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 $2S$ $\mathrm{SU}\left(2\right)\mathrm{}$ Wess-Zumino-Witten (WZW) model. This effective theory is then mapped into a compact $U\left(1\right)\mathrm{}$ 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 and the ${\mathrm{SU}\left(2\right)\mathrm{}}_{2S}$ WZW model flows towards the ${\mathrm{SU}\left(2\right)\mathrm{}}_1$ stable fixed point. This gives a field theory treatment of the so-called Haldane’s conjecture for arbitrary values of the spin $S$.