Novel magnetic phases for two-component Bose-Hubbard model with synthetic spin-orbit coupling in one dimension

We present a new phase diagram for the two-component Bose-Hubbard model with a synthetic spin-orbit coupling in one dimension by employing the density-matrix renormalization group method. A ferromagnetic long-range order emerges in both the Mott insulator and superfluid phases. It results from the spontaneous breaking of the $Z_2$ symmetry, when the spin-orbit coupling term becomes comparable to the hopping kinetic energy and the inter-component interaction is smaller than the intra-component one. These novel effects are expected to be detectable with the present realization of synthetic spin-orbit coupling in experiments.