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In this work, we study strongly interacting spinor atoms in a lattice subjectto a 2 dimensional (2d) anisotropic Rashba type of spin orbital coupling (SOC)and an Zeeman field. We find the interplay between the Zeeman field and the SOCprovides a new platform to host rich and novel classes of quantum commensurateand in-commensurate phases, excitations and phase transitions. Thesecommensurate phases include two collinear states at low and high Zeeman field,two co-planar canted states at Mirror reflected SOC parameters respectively.Most importantly, there are non-coplanar incommensurate Skyrmion (IC-SkX)crystal phases surrounded by the 4 commensurate phases. New excitation spectraabove all the 5 phases, especially on the IC-SKX phase are computed. Threedifferent classes of quantum commensurate to in-commensurate transitions fromthe IC-SKX to its 4 neighboring commensurate phases are identified. Finitetemperature behaviors and transitions are discussed. The critical temperaturesof all the phases can be raised above that reachable by current cold atomcooling techniques simply by tuning the number of atoms $ N $ per site. In viewof recent impressive experimental advances in generating 2d SOC for cold atomsin optical lattices, these new many-body phenomena can be explored in thecurrent and near future cold atom experiments. Applications to variousmaterials such as MnSi, Fe$_{0.5}$Co$_{0.5}$Si, especially the complexincommensurate magnetic ordering in Li$_2$IrO$_3$ are given.

Quantum incommensurate Skyrmion crystals and Commensurate to  In-commensurate transitions in cold atoms and materials with spin orbit  couplings in a Zeeman field