Phase diagrams of the Bose-Hubbard model and the Haldane-Bose-Hubbard model with complex hopping amplitudes

In this paper, we study Bose-Hubbard models on square and honeycomb lattices with complex hopping amplitudes, which are feasible by recent experiments of cold atomic gases in optical lattices. To clarify phase diagrams, we use extended quantum Monte Carlo simulations (eQMC). For the system on the square lattice, the complex hopping is realized by an artificial magnetic field. We found that vortex-solid states form for certain set of magnetic field, i.e., the magnetic field with the flux quanta per plaquette $f=p/q$, where $p$ and $q$ are co-prime natural numbers. For the system on the honeycomb lattice, we add the next-nearest-neighbor complex hopping. The model is a bosonic analog of the Haldane-Hubbard model. By means of eQMC, we study the model with both weak and strong onsite repulsions. Numerical study shows that the model has a rich phase diagram. We also found that in the system defined on the honeycomb lattice of the cylinder geometry, an interesting edge state appears.

Figure 1

Phase diagram in $f\text{−}U$ plane. The red (light gray) dotted lines indicate the $f=1/q$ vortex solid states and the blue (dark gray) dotted lines indicate the $f=p/q$ vortex solid states. The phase boundary between the solid states and the vortex quantum-liquid states is determined by the behavior of vortex lines in the imaginary time direction. See Fig. 5. $J=5$.

Figure 2

The internal energy $E$ as a function of the strength of the magnetic flux $f$. At $f\simeq 0.02$, there exists a sharp peak that indicates a phase transition. Correlation function indicates that the system loses the SF at that point. $J=5$ and $U=10$. Calculated $E$ indicates the existence of stable states (local minimums) for certain values of $f$ like $f=\frac{1}{2}$ and $\frac{1}{3}$, whereas at $f=\frac{1}{4},\frac{1}{6}$ and $\frac{1}{12}$, a plateau appears. System size $L=12$.

Figure 3

Snapshots of vortex $\mathrm{\Omega }\left(r^{\prime }\right)$ for $\overline{\rho }=2.0$ and $J=5$. They show vortex solid states in $f=p/q \left(1/9\le f\le 1/2\right)$. The results indicates 14 patterns of the vortex solid states, which are all of possible combinations of co-prime numbers $p$ and $q$. For $f=\frac{1}{2}$ and $\frac{1}{3}$, the rigid patterns appear, whereas for $f=\frac{1}{4}$ and $\frac{1}{6}$ locations of vortices are sightly loose. On the other hand for incommensurate $f$'s (0.23 and 0.27), vortices do not form a lattice.

Figure 4

Specific heat $C$ for the system size $L=12,18$ and 24. Scaling function of the FSS for $f=\frac{1}{2}$ and $f=\frac{1}{3}$.

Figure 5

The typical vortex lines in $f=1/3$ and $1/4$ along the imaginary time direction $\tau$. The red (gray) dots are the vortices residing in the dual site $r$. As the onsite repulsive interaction $U$ increases, the vortex lines are entangled, and the quantum tunneling from the vortex lattice shown in Fig. 3 to another parallel-translated one takes place. When $U$ is small, the vortex solid states do not change along imaginary time direction.

Figure 6

Numerical results of the dilute density regime. (Left panel) The phase diagram in the $\overline{\rho }\text{−}f$ plane. The red (gray) broken lines show the observed vortex solid states in which the vortices are pinned on the dual lattice. (Right panel) Snapshots for $f=1/3$. At $\overline{\rho }=2$, the robust vortex solid forms. As the density is decreased, so-called melting of the solid takes place. At present, we do not have clear understanding of the states in the dilute regime. However, we expect that they are a kind of vortex-liquid state. In the right column, we show the results for $f=1/2$ and $\overline{\rho }=0.25$, in which a bosonic fractional Hall state is expected to form. The state is rather homogeneous but slightly unstable against the MC updates.

Figure 7

Honeycomb lattice with the periodic boundary condition (torus) and also that of the cylinder geometry. The system size can be defined from $A$ sublattice size; its lattice is a square $L\times L$ lattice. The imaginary-time $\tau$ lattice is fixed $L_{\tau }=8$ and has periodic boundary conditions.

Figure 8

Phase diagrams of HBHM with $U=0.1$ (top left) and $U=10$ (top middle). Calculations of the various physical quantities are shown, which are used for identification of the phases. Between the SF and CSF, there exists a coexisting SF+CSF phase. Specific heat $C$ along the line (c) in the phase diagram (bottom right) exhibits two peaks at $J_1\simeq 1.6$ and $2.3$. The system size is $L=6$.

Figure 9

Specific heat $C$ and the scaling function of the FSS $\mathrm{\Phi }\left(x\right)$. Upper panels, the PMI-CSF phase transition. Lower panels, the PMI-SF phase transition. FSS indicates that both phase transitions are of second order. $U=10$.

Figure 10

Snapshots of vortex ${\mathrm{\Omega }}_A$ and ${\mathrm{\Omega }}_B$ in the SF, CSF, and SF+CSF phases. In the SF, the phase of BEC has a uniform distribution. On the other hand, in the CSF, a ${120}^{\circ }$ structure forms. In the SF+CSF phase, a spatial mixture of them appears.

Figure 11

Phase diagrams of HBHM in the cylinder geometry. Calculations of the various physical quantities are shown, which are used for identification of the phases. Near zigzag edges, the quasi-one-dimensional SF state forms in the bulk CSF state. Upper panel for $U=0.5$ and lower panel $U=10$. The system size is $L=6$.