Supersolid Hard-Core Bosons on the Triangular Lattice

We determine the phase diagram of hard-core bosons on a triangular lattice with nearest-neighbor repulsion, paying special attention to the stability of the supersolid phase. Similar to the same model on a square lattice we find that for densities $\rho <1/3$ or $\rho >2/3$ a supersolid phase is unstable and the transition between a commensurate solid and the superfluid is of first order. At intermediate fillings $1/3<\rho <2/3$ we find an extended supersolid phase even at half filling $\rho =1/2$. The emergence of the supersolid on the triangular lattice reflects a novel and interesting way for a quantum system to avoid classical frustration, similar to an order-by-disorder mechanism. It also offers an exciting possibility of realizing such phenomena in ultracold atoms on optical lattices.

Figure 1

Zero-temperature phase diagram of hard-core bosons on the triangular lattice in the canonical ensemble obtained from quantum Monte Carlo simulations. The regions of phase separation are denoted by PS. The insets exhibit the density distribution inside the solid phases for $\rho =1/3$ (lower panel) and $\rho =2/3$ (upper panel).

Figure 2

Zero-temperature phase diagram of hard-core bosons on the triangular lattice in the grand-canonical ensemble obtained from quantum Monte Carlo simulations. Second-order phase transitions are denoted by solid lines, whereas first-order transitions are denoted by dashed lines.

Figure 3

Density of hard-core bosons on the triangular lattice as a function of $\mu$ along lines of constant values of $t/V$. The inset displays the jump in the density as a function of $t$ for $\mu /V=4$ at $t/V\approx 0.165$.

Figure 4

Static structure factor $S(\mathbf{Q})$ for hard-core bosons on the triangular lattice as a function of $\mu$ along a line of constant $t/V=0.1$. The inset shows the behavior of the superfluid density ${\rho }_S$.

Figure 5

The $\rho =2/3$ solid doped with bosons. (a) Additional bosons (open circles) added on top of the solid. (b) Lining the bosons up costs no additional potential energy. (c) Shifting the lower half of the lattice introduces a domain wall (dashed line) at no cost, but now (d) the additional particles can hop freely across the domain wall, gaining additional kinetic energy.

Figure 6

Finite size scaling of the static structure factor $S(\mathbf{Q})$ and the superfluid density ${\rho }_S$ for hard-core bosons on the triangular lattice at $t/V=0.1$ and $\mu /V=3$. Dashed lines are extrapolations to the infinite lattice. The inset shows the finite size scaling of the order parameter $⟨\cos (6\theta )⟩$.

Figure 7

Static structure factor $S(\mathbf{Q})$ for hard-core bosons on the triangular lattice as a function of $t$ at half filling $(\mu /V=3)$. The inset shows the behavior of the superfluid density ${\rho }_S$ with a kink at $t/V\approx 0.12$, indicated by an arrow.

Figure 8

Static structure factor $S(\mathbf{Q})$ for hard-core bosons on the triangular lattice as a function of $t$ along lines of constant $\mu /V=3.4$ and $\mu /V=4$. The inset shows the superfluid density ${\rho }_S$, exhibiting a kink at $t/V\approx 0.125$ for $\mu /V=3.4$.