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arXiv:cond-mat/0511604 · Physical Review B

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

Dirk Schuricht, Martin Greiter

Published Jun 7, 2006• 13 citations
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Abstract

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 $\mathrm{SU}(n)$.

DOI
10.1103/physrevb.73.235105

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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 3‾\overline{3}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∕3g=2∕3g=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 SU(n)\mathrm{SU}(n)SU(n).
We propose two possible trial wave functions for the elementary excitations of the SU (3) Haldane-Shastry model, but then argue on very general grounds that only one or the other can be a valid excitation. We then prove explicitly that the trial wave function describing a coloron excitation which transforms according to representation under SU (3) rotations if the spins of the original model transform according to representation 3, is exact. If a basis for the spins on the chain is spanned by the colors blue, red, and green, a basis for the coloron excitations is hence given by the complementary colors yellow, cyan, and magenta. We derive the dispersion and show that colorons exist only in one-third of the Brillouin zone. We further obtain the exclusion statistics among polarized colorons and discuss the generalization to SU ( n ).
We study the one- and two-holon excitations of the SU(3) Kuramoto-Yokoyama model on the level of explicit wave functions, and generalize the calculations to the case of SU(n)\mathrm{SU}(n)SU(n). We obtain the exact energies and the single-holon momenta, which we find fractionally spaced according to fractional statistics with statistical parameter g=1∕ng=1∕ng=1∕n.
To begin with, we introduce several exact models for SU(3) spin chains: First is a translationally invariant parent Hamiltonian involving four-site interactions for the trimer chain, with a threefold degenerate ground state. We provide numerical evidence that the elementary excitations of this model transform under representation 3‾\overline{\mathbf{3}}3 of SU(3) if the original spins of the model transform under representation 3. Second is a family of parent Hamiltonians for valence bond solids of SU(3) chains with spin representations 6, 10, and 8 on each lattice site. We argue that of these three models, only the latter two exhibit spinon confinement and, hence, a Haldane gap in the excitation spectrum. We generalize some of our models to SU(n)\mathrm{SU}(n)SU(n). Finally, we use the emerging rules for the construction of valence bond solid states to argue that models of antiferromagnetic chains of SU(n)\mathrm{SU}(n)SU(n) spins, in general, possess a Haldane gap if the spins transform under a representation corresponding to a Young tableau consisting of a number of boxes $\ensuremath{\lambda}$ which is divisible by nnn. If $\ensuremath{\lambda}$ and nnn have no common divisor, the spin chain will support deconfined spinons and not exhibit a Haldane gap. If $\ensuremath{\lambda}$ and nnn have a common divisor different from nnn, it will depend on the specifics of the model including the range of the interaction.
We introduce several exact models for SU(3) spin chains: (1) a translationally invariant parent Hamiltonian involving four-site interactions for the trimer chain, with a threefold degenerate ground state. We provide numerical evidence that the elementary excitations of this model transform under representation 3‾\overline{\mathbf{3}}3 of SU(3) if the original spins of the model transform under representation 3. (2) a family of parent Hamiltonians for valence bond solids of SU(3) chains with spin representations 6, 10, and 8 on each lattice site. We argue that of these three models, only the latter two exhibit spinon confinement, and a Haldane gap in the excitation spectrum.
While the Bethe ansatz solution of the Haldane-Shastry model appears to suggest that the spinons represent a free gas of half-fermions Bernevig, Giuliano, and Laughlin (BGL) [Phys. Rev. Lett. 86, 3392 (2001); Phys. Rev. B 64, 24425 (2001)] have concluded recently that there is an attractive interaction between spinons. We argue that the dressed scattering matrix obtained with the asymptotic Bethe ansatz is to be interpreted as the true and physical scattering matrix of the excitations, and hence, that the result by BGL is inconsistent with an earlier result by Essler [Phys. Rev. B 51, 13357 (1995)]. We critically reexamine the analysis of BGL, and conclude that there is no interaction between spinons or spinons and holons in the Haldane-Shastry model.