Magnetic and Superfluid Transitions in the One-Dimensional Spin-1 Boson Hubbard Model

Recent progress in experiments on trapped ultracold atoms has made it possible to study the interplay between magnetism and superfluid-insulator transitions in the boson Hubbard model. We report on quantum Monte Carlo simulations of the spin-1 boson Hubbard model in the ground state. For antiferromagnetic interactions favoring singlets, we present exact numerical evidence that the superfluid-insulator transition is first (second) order for even (odd) Mott lobes. Inside even lobes, we search for nematic-to-singlet first order transitions. In the ferromagnetic case where transitions are all continuous, we map the phase diagram and show the superfluid to be ferromagnetic. We compare the quantum Monte Carlo phase diagram with a third order perturbation calculation.

Figure 1

$\rho (\mu )$ exhibits Mott plateaux: gapped, insulating phases at commensurate fillings. Inset: $N_{+}/N$ and $N_0/N$ vs $\rho =N/L$. Singlets form where $N_{+}/N$ and $N_0/N$ are equal (see text).

Figure 2

The square of the local moment, $⟨F^2⟩$, and the superfluid density, ${\rho }_s$, vs $t/U_0$ for $\rho =2$. In the Mott lobe, $⟨F^2⟩\to 0$ signaling singlet formation and ${\rho }_s\to 0$ indicating an insulator. The dashed line indicates the critical $t/U_0$ from Rizzi et al. [11]. Inset: Same but with $U_2/U_0=0.01$.

Figure 3

The density $\rho$ vs chemical potential $\mu$ exhibits the usual Mott plateaux at commensurate filling for $U_2<0$. Inset: The spin population fractions showing ${\rho }_0=2{\rho }_{+}$. They do not exhibit oscillations like in Fig. 1.

Figure 4

The magnetic structure factor $S_{xx}(q)$ for $U_2<0$ and $\rho =0.5$ exhibits a sharp $q=0$ peak indicating ferromagnetic order.

Figure 5

Phase diagram of the spin-1 BHM for $U_2/U_0=-0.1$.