Ultracold bosons with the synthetic three-dimensional spin-orbit coupling in an optical lattice

We study ultracold bosonic atoms with the synthetic three-dimensional spin-orbit (SO) coupling in a cubic optical lattice. In the superfluidity phase, the lowest energy band exhibits one, two or four pairs of degenerate single-particle ground states depending on the SO-coupling strengths, which can give rise to the condensate states with spin-stripes for the weak atomic interactions. In the deep Mott-insulator regime, the effective spin Hamiltonian of the system combines three-dimensional Heisenberg exchange interactions, anisotropy interactions and Dzyaloshinskii-Moriya interactions. Based on Monte Carlo simulations, we numerically demonstrate that the resulting Hamiltonian with an additional Zeeman field has a rich phase diagram with spiral, stripe, vortex crystal, and especially Skyrmion crystal spin-textures in each xy-plane layer. The obtained Skyrmion crystals can be tunable with square and hexagonal symmetries in a columnar manner along the z axis, and moreover are stable against the inter-layer spin-spin interactions in a large parameter region.