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We examine the effective field theory of the Bethe ansatz integrableHeisenberg antiferromagnetic spin chains. It shows that the quantum criticaltheories for the integer spin-S chains should be characterized by theSO(3)level-S Wess-Zumino-Witten model, and classified by the third cohomologygroup $H^{3}(SO(3),Z)=Z$. Depending on the parity of spin S, this integerclassification is further divided into two distinct universality classes, whichare associated with two completely different conformal field theories: theeven-S chains have gapless bosonic excitations and the odd-S chains have bothbosonic and fermionic excitations. We further show that these two classes ofcritical states correspond to the boundary states of two distinct topologicalphases in two dimension, which can be described by two-dimensional doubledSO(3) topological Chern-Simons theory and topological spin theory,respectively.

Classification of quantum critical states of integrable antiferromagnetic spin chains and their correspondent two-dimensional topological phases