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We construct a family of 1D and 2D long-range SU(2) spin models as parentHamiltonians associated with infinite dimensional matrix product states thatarise from simple current correlation functions in $SU(2)_k$ WZW models. Ourresults provide a conformal field theory foundation for recent proposals byGreiter and coauthors regarding the realization of non-abelian chiral spinliquids. We explain, in particular, how the symmetrization procedure for theamplitudes of the ground state wave function suggested in Ref. arXiv:0903.4547originates from the conformal field theory description.

Conformal field theory and the non-abelian $SU(2)_k$ chiral spin liquid