  In this work, we study strongly interacting spinor atoms in a lattice subject
to 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 SOC
provides a new platform to host rich and novel classes of quantum commensurate
and in-commensurate phases, excitations and phase transitions. These
commensurate 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 spectra
above all the 5 phases, especially on the IC-SKX phase are computed. Three
different classes of quantum commensurate to in-commensurate transitions from
the IC-SKX to its 4 neighboring commensurate phases are identified. Finite
temperature behaviors and transitions are discussed. The critical temperatures
of all the phases can be raised above that reachable by current cold atom
cooling techniques simply by tuning the number of atoms $ N $ per site. In view
of recent impressive experimental advances in generating 2d SOC for cold atoms
in optical lattices, these new many-body phenomena can be explored in the
current and near future cold atom experiments. Applications to various
materials such as MnSi, Fe$_{0.5}$Co$_{0.5}$Si, especially the complex
incommensurate magnetic ordering in Li$_2$IrO$_3$ are given.
