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We considered symplectic quaternions sitting on a $(2+1)D$ lattice inmomentum space which interact with nearest neighbor interactions. The action isexpressed by fixed point actions given by the superposition of blocks whoselength of the contour is 4, 6 or 8 lattice spacings.  The spinors are on a finite $2D$ plane expressed as $u_1 a e_1+ u_2 a e_2$ (Atype) and/or on two $2D$ planes separated by $\pm a e_1\wedge e_2$ (B type). Inthe A type, link actions on counter-clockwise rotating loop and on clockwiserotating loop cancel, however in the B type, the direction of links between thelower $2D$ plane and the upper $2D$ plane induces differences in the action.The time-reversal conserving and spin rotational symmetry breaking actions canbe measured. Result of Monte-Carlo simulations of $64\times 64$, $128\times128$ and $256\times 256$ spatial lattices are presented.

Monte Carlo simulations of phonon propagation in the Fermi-sea of Weyl spinors and detection of hysteresis effects using Groupoids