Non-Abelian bosonization and Haldane’s conjecture

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 (WZW) model. This effective theory is then mapped into a compact $U(1)$ boson interacting with Z2S 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 and the SU(2)2S WZW model flows towards the SU(2)₁ stable fixed point. This gives a field theory treatment of the so-called Haldane's conjecture for arbitrary values of the spin S.