fillings $\rho$ for $L=40, U/t=2$, and $|t^{\prime }|/t=1/\sqrt{2}$ (blue circles) and $1.5$ (red squares).

Figure 11

Sketch of the ${\mathrm{SF}}_0$ (a), ${\mathrm{SF}}_{\pi /2}$ (b), and SS (c). The circle size is proportional to the local density, whereas $\pm$ denote the sign. (d) Phase diagram in the weak-coupling limit as function $U\rho$ and $|t^{\prime }|/t$ for a fixed (but arbitrary) filling. The ${\mathrm{SF}}_0$ to the SS phase is shown as a solid line. The dotted and dashed lines depict, respectively, the ${\mathrm{SF}}_{\pi }/2$-SS transition line obtained from the variational approach (6) and the roton instability. At $t=0$ SS-order vanishes (thick solid line) resulting in a highly degenerate ground state, since particles may occupy the uncoupled $B$ sites in an arbitrary configuration.

Figure 12

Densities of the classical model for $\rho U=2$ and (a) $t=1$ and $t^{\prime }=-0.5$ and (b) $t=0.8$ and $t^{\prime }=-1$ as a function of the chemical potential $\mu$. The dotted (blue) line depicts the average density of the $B$ sites; the solid (black) and dashed (red) lines show densities of odd and even $A$ sites.

Figure 13

Bogoliubov spectrum $\epsilon \left(k\right)$ for $|t^{\prime }|/t=1$ and (from top to bottom) $\rho U=0, 0.1, 0.2, 0.3, 0.4, 0.5, 0.5278$. At $k=0$ a rotonlike minimum develops, which touches zero at $\rho U\simeq 0.5278$.

Figure 14

Sawtooth lattice formed by an incoherent superposition between a kagome lattice [20] and an additional lattice $V\left(\vec{r}\right)={sin}^2\left(\frac{\sqrt{3}}{4}ky\right)+\frac{1}{4}{sin}^2\left(\frac{\sqrt{3}}{2}ky\right)$ with $k$ the laser wave number. Darker regions mean lower potential.

Figure 15

Momentum distribution $N\left(k\right)$ in different regions of the phase diagram shown in Fig. 3 for $U=3$.

Figure 16

$N\left(0\right)$ (red squares) and $N(\pi /2)$ (blue circles) in different regions of the phase diagram shown in Fig. 3 at $U=3$.