A spin-1 dipolar Bose-Einstein condensate with spin-orbit coupling (SOC) in a quasi-one-dimensional harmonic trap is studied. We focus on ground state domain distribution calculated numerically with spinor Gross-Pitaevskii equation. It was found that the domain structures are greatly affected by dipole-dipole interactions (DDIs). Spatial symmetry is broken and transition behaviors of distribution pattern are shown as SOC strength and magnetization are increased.

Two-dimensional Compass-XY model on a square lattice is investigated by means of Monte Carlo method. Effects of decreasing frustration of exchange interactions on directional and topological orders are studied by adjusting crossover parameter α. It is demonstrated that crossover between directional and topological orders is a gradual process,which means that there is no distinct barrier. Two kinds of ordering are partly overlapped within a certain range,making it difficult to tell them apart. Decrease of frustration facilitates topological order while suppressing directional order. Directional order may be destroyed as frustration is weak enough.