Clifford Fourier Transforms in (2+1)D Lattice Simulations of Soliton Propagations

Monte Carlo simulation of solitonic phonon propagation on a 2D plane in Weyl fermion-sea is analyzed. We assume materials are filled with Weyl spinors located on $256\times 256$ 2D lattice points, which are expressed by quaternions ${\bf H}$. The topology of solitonic phonon propagation is defined by modifying fixed point actions of 4D Quantum Chromo Dynamics action to (2+1)D action, replacing Dirac fermions by Weyl fermions, and changing the electric charge current flow to the energy flow. We consider $A$ type loops whose path are on a 2D plane, and $B$ type loops which contain two parallel links that connect two 2D plane on different time slices. The length of loops are restricted to be less than or equal to 8 lattice units. At the moment spatial lattice unit and time lattice unit are same. They can be chosen arbitrarily when one compares hysteresis effects with experimental data. Using the quaternion expression of Porteous, we calculate the plaquette part of loop actions and the link part of loop actions. Link actions of $A$ type loops cancel with each other, but those of $B$ type loops depends on whether the spin rotation is clockwise or that is counterclockwise. In the present work we consider average of clockwise rotating and counterclockwise rotating loop contributions.