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

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