Supersolid phases in the bosonic extended Hubbard model

We present a comprehensive numerical study on the ground state phase diagram of the two-dimensional hard core bosonic extended Hubbard model with nearest $\left(V_1\right)$ and next nearest neighbor $\left(V_2\right)$ repulsions. In addition to the quantum solid and superfluid phases, we report the existence of a striped supersolid and three-quarter (quarter)-filled supersolid at commensurate density $\rho =0.75$ (0.25) due to the interplay of $V_1$ and $V_2$ interactions. The nature of the three-quarter-filled supersolid and the associated quantum solid will be discussed. The observed quantum phase transition between the two supersolids of different symmetries is found to be first order.

Figure 1

(Color online) Ground state phase diagram of $V_2$ as a function of $V_1$ for density $\rho =0.75$. The first (second) order phase transition is denoted by dotted (solid) lines. The inset shows the boson occupation profile (the star pattern) of the QS2 and SS2 phases. The cross shows a representative point of the QS2 phase at $\rho =0.75$, where the order parameters are plotted in Fig. 2. Lattice sizes of $36\times 36$ and $28\times 28$ are used with temperature $\beta =1∕2L$.

Figure 2

(Color online) QMC result of (a) boson density $\rho$, (b) superfluidity ${\rho }_x$ and ${\rho }_y$, and (c) structure factor of wave vectors $(\pi ,\pi )$, $(0,\pi )$, and $(\pi ,0)$, as functions of chemical potential $\mu -2V_1-2V_2$. $V_1=2.0$, $V_2=6.0$, and lattice size is $28\times 28$.

Figure 3

(Color online) Finite-size scaling of QMC data for ${\rho }_x$, ${\rho }_y$, $S(0,\pi )∕N$, $S(\pi ,0)∕N$, and $S(\pi ,\pi )∕N$ of the SS2 phase with $V_1=2.0$, $V_2=6.0$, and $\mu -2V_1-2V_2=14.5$.

Figure 4

(Color online) Ground state order parameters of a $28\times 28$ lattice at $\rho =0.75$ for different $V_2$. Inset shows the first order phase transition at $V_2=6.0$.

Figure 5

(Color online) Phase diagram of $\mu$ vs $t$ for fixed $V_1=2.0$. The lines are drawn to guide the eyes, with a dashed line (solid line) for first (second) order phase transitions. The change of order parameters along the black dotted line at $V_2=6.0$ is plotted in Fig. 2, while that along the red dottedline at fixed $\rho =0.75$ is displayed in Fig. 4. MI denotes the fully occupied Mott insulating phase.

Figure 6

(Color online) VMC results of (a) the condensate $N_{k=0}$ (square) and structure factor of wave vectors $(\pi ,\pi )$ (circle) and $(0,\pi )$ (triangle) as functions of density, and (b) boson density $\rho$ as a function of the chemical potential. Here, $V_1=4$, $V_2=6$, and lattice size is $24\times 24$.

Figure 7

(Color online) (a) VMC results of the condensate $N_{k=0}$, structure factor of wavevectors $(\pi ,\pi )$ and $(0,\pi )$ as functions of $V_2$ with $V_1=4$ and the boson density $\mathrm{n}=0.25$. The inset shows the corresponding QMC data for comparison. Note that the black squares denote the superfluidity ${\rho }_y$ in the QMC data. (b) The variational parameters $v_{i,j}$ vs $r_{ij}$ for $V_2=2$, 5.5 and 8. The solid lines are the fitting functions of $A{e}^{-r_{ij}∕\xi }$. The lattice size is $24\times 24$.