Renormalization of $(2+1)D$ scalar Weyl spinors interactions on lattices using the Clifford groups

TL;DR

This paper explores the renormalization of $(2+1)D$ scalar Weyl spinor interactions on lattices using Clifford groups, aiming to model phonon propagation in materials through a novel lattice approach and renormalization techniques.

Contribution

It introduces a lattice-based renormalization method for $(2+1)D$ scalar Weyl spinors using symplectic quaternions and applies the renormalization group to simulate phonon propagation.

Findings

Fixed point Wilson action calculated on 2D lattices.

Feasibility of numerical simulation for phonon propagation discussed.

Renormalization group method successfully applied to scalar $^4$ system.

Abstract

We consider symplectic quaternions instead of unitary spinors sitting on a lattice, and calculate the fixed point Wilson action on a finite $2D$ plane expanded by $u_1 a e_1+ u_2 a e_2$ and on two $2D$ planes separated by $a e_1\wedge e_2$. Only the nearlest neighbor interactions are considered. Following Migdal and Kadanoff, we perform the renormalization of the Wilson action by making the lattice spacing $(\frac{1}{2})^{h}a$ $(h=0,\cdots,11 )$, in order to simulate bosonic and solitonic phonon propagation in materials. Renormalization group method of Benfatto and Gallavotti for $(2+1)D$ scalar $\phi^4$ system for sound propagation in Fermi sea is applied and feasibility of numerical simulation is discussed.