Triplet states, Dicke states and symmetric spin-1 states

Question

When adding two spin-1/2 particles, it is well known that the basis of the composite system can be written in terms of the spin singlet and triplet states. These sates have a well defined symmetry under particle exchange: the singlet is fully antisymmetric, while the triplet is fully symmetric. The same idea can be generalised to more than two spin-1/2 particles (e.g. Adding 3 electron spins for 3 and 4), where the fully symmetric states take in general the name of Dicke states. Dicke states are states $\vert S, m\rangle$, such that $S^2 \vert S, m\rangle = S(S+1)\vert S, m\rangle$, and $S_z\vert S, m\rangle = m \vert S, m\rangle$. Here, symmetry requires $S=N/2$ where $N$ is the number of spin-1/2 particles considered. When $N=2$, $\vert 1, -1\rangle$, $\vert 1, 0\rangle$, $\vert 1, +1\rangle$ are the triplet states.

Question: how is this generalises to spin-1? Fist, if I have two spin-1, how do I write down a basis with a well defined symmetry (analogous to singlet/triplet for spin-1/2)? How is this generalised to three or more spin-1 particles? I know that SU(3) has 8 generators (instead of just $S_{x,y,z}$), therefore I imagine I need one more quantum number to characterise fully symmetric states, with respect to just $S, m$.

I appreciate any help and clarification.

Answer 1

This does not generalize so easily, although there are special cases.

If you are looking at the $n$-fold coupling of the fundamental representation of any $SU(n)$ group, you can use Schur-Weyl duality and deduce immediately the symmetry of the resulting states from the symmetry of the dual representation of the symmetric group. For $SU(2)$ see this related answer this question.

When coupling any 2 identical angular momenta, the permutation symmetry can be deduced from the symmetry of the corresponding Clebsch-Gordan coefficient: \begin{align} C^{LM}_{\ell_1m_1;\ell_2m_2}=(-1)^{\ell_1+\ell_2-L}C^{LM}_{\ell_2m_2;\ell_1m_1} \end{align} Thus in your example the coupling of two $\ell_1=\ell_2=\ell$ angular momentum states will be symmetric if $2\ell-L$ is even and antisymmetric if $2\ell-L$ is odd. This applies to any value of $\ell$.

The strategy would hold for any two irreps of anything, but the CGs aren't so readily available so not a practical route.

For the coupling of multiple $\ell=1$ states, one can use the Schur-Weyl trick because the $\ell=1$ states are a basis for the fundamental representation of $(1,0)$ (or $\mathbf{3}$) of $SU(3)$. Then the double coupling will yield the $SU(3)$ irreps $(2,0)\oplus (0,1)$ and the branching rules give \begin{align} (2,0)\downarrow L=2\oplus L=0\, ,\qquad (0,1)\downarrow L=1 \end{align} Thus, states in $(2,0)$ are symmetric and states in $(0,1)$ antisymmetric. The branching rules were originally derived by Elliott for the nuclear $\mathfrak{su}(3)$ model. It looks like

Elliott, J.P., 1958. Collective motion in the nuclear shell model. I. Classification schemes for states of mixed configurations. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences, 245(1240), pp.128-145

deals with this but looking back at Phil Elliott's paper of that era you will certainly find what you're looking for.

One can continue in this way for the particles, which yields \begin{align} (3,0)\oplus (1,1)\oplus (1,1)\oplus (0,0) \end{align} with $(3,0)\downarrow L=3\oplus L=1$ the symmetric part, $(0,0)\downarrow L=0$ fully antisymmetric, and both copies of $(1,1)\downarrow L=2\oplus L=1$ of mixed symmetry.

In general, the decomposition of this types into irreps of specific symmetries requires the notion of plethysm and Schur functions. A reasonable review with deals with some aspects of this is

Rowe, D.J., Carvalho, M.J. and Repka, J., 2012. Dual pairing of symmetry and dynamical groups in physics. Reviews of Modern Physics, 84(2), p.711.

The basic idea is to use a substitution rule in Schur functions for a larger group so that Schur-Weyl duality can be exploited. The example of decomposing multiple copies of an $\ell=1$ states by passing to multiple copies of the fundamental of $SU(3)$ is an example of this type of procedure. This machinery requires the expansion of a (usually pretty long) polynomial and is not easy to do by hand.