Stability of supercurrents in a superfluid phase of spin-1 bosons in an optical lattice

We study collective modes and superfluidity of spin-1 bosons with antiferromagnetic interactions in an optical lattice based on the time-dependent Ginzburg-Landau (TDGL) equation derived from the spin-1 Bose-Hubbard model. Specifically, we examine the stability of supercurrents in the polar phase in the vicinity of the Mott insulating phase with even filling factors. Solving the linearized TDGL equation, we obtain gapless spin-nematic modes and gapful spin-wave modes in the polar phase that arise due to the breaking of $S^2$ symmetry in spin space. Supercurrents exhibit dynamical instabilities induced by growing collective modes. In contrast to the second-order phase transition, the critical momentum for mass currents is finite at the phase boundary of the first-order superfluid-Mott insulator (SF-MI) phase transition. Furthermore, the critical momentum remains finite throughout the metastable SF phase and approaches zero towards the phase boundary, at which the metastable SF state disappears. We also study the stability of spin currents motivated by recent experiments for spinor gases. The critical momentum for spin currents is found to be zero, where a spin-nematic mode causes the dynamical instability. We investigate the origin of the zero critical momentum for spin currents and find it attributed to the fact that the polar state becomes energetically unstable even in the presence of an infinitesimal spin current. We discuss implications of the zero critical momentum for spin currents for the stability of the polar state.