  In this work, we investigate the possible dramatic effects of Rashba or
Dresselhaus spin-orbit coupling (SOC) on fermionic Hubbard model in a 2d square
lattice. In the strong coupling limit, it leads to the Rotated
Anti-ferromagnetic Heisenberg model which is a new class of quantum spin model.
For a special equivalent class, we identify a new spin-orbital entangled
commensurate ground ( Y-y ) state subject to strong quantum fluctuations at
$T=0$. We evaluate the quantum fluctuations by the spin wave expansion up to
order $ 1/S^2 $. In some SOC parameter regime, the Y-y state supports a massive
relativistic in-commensurate magnon ( C-IC ) with its two gap minima positions
continuously tuned by the SOC parameters. The C-IC magnons dominate all the low
temperature thermodynamic quantities and also lead to the separation of the
peak positions between the longitudinal and the transverse spin structure
factors. In the weak coupling limit, any weak repulsive interaction also leads
to a weak Y-y state. There is only a crossover from the weak to the strong
coupling. High temperature expansions of the specific heats in both weak and
strong coupling are presented. The dramatic roles to be played by these C-IC
magnons at generic SOC parameters or under various external probes are hinted.
Experimental applications to both layered noncentrosymmetric materials and cold
atom are discussed.
