Phase diagrams of Bose-Hubbard model and antiferromagnetic spin-1/2 models on a honeycomb lattice

Motivated by the recent experimental realization of the Haldane model by ultracold fermions in an optical lattice, we investigate phase diagrams of the hard-core Bose-Hubbard model on a honeycomb lattice. This model is closely related with a spin-1/2 antiferromagnetic (AF) quantum spin model. Nearest-neighbor (NN) hopping amplitude is positive and it prefers an AF configuration of phases of Bose-Einstein condensates. On the other hand, an amplitude of the next-NN hopping depends on an angle variable as in the Haldane model. Phase diagrams are obtained by means of an extended path-integral Monte Carlo simulation. Besides the AF state, a ${120}^{\circ }$-order state, there appear other phases including a Bose metal in which no long-range orders exist.

Figure 1

Phase diagrams of the Bose-Hubbard model on the honeycomb lattice with $V=\varphi =0$. (a) $V_0=5$. (b) $V_0=0.5$. There are four phases, antiferromagnetic (AF), Bose metal (BM), collinear (CL), and ${120}^{\circ }$-order state. Lattice size is small, $(L_x,L_y)=(3,4)$.

Figure 2

$x$, $y$, and $x+y$ directions on the honeycomb lattice.

Figure 3

Spin (phase ${\theta }_i$) configurations of the three ordered states, antiferromagnetic (AF), collinear (CL), and ${120}^{\circ }$-order phase in the phase diagram in Fig. 1.

Figure 4

Phase diagrams of the Bose-Hubbard model on the honeycomb lattice with $V=\varphi =0$. (a) $V_0=5$. (b) $V_0=0.5$. There are unidentified phases besides antiferromagnetic (AF), Bose metal (BM), collinear (CL), and phase with the ${120}^{\circ }$ order. Lattice size, $(L_x,L_y)=(6,6)$.

Figure 5

Specific heat $C$ for the case of $V=\varphi =0$. (a) $(L_x,L_y)=(3,4)$ and $V_0=5$, (b) $(L_x,L_y)=(3,4)$ and $V_0=0.5$, (c) $(L_x,L_y)=(6,6)$ and $V_0=5$, and (d) $(L_x,L_y)=(6,6)$ and $V_0=0.5$. Sharp peaks in the system with $(L_x,L_y)=(6,6)$ indicate that the transition is of first order. This is verified by the measurement of the internal energy $E$.

Figure 6

Correlation functions in Eq. (16) for the phases in Fig. 4. (a) AF, (b) BM, (c) CL, (d) ${120}^{\circ }$-order state, and (e, f) are phases the existence of which is indicated by the calculation of the specific heat.

Figure 7

Density distribution in the momentum space $n\left(\mathbf{k}\right)$ defined by Eq. (17) for the phases in Fig. 4. (a) AF, (b) BM, (c) CL, (d) ${120}^{\circ }$-order state, and (e, f) are phases the existence of which is indicated by the calculation of the specific heat. The dotted line denotes the boundary of the Brillouin zone.

Figure 8

Finite-$T$ phase transitions for the (a) AF, (b) BM, and (c) ${120}^{\circ }$-order state phases, respectively. The thermal specific heat $C_T$ exhibits a sharp peak for the AF and the ${120}^{\circ }$-order phase, whereas there is no signal of the phase transition for the BM. This result indicates that the BM does not have any long-range orders. $C_T$ for the AF and the ${120}^{\circ }$-order phase approaches to a constant close to unity at low-$T$ limit. $V_0=5$.

Figure 9

Hopping terms in the BHHM. The NNN hopping amplitudes with an arrow are $J_2{e}^{i\varphi }$. Case $O$ ($M$) refers to the original (modified) model.

Figure 10

Phase diagrams of the original (left) and modified (right) BHHMs. Finite $\varphi$ makes the system less frustrated, and clear phase boundaries are obtained. $V_0=5$ and the system size $(L_x,L_y)=(6,6)$.

Figure 11

Properties of the $\mathbf{k}=(-\pi /\sqrt{3},0)$ phase. (a) Spin configuration, (b) correlation function, and (c) density distribution in the momentum space.

Figure 12

Phase diagrams of the BHHM with the NN interaction $V>0$. $J_1=10$ and $V_0=5$. (a) Original BHHM with $J_2=3$, (b) original BHHM with $J_2=20$, (c) modified BHHM with $J_2=3$, and (d) modified BHHM with $J_2=20$. For $J_2=20$, the phase diagrams of the original and modified BHHM are almost the same as the NNN hopping dominating over the NN hopping. The system is at half filling with $\mu /V=1.5$.

Figure 13

Internal energy $E$ and specific heat $C$ measured in the phases in Fig. 12. $V_0=5$. (a) $E$ as a function of $\varphi$ in the original BHHM with $J_1=10,J_2=3$, and $V=25$. (b) $C$ as a function of $\varphi$ in the original BHHM with $J_1=10,J_2=3$, and $V=25$. (c) $C$ as a function of $V/J_1$ of the original BHHM with $J_2=3.0$. Phase transition between the AF and CDW is of first order, and that between the BM and CDW is of second order. The system is at half filling with $\mu /V=1.5$, where $\mu$ is the chemical potential of the BHHM.