Spin chain techniques for angular momentum 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.