Many-Spinon States and the Secret Significance of Young Tableaux

We establish a one-to-one correspondence between the Young tableaux classifying the total spin representations of $N$ spins and the exact eigenstates of the Haldane-Shastry model for a chain with $N$ sites classified by the total spins and the fractionally spaced single-particle momenta of the spinons.

Figure 1

Total spin representations of three $S=\frac{1}{2}$ spins with Young tableaux. For $\mathrm{SU}(n)$ with $n>2$, the tableaux with three boxes on top of each other exists as well.

Figure 2

Young tableau decomposition and the corresponding spinon states for an $S=\frac{1}{2}$ spin chain with $N=4$ sites. The dots represent the spinons. For SU(2), the spinon momentum numbers $a_i$ are given by the numbers in the boxes of the same column. Note that $\sum _{}^{}(2S_{\mathrm{tot}}+1)=2^N$.

Figure 3

Examples of eigenstates of the SU(3) HSM with $N=6$ sites in terms of colorons.

Figure 4

Examples of allowed SU(3) spin representations for given SMNs in a chain with $N=6$ sites.