  We study the evolution of magnetic structure driven by a synthetic spin-orbit
coupling in a one-dimensional two-component Bose-Hubbard model. In addition to
the Mott insulator-superfluid transition, we found in Mott insulator phases a
transition from a gapped ferromagnetic phase to a gapless chiral phase by
increasing the strength of spin-orbit coupling. Further increasing the
spin-orbit coupling drives a transition from the gapless chiral phase to a
gapped antiferromagnetic phase. These magnetic structures persist in superfluid
phases. In particular, in the chiral Mott insulator and chiral superfluid
phases, incommensurability is observed in characteristic correlation functions.
These unconventional Mott insulator phase and superfluid phase demonstrate the
novel effects arising from the competition between the kinetic energy and the
spin-orbit coupling.
