Dynamical instability in the $S=1$ Bose-Hubbard model

We study the dynamical instabilities of superfluid flows in the $S=1$ Bose-Hubbard model. The time evolution of each spin component in a condensate is calculated based on the dynamical Gutzwiller approximation for a wide range of interactions, from a weakly correlated regime to a strongly correlated regime near the Mott-insulator transition. Owing to the spin-dependent interactions, the superfluid flow of the spin-1 condensate decays at a different critical momentum from a spinless case when the interaction strength is the same. We furthermore calculate the dynamical phase diagram of this model and clarify that the obtained phase boundary has very different features depending on whether the average number of particles per site is even or odd. Finally, we analyze the density and spin modulations that appear in association with the dynamical instability. We find that spin modulations are highly sensitive to the presence of a uniform magnetic field.

Figure 1

Time evolution of condensate fraction $n_{k=p}$ corresponding to three different cases: the $S=1$ Bose-Hubbard model (BHM) with $U_2/U_0=0.3$ (solid line) and $U_2/U_0=-0.3$ (dashed line), and the $S=0$ BHM (dotted line). The momentum $p$ is increased almost adiabatically in proportion to time $t$ such that $p\left(t\right)=0.005t$. We employ the system parameters $U_0=10$ and $n=1$. The decay momenta are correspondingly $p_d=0.45$ for $U_2/U_0=0.3$ and 0.44 for $U_2/U_0=-0.3$ in the $S=1$ BHM, and 0.38 in the $S=0$ BHM.

Figure 2

Time evolution of condensate fraction $n_{k=p}$ and population of each spin component $n_{\gamma }/n$ ($\gamma =0,\pm 1$) in the $S=1$ BHM with antiferromagnetic interaction $U_2/U_0=0.3$: (a) $U_0/U_{0c}=0.2$ and (b) $U_0/U_{0c}=0.8$. Here $U_{0c}(=37.9)$ is the repulsive interaction strength at the Mott-insulator transition point. We set the filling at $n=1$.

Figure 3

Dynamical phase diagrams of the $S=1$ BHM with ferromagnetic interaction $U_2/U_0=-0.3$ and the $S=0$ BHM as a function of (a) $U_0$ and (b) $U_0/U_{0c}$. A superfluid becomes dynamically unstable in the region above the phase boundary. The arrow in (a) corresponds to the horizontal axis in Fig. 1, and the cross indicates the decay momentum of the dashed line in Fig. 1. The phase boundaries of both models in (b) are completely identical. We set the filling at $n=1$.

Figure 4

Dynamical phase diagrams of the $S=1$ BHM with antiferromagnetic interaction $U_2/U_0=0.3$ and the $S=0$ BHM as a function of (a) $U_0$ and (b) $U_0/U_{0c}$. In (b), we divide the phase diagram into two regions: region 1 for $U_0/U_{0c}>0.6$ and region 2 for $U_0/U_{0c}<0.6$. We set the filling at $n=1$.

Figure 5

(a) Populations of the local spin singlet state $|S_j=0\rangle$ at $p=0$ (dashed line) and $p=p_c$ (solid line). (b) Total population of $n_j\ge 3$ states within our truncated Fock space at $p=0$ (dashed line) and $p=p_c$ (solid line). We employ $U_2/U_0=0.3$ and $n=1$ as in Fig. 4.

Figure 6

Dynamical phase diagrams in the $S=1$ and 0 BHMs for three different fillings: (a) $n=2$, (b) $n=3$, and (c) $n=4$. In (d), the results of (a) are presented as a function of $U_0/U_{0c}$. $U_2/U_0$ is fixed at 0.3 in all cases for the $S=1$ model. Critical interaction strengths are $U_{0c}=39.6(n=2), 55.2(n=3)$, and $70.7(n=4)$ for $S=0$ and $U_{0c}=11.2(n=2), 68.45(n=3)$, and $18.0(n=4)$ for $S=1$.

Figure 7

Dynamical phase diagram for incommensurate fillings: (a) $n=0.8$, (b) $n=1.2$, (c) $n=1.8$, and (d) $n=2.2$. We employ $U_2/U_0=0.3$ in all cases for the $S=1$ model.

Figure 8

Density modulation associated with the dynamical instability with no magnetic field. (a) Density distributions of $S_z=1$ (triangles), $S_z=0$ (circles), and $S_z=-1$ (squares) components over the lattice sites. The density distributions of $S_z=\pm 1$ are fully identical. (b) Deviation of densities from the mean values: particle number (light line) and magnetization (dark line). The parameters are fixed at $n=1, U_0=10$, and $U_2/U_0=0.3$. These results correspond to the point at $p/\pi =0.46$ after the decay at $p_d/\pi =0.45$ on the solid line in Fig. 1.

Figure 9

Density modulation associated with dynamical instability under a uniform magnetic field. (a) Density distributions of $S_z=1$ (triangles), $S_z=0$ (circles), and $S_z=-1$ (squares) components over the lattice sites. (b) Deviation of densities from the mean values: particle number (light line) and magnetization (dark line). The parameters are $n=1, U_0=10$, and $U_2/U_0=0.3$ as in Fig. 8. A uniform magnetic field is used to adjust the initial populations of the spin components to $n_1:n_{-1}\sim 7:3$ under the condition of $p=0$. These results correspond to the point at $p/\pi =0.44$ after the decay at $p_d/\pi =0.427$ as the momentum is increased such that $p\left(t\right)=0.005t$.