First- and second-order superfluid–Mott-insulator phase transitions of spin-1 bosons with coupled ground states in optical lattices

We investigate the superfluid–Mott-insulator quantum phase transition of spin-1 bosons in an optical lattice created by pairs of counterpropagating linearly polarized laser beams, driving an $F_g=1$ to $F_e=1$ internal atomic transition. The whole parameter space of the resulting two-component Bose-Hubbard model is studied. We find that the phase transition is not always second order as in the case of spinless bosons, but can be first order in certain regions of the parameter space. The calculations are done in the mean-field approximation by means of exact numerical diagonalization as well as within the framework of perturbation theory.

Figure 1

Ground-state energy of the Hamiltonian (4) for ${}^{23}\mathrm{Na}$ ($U_a∕U_s\approx 0.037$ [7]). $\mu ∕U_s=1.5,2dJ∕U_s=0.125\left(1\right)$, 0.148(2), 0.157(3), 0.167(4).

Figure 2

In the shaded regions of this diagram, $a_4$ is negative and $E_g\left(\psi \right)$ has two minima at certain values of $J$. In the remaining part $a_4$ is positive and $E_g\left(\psi \right)$ has only one minimum.

Figure 3

Phase diagram for ${}^{23}\mathrm{Na}$ ($U_a∕U_s\approx 0.037$ [7]). The regions of metastable superfluid phase coexisting with the stable Mott phase and that of metastable Mott phase coexisting with the stable superfluid phase are denoted by MS and MM, respectively.