Phase Separation in the Two-Dimensional Bosonic Hubbard Model with Ring Exchange

We show that soft-core bosons in two dimensions with a ring exchange term exhibit a tendency for phase separation. This observation suggests that the thermodynamic stability of normal Bose liquid phases driven by ring exchange should be carefully examined.

Figure 1

Average ground state separation $\langle {\Phi }_0|r^2|{\Phi }_0\rangle$ of two bosons, normalized to the system size, as a function of the magnitude of the ring exchange energy scale $K$. Here the hopping $t=1$ and the soft-core repulsion $U=12$. At small $K$, the normalized separation is independent of lattice size, indicating that the two bosons are spread independently throughout the lattice. At larger $K$ the bosons prefer smaller separation to optimize the ring exchange energy, indicating the formation of a bound state.

Figure 2

Typical QMC results for the average density distribution in the phase separated region. Panel (a): $U=4$, $K=2.5$, and $\rho =50/256$; panel (b): $U=4$, $K=2.5$, and $\rho =128/256$ (half filling); panel (c): $U=4$, $K=10$, and $\rho =128/256$ (half filling). At half filling, a stripe across our periodic boundary condition lattice maximizes the number of occupied plaquettes, and hence minimizes the ring exchange energy. As $K$ increases, the stripe becomes narrower, reflecting the tendency of $K$ to localize the bosons.

Figure 3

The average structure factor, $[S(2\pi /L,0)+S(0,2\pi /L)+S(2\pi /L,2\pi /L)]/3$ versus $K$. We see a sharp increase in $S$ at $K=2$ for $U=4$ and $K\approx 4$ for $U=8$. The simulations were done for $\beta =8$ and $\rho =0.5$.

Figure 4

The superfluid density as a function of $K$ for $U=8$. ${\rho }_s$ remains finite in the phase separated region, indicating that the bosons are delocalized across the clump of occupied sites. Solid circles are $16\times 16$ lattices and open squares are $24\times 24$ lattices. Inset: The hard-core limit for which, instead, ${\rho }_s\to 0$ at large $K$.

Figure 5

The phase diagram of the half-filled Bose-Hubbard model in the $U\mathrm{\text{−}}K$ plane. Below the solid line, the system is superfluid while above the line it phase separates; see text.

Figure 6

The boson density, $\rho$, versus the chemical potential, $\mu$, for hard-core bosons at $K=10$. The jump in $\mu$ at $\rho =\frac{1}{2}$ is associated with the nonzero gap in the phase with long range striped plaquette order. There are indications of a region of negative compressibility $\kappa =\partial \rho /\partial \mu$ immediately adjacent to the gapped phase.