Spin chain techniques for angular momentum quasicharacters

TL;DR

This paper extends the concept of group characters to multiple copies of SU(2), introducing irreducible quasicharacters linked to angular momentum recoupling coefficients, and provides explicit constructions and product rules for low N cases.

Contribution

It introduces irreducible quasicharacters for multiple SU(2) copies, connecting them to angular momentum recoupling coefficients and offering explicit constructions and product rules.

Findings

Quasicharacters relate to Racah and 9j coefficients.

Explicit constructions for N=2, 3, 4 quasicharacters.

Derived product rules for N=2 quasicharacters.

Abstract

We study the ring of invariant functions over the $N$-fold Cartesian product of copies of the compact Lie group $G=SU(2)$, modulo the action of conjugation by the diagonal subgroup, generalizing the group character ring. For $N=1$, an orthonormal basis for the space of invariant functions is given by the irreducible characters, and the structure constants under pointwise multiplication are the coefficients of the Clebsch-Gordan series for the reduction of angular momentum tensor products ($3j$ coefficients). For $N \ge 2$, the structure constants under pointwise multiplication of the corresponding invariants, which we term irreducible quasicharacters, are Racah $3(2N\!-\!1)j$ recoupling coefficients, which can be decomposed as products of $9j$ coefficients (for $N=2$, they are squares thereof). We identify the irreducible quasicharacters for $\times^N\! SU(2)$ with traces of representations of group elements, over totally coupled angular momentum states labelled by binary coupling trees $T$ with $N$ leaves, $N\!-\!1$ internal vertices and associated intermediate edge labels. Using concrete spin chain realizations and projection techniques, we give explicit constructions for some low degree $N=2, 3$ and $4$ quasicharacters. In the case $N=2$, related methods are used to work out the expansions of products of generic, with elementary spin-$\textstyle{\frac 12}$, quasicharacters (equivalent to an \emph{ab initio} evaluation of certain basic $9j$ coefficients). We provide an appendix which summarizes formal properties of the quasicharacter calculus known from our previous work for both $SU(2)$ and for compact $G$ (J Math Phys 59 (8) 083505 (2018) and 62(3) 033514 (2021). In particular, we provide an explicit derivation for the $N=2$ angular momentum quasicharacter product rule.