Hard-Core Bosons on the Kagome Lattice: Valence-Bond Solids and Their Quantum Melting

Using large scale quantum Monte Carlo simulations and dual vortex theory, we analyze the ground state phase diagram of hard-core bosons on the kagome lattice with nearest-neighbor repulsion. In contrast with the case of a triangular lattice, no supersolid emerges for strong interactions. While a uniform superfluid prevails at half filling, two novel solid phases emerge at densities $\rho =1/3$ and $\rho =2/3$. These solids exhibit an only partial ordering of the bosonic density, allowing for local resonances on a subset of hexagons of the kagome lattice. We provide evidence for a weakly first-order phase transition at the quantum melting point between these solid phases and the superfluid.

Figure 1

Ground state phase diagram of hard-core bosons on the kagome lattice (inset). The primitive vectors ${\mathbf{a}}_1$ and ${\mathbf{a}}_2$ are constrained on a (periodic) torus spanned by ${\mathbf{L}}_1=n_1\times {\mathbf{a}}_1$ and ${\mathbf{L}}_2=n_2\times {\mathbf{a}}_2$ (where $a_1=a_2=2$). The circles illustrate the subset of hexagons with a resonating boson occupation of three bosons per hexagon in the $\rho =2/3$ solid. The remaining bosons localize to form a solid backbone on the sites that do not belong to any of these hexagons.

Figure 2

Contour plots of the structure factor (left panel) and the static susceptibility (right panel) in the VBS close to the phase boundary for $\mu /V=11/12$, $t/V=1/8$, and $T=t/24$. The BZ of the underlying triangular lattice is hexagonal (see Fig. 1), with corners at $\mathbf{Q}=(2\pi /3,0)$ and symmetry related momenta. In both panels, the axes range from $-2\pi$ to $2\pi$, and ferromagnetic peaks at reciprocal lattice vectors have been suppressed.

Figure 3

The superfluid density ${\rho }_s$ as a function of $t/V$ at half filling for $\mu /V=2$ and $T=t/5$. The lattice linear dimension is denoted $L$ (see Fig. 1), so that the number of sites is $N=L\times L\times 3$. Inset: The static structure factor $S(\mathbf{Q})$, superfluid density ${\rho }_s$, and boson density $\rho$ as a function of $\mu /V$ at $t/V=1/10$, for $L=24$ and $T=t/24$.

Figure 4

Finite-size scaling of the bond structure factor $S_b(\mathbf{Q})$ in the VBS phase for $\mu /V=11/12$, $t/V=1/8$, and $T=t/12$. Error bars are smaller than the symbol sizes, and the line shows a linear extrapolation to the thermodynamic limit.

Figure 5

The correlation function $C_b({\mathbf{r}}_1-{\mathbf{r}}_{\delta })$ between the bond indicated by large (red) text and the other bonds for $L=24$, $\mu /V=11/12$, $t/V=1/8$, and $T=t/12$.

Figure 6

Data collapse of the superfluid density ${\rho }_s$ and static susceptibility $\chi (\mathbf{Q})$ at $\mu /V=11/12$ for $\beta /L=1/t$.

Figure 7

Distribution (arbitrary units) of the kinetic energy close to the transition point for various system sizes, $\mu /V=11/12$, $t/V=1/7.796$, and $T=t/60$. As the system size increases, the double-peaked features become more pronounced.