Extended Bose-Hubbard model on a honeycomb lattice

We study the extended Bose-Hubbard model on a two-dimensional honeycomb lattice by using large-scale quantum Monte Carlo simulations. We present the ground-state phase diagrams for both the hard-core and the soft-core bosons. For the hard-core case, the transition between the $\rho =1∕2$ solid and the superfluid is first order, and the supersolid state is unstable toward phase separation. For the soft-core case, due to the presence of multiple occupation, a stable particle-induced supersolid (SS-p) phase emerges when $1∕2<\rho <1$. The transition from the solid at $\rho =1∕2$ to the SS-p phase is second order with the superfluid density scaling as ${\rho }_s\sim \rho -1∕2$. The SS-p phase has the same diagonal order as the solid at $\rho =1∕2$. As the chemical potential increases further, the SS-p phase turns into a solid where two bosons occupy each site of one sublattice through a first-order transition. We also calculate the critical exponents of the transition between the $\rho =1∕2$ solid and superfluid at the Heisenberg point for the hard-core case. We find the dynamical critical exponent $z=0.15$, which is smaller than results obtained on smaller lattices. This indicates that $z$ approaches zero in the thermodynamic limit, and thus the transition is also first order even at the Heisenberg point.

Figure 1

Ground-state phase diagram for two-dimensional (2D) hard-core Bose-Hubbard model on a honeycomb lattice in the grand canonical ensemble obtained from quantum Monte Carlo simulation. SF, superfluid phase. The solid line is a first-order transition, while the dashed line is second order.

Figure 2

(Color online). (a) Boson density $\rho$ as a function of chemical potential $\mu$ for $\beta =100V$ and $L=24$. (b), (c) Superfluid density ${\rho }_s$ and ${\rho }_{A-B}$ as functions of $\mu$ for $t∕V=0.4$, $\beta =100V$, and $L=12,24,30$, respectively.

Figure 3

Ground-state phase diagram for 2D soft-core Bose-Hubbard model on the honeycomb lattice obtained from quantum Monte Carlo simulation. SF, superfluid phase; SS, supersolid phase; PS, phase separation; CDW I, $\rho =1∕2$ solid with one sublattice occupied and one boson per site; CDW II, $\rho =1$ solid with one sublattice occupied and two bosons per site; MI, Mott-insulator phase with one boson per site. The solid line is first order. The dashed line is second order.

Figure 4

(Color online) (a) Boson density $\rho$ as a function of $\mu$ for $V=4t,5t,6t$ at $U=15t$ and $\beta =2L$ for soft-core case. (b), (c) Finite-size scaling of superfluid density ${\rho }_s$ and diagonal long-range order ${\rho }_{A-B}$ as a function of $\rho$ for $V=4t,6t$, respectively, for soft-core case.

Figure 5

(Color online) ${\rho }_s$ as a function of $t∕V$ for $L=16,18,24,30,36$ for $\beta =100V$ at $\mu ∕V=1.5$. The intersection at $t∕V=0.5$ is the critical point.

Figure 6

(Color online) Data collapse of the superfluid density ${\rho }_s$ for $\beta =100V$ at $\mu ∕V=1.5$. This yields the critical exponents $z=0.15$ and $\nu =0.38$.