Coloron excitations of the SU(3) Haldane-Shastry model

In a recent publication, we proposed two possible wave functions for the elementary excitations of the SU(3) Haldane-Shastry model, but argued on very general grounds that only one or the other can be a valid excitation. Here we provide the explicit details of our calculation proving that the wave function describing a coloron excitation which transforms according to representation $\overline{3}$ under SU(3) rotations if the spins of the original model transform according to representation 3, is exact. We further provide an explicit construction of the exact color-polarized two-coloron eigenstates, and thereby show that colorons are free but that their relative momentum spacings are shifted according to fractional statistics with parameter $g=2∕3$. We evaluate the SU(3) spin currents. Finally, we interpret our results within the framework of the asymptotic Bethe ansatz and generalize some of them to the case of $\mathrm{SU}\left(n\right)$.

Figure 1

(Color online) Weight diagrams of the SU(3) representations $3$ and $\overline{3}$ (reproduced with permission from Ref. 14). $J^3$ and $J^8$ denote the diagonal generators.[25]

Figure 2

(Color online) (a) Total antisymmetric $N$-particle state. (b) Typical configuration in $\mid {\Psi }_0⟩$ (reproduced with permission from Ref. 14).

Figure 3

One-coloron dispersion relation (reproduced with permission from Ref. 14).

Figure 4

Weight diagrams of the SU(3) representations $\overline{3}$ and $\overline{6}$.

Figure 5

$\mathrm{SU}\left(n\right)$ spinon dispersion relations. (a) $n$ even. The allowed momenta fill the interval $[\pi -\frac{\pi }{n},\pi +\frac{\pi }{n}]$ for $M$ even and $[-\frac{\pi }{n},\frac{\pi }{n}]$ for $M$ odd. (b) $n$ odd. The allowed momenta fill the interval $[\pi -\frac{\pi }{n},\pi +\frac{\pi }{n}]$.

Figure 6

Contours for integrations.