Bose-Hubbard model on a triangular lattice with diamond ring exchange

Ring-exchange interactions have been proposed as a possible mechanism for a Bose-liquid phase at zero temperature, a phase that is compressible with no superfluidity. Using the stochastic Green function algorithm (SGF), we study the effect of these interactions for bosons on a two-dimensional triangular lattice. We show that the supersolid phase that is known to exist in the ground state for a wide range of densities is rapidly destroyed as the ring-exchange interactions are turned on. We establish the ground-state phase diagram of the system, which is characterized by the absence of the expected Bose-liquid phase.

Figure 1

The triangular lattice and the effect of different terms of the Hamiltonian. The usual kinetic term $t$ allows the particles to hop between near-neighboring sites. The ring-exchange term $K$ performs a correlated hopping of two particles within the same diamond. This process is possible only if the diamond contains exactly two particles that are either opposite first neighbors (green diamond) or second neighbors (blue and red diamonds). The presence of a pair of first-neighboring particles is penalized by the potential $V$ (yellow). For a given diamond $\diamond$ we label the sites in a counterclockwise fashion, $\diamond 1, \diamond 2, \diamond 3, \diamond 4$, starting from one of the opposite first neighbors (cyan diamond).

Figure 2

The superfluid density ${\rho }_s$ as a function of $K/t$ for $V/t=0,10$ and $\rho =\frac{1}{3},\frac{1}{2}$.

Figure 3

The density as a function of the chemical potential for $V/t=10$ and different values of $K/t$. The error bars are about the size of the purple symbols.

Figure 4

The solid and the supersolid phases. The solid phase ($\rho =\frac{1}{3}$) is formed by distributing a maximum number of particles without creating pairs of first neighbors. Any attempt to dislocate one particle (white $\to$ cyan) has an energy cost of $2V$, since two sites would have occupied first neighbors which we term links. The supersolid phase is obtained by adding extra particles to the solid phase ($\rho >\frac{1}{3}$). The presence of an extra particle (red) is associated with the formation of three links with a total energy of $3V$. This extra particle is unable to break the underlying solid structure, because moving one of the extra particle's neighbors (white $\to$ cyan) away would destroy only one link while creating two new extra links, resulting in an energy increase of $V$. However, the extra particle is able to move freely across the solid structure, and can thus generate winding associated with superfluidity.

Figure 5

Intensity plots of the structure factor for $K/t=0$ for a $24\times 24$ lattice. The nature of the phase transitions can be directly seen. The sudden change in the symmetry of $S\left(\vec{k}\right)$ from $\mu /V=0.4$ to $\mu /V=0.5$ is the signature of a first-order transition, while the continuous change of the intensity from $\mu /V=2.4$ to $\mu /V=3$ reveals a second-order transition.

Figure 6

Structure factor for $K/t=2$ for a $24\times 24$ lattice. The sudden change in the intensity of $S\left(\vec{k}\right)$ from $\mu /V=0.6$ to $\mu /V=0.7$ and from $\mu /V=1.5$ to $\mu /V=1.6$ is the signature of a first-order transition.

Figure 7

Structure factor for $K/t=4$ for a $24\times 24$ lattice. The symmetry of $S\left(\vec{k}\right)$ remains the same for all fillings, with a slight dependence in $\mu$ of the intensity.

Figure 8

The superfluid density and the structure factor as functions of the inverse system size, for sizes up to $42\times 42$ and $K/t=0.1$ at half-filling. Both quantities extrapolate to a finite value in the thermodynamic limit, which is the signature of the supersolid phase.

Figure 9

The superfluid density and the structure factor as functions of the inverse system size, for sizes up to $42\times 42$ and $K/t=0.5$ at half-filling. While the superfluid density extrapolates to a finite value in the thermodynamic limit, the structure factor decays as a power law yielding a superfluid phase.

Figure 10

The zero-temperature phase diagram for $V/t=10$. The border (dashed line) between the supersolid and superfluid regions is symbolic, only the point $K_c$ on this border is obtained from QMC simulations. The exact shape will be determined in further work.