Non-Abelian Bosonization and Haldane's Conjecture

TL;DR

This paper provides a field theory approach to Haldane's conjecture, showing that integer spins in antiferromagnetic chains have a mass gap, while half-integer spins do not, using non-Abelian bosonization techniques.

Contribution

It introduces a novel field theory framework for analyzing spin chains, extending Haldane's conjecture to arbitrary spin values via non-Abelian bosonization.

Findings

Integer spins have a mass gap in the excitation spectrum.

Half-integer spins lack a mass gap, exhibiting gapless excitations.

The effective theory maps to a U(1) boson interacting with Z_{2S} parafermions.

Abstract

We study the long wavelength limit of a spin S Heisenberg antiferromagnetic chain. The fermionic Lagrangian obtained corresponds to a perturbed level 2S SU(2) Wess-Zumino-Witten model. This effective theory is then mapped into a compact U(1) boson interacting with Z_{2S} parafermions. The analysis of this effective theory allows us to show that when S is an integer there is a mass gap to all excitations, whereas this gap vanishes in the half-odd-integer spin case. This gives a field theory treatment of the so-called Haldane's conjecture for arbitrary values of the spin S.