Valence bond solids for SU(n) spin chains: exact models, spinon confinement, and the Haldane gap

TL;DR

This paper introduces exact models for SU(3) and SU(n) spin chains, demonstrating conditions for spinon confinement and the presence of a Haldane gap, advancing understanding of quantum spin chain excitations.

Contribution

The paper constructs exact SU(3) spin chain models, analyzes spinon confinement, and generalizes results to SU(n), providing criteria for the Haldane gap based on representation properties.

Findings

Certain SU(3) models exhibit a Haldane gap due to spinon confinement.

Models with specific Young tableau divisibility conditions support a Haldane gap.

Deconfined spinons occur when the representation's Young tableau has no common divisor with n.

Abstract

To begin with, 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 three-fold degenerate ground state. We provide numerical evidence that the elementary excitations of this model transform under representation 3bar of SU(3) if the original spins of the model transform under rep. 3. (2) a family of parent Hamiltonians for valence bond solids of SU(3) chains with spin reps. 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). Finally, we use the emerging rules for the construction of VBS states to argue that models of antiferromagnetic chains of 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 \lambda which is divisible by n. If \lambda and n have no common divisor, the spin chain will support deconfined spinons and not exhibit a Haldane gap. If \lambda and n have a common divisor different from n, it will depend on the specifics of the model including the range of the interaction.