We study the long-wavelength limit of a spin $S$ Heisenberg antiferromagnetic chain. The fermionic Lagrangian obtained corresponds to a perturbed level $2S$ $\mathrm{SU}(2)$ Wess-Zumino-Witten (WZW) 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 and the ${\mathrm{SU}(2)}_{2S}$ WZW model flows towards the ${\mathrm{SU}(2)}_{1}$ stable fixed point. This gives a field theory treatment of the so-called Haldane's conjecture for arbitrary values of the spin $S$ .We study the long-wavelength limit of a spin $S$ Heisenberg antiferromagnetic chain. The fermionic Lagrangian obtained corresponds to a perturbed level $2S$ $\mathrm{SU}(2)$ Wess-Zumino-Witten (WZW) 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 and the ${\mathrm{SU}(2)}_{2S}$ WZW model flows towards the ${\mathrm{SU}(2)}_{1}$ stable fixed point. This gives a field theory treatment of the so-called Haldane's conjecture for arbitrary values of the spin $S$.

arXiv:hep-th/9802014Physical review. B, Condensed matterJul 1, 199824 citations

W Algebra in the SU(3) Parafermion Model

Xiang-mao Ding A, Heng Fan, Kangjie Shi +2 more

Physical Review LettersAug 16, 19930 citations

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}(2)

Wess-Zumino-Witten (WZW) 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 and the

{\mathrm{SU}(2)}_{2S}

WZW model flows towards the

{\mathrm{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