Composition of many spins, random walks and statistics

TL;DR

This paper analyzes the decomposition of many spins into irreducible representations using random walks and partition functions, deriving large-scale behavior and applications to ferromagnetism.

Contribution

It introduces two methods for computing spin multiplicities and explores their large-$n$ limits, including nonperturbative effects and duality relations.

Findings

Derived large-$n$ scaling limits with nonperturbative corrections

Established duality and bosonization relations for symmetric spin composition

Applied results to ferromagnetism and generalized to multiple spin types

Abstract

The multiplicities of the decomposition of the product of an arbitrary number $n$ of spin $s$ states into irreducible $SU(2)$ representations are computed. Two complementary methods are presented, one based on random walks in representation space and another based on the partition function of the system in the presence of a magnetic field. The large-$n$ scaling limit of these multiplicities is derived, including nonperturbative corrections, and related to semiclassical features of the system. A physical application of these results to ferromagnetism is explicitly worked out. Generalizations involving several types of spins, as well as spin distributions, are also presented. The corresponding problem for (anti-)symmetric composition of spins is also considered and shown to obey remarkable duality and bosonization relations and exhibit qualitatively different large-$n$ scaling properties.