We study the odd-integer filled Mott phases of a spin-1 Bose-Hubbard chain and determine their fate in the presence of a Raman induced spin-orbit coupling which has been achieved in ultracold atomic gases; this system is described by a quantum spin-1 chain with a spiral magnetic field. The spiral magnetic field initially induces helical order with either ferromagnetic or dimer order parameters, giving rise to a spiral paramagnet at large field. The spiral ferromagnet-to-paramagnet phase transition is in a universality class with critical exponents associated with the divergence of the correlation length $\ensuremath{\nu}\ensuremath{\approx}2/3$ and the order-parameter susceptibility $\ensuremath{\gamma}\ensuremath{\approx}1/2$. We solve the effective spin model exactly using the density-matrix renormalization group, and compare with both a large-
S classical solution and a phenomenological Landau theory. We discuss how these exotic bosonic magnetic phases can be produced and probed in ultracold atomic experiments in optical lattices.
We study the zero-temperature phase diagram of the disordered spin-
1 Bose-Hubbard model in a two-dimensional square lattice. To this aim, we use a mean-field Gutzwiller ansatz and a probabilistic mean-field perturbation theory. The spin interaction induces two different regimes, corresponding to a ferromagnetic and antiferromagnetic order. In the ferromagnetic case, the introduction of disorder reproduces analogous features of the disordered scalar Bose-Hubbard model, consisting in the formation of a Bose glass phase between Mott insulator lobes. In the antiferromagnetic regime, the phase diagram differs more from the scalar case. Disorder in the chemical potential can lead to the disappearance of Mott insulator lobes with an odd-integer filling factor and, for sufficiently strong spin coupling, to Bose glass of singlets between even-filling Mott insulator lobes. Disorder in the spinor coupling parameter results in the appearance of a Bose glass phase only between the
n and the
n+1 lobes for
n odd. Disorder in the scalar Hubbard interaction inhibits Mott insulator regions for occupation larger than a critical value.
In this study, we perform a strong-coupling expansion up to third order of the hopping parameter
t for the spin-1 Bose-Hubbard model with antiferromagnetic interaction. As expected from previous studies, the Mott-insulator phase is considerably more stable against the superfluid phase when filling with an even number of bosons than when filling with an odd number of bosons. The phase-boundary curves are consistent with the perturbative mean-field theory in the limit of infinite dimensions. The critical value of the hopping parameter
tC at the peak of the Mott lobe depends on the antiferromagnetic interaction. This result indicates the reliability of the strong-coupling expansion when
U2 possesses large (intermediate) values for Mott lobe with an even (odd) number of bosons. Moreover, in order to improve our results, we apply a few extrapolation methods up to infinite order in
t. The fitting results of the phase-boundary curves agree better with those of the perturbative mean-field approximation. In addition, the linear fit error of
tC is very small for the strong antiferromagnetic interaction.