Bijection between spin $S=\frac{p^{M}-1}{2}$ and a cluster of $M$ spins $\sigma=\frac{p-1}{2}$

TL;DR

This paper introduces an exact method to decompose a high-spin into multiple lower spins, enabling dimensional reduction of complex spin models for simplified analysis.

Contribution

It presents a novel exact mapping between a high-spin $S=rac{p^{M}-1}{2}$ and a cluster of $M$ lower spins $\sigma=rac{p-1}{2}$, facilitating model simplification.

Findings

Derived an exact spin mapping for specific high-spin values.

Showed how to reduce a $d+1$ dimensional model to a $d$ dimensional one.

Discussed potential applications of the transformation in spin model analysis.

Abstract

We propose a general method by which a spin-$S$ is decomposed into spins less than $S$. We have obtain the exact mapping between spin $S=\frac{p^{M}-1}{2}$ and a cluster of $M$ spins $\sigma=\frac{p-1}{2}$. We have discuss the possible applications of such transformations. In particular we have show how a general $d+1$ dimensional spin-$\frac{p-1}{2}$ model with general interactions can be reduced to $d$-dimensional spin-$S$ model with $S=\frac{p^{M}-1}{2}$.