Persistent Supersolid Phase of Hard-Core Bosons on the Triangular Lattice

We study hard-core bosons with unfrustrated hopping ($t$) and nearest neighbor repulsion ($U$) (spin $S=1/2$ $XXZ$ model) on the triangular lattice. At half filling, the system undergoes a zero temperature ($T$) quantum phase transition from a superfluid phase at small $U$ to a supersolid at $U_c\approx 4.45$ in units of $2t$. This supersolid phase breaks the lattice translation symmetry in a characteristic $\sqrt{3}\times \sqrt{3}$ pattern, and is remarkably stable—indeed, a smooth extrapolation of our results indicates that the supersolid phase persists for arbitrarily large $U/t$.

Figure 1

(a) A flippable pair of spins, and mapping to dimers. (b) Actual $T=0$ phase diagram and nature of spatial symmetry breaking in the supersolid phase. Darker bonds and sites indicate higher kinetic energy and density, respectively, and the state shown corresponds to ${\theta }_K={\theta }_n=0$.

Figure 2

Superfluid density ${\rho }_s$ and density wave order parameter $m^2\equiv S_{\rho }(\overrightarrow{Q},{\omega }_n=0)/\beta L^2$ extrapolated to $T\to 0$ and $L\to \infty$.

Figure 3

Extrapolations implicit in Fig. 2 shown here for two values of $U$.

Figure 4

Binder cumulant $g=1-⟨m^4⟩/3⟨m^2{⟩}^2$ as a function of $U$ in the transition region. From the location of the crossing point we identify a $T=0$ phase transition to the supersolid phase at $U\approx 4.45$. Inset: histogram of $|{\psi }_n|$ has a single peak, indicating a second-order transition.

Figure 5

Top panels: histogram of $\rho -1/2$; ${\theta }_n$ dependence of $\rho$. Bottom panels: probability distribution of ${\theta }_n$ ${\theta }_K+2{\theta }_n$, and ${\theta }_K$.