Bridging conformal field theory and parton approaches to SU(n)_k chiral spin liquids

Abstract: We employ the $\mathrm{SU}(n)_k$ Wess-Zumino-Witten (WZW) model in conformal field theory to construct lattice wave functions in both one and two dimensions. The spins on all lattice sites are chosen to transform under the $\mathrm{SU}(n)$ irreducible representation with a single row and $k$ boxes in the Young tableau. It is demonstrated that the wave functions can be reinterpreted as parton states, which enables efficient conversion to matrix product states such that many physical properties can be evaluated directly. In one dimension, these wave functions describe critical spin chains whose universality classes are in one-to-one correspondence with the WZW models used in the construction. In two dimensions, our constructions yield model wave functions for chiral spin liquids, and we show how to find all topological sectors of them in a systematic way. Using the null vectors of Kac-Moody algebras, parent Hamiltonians of the $\mathrm{SU}(3)_k$ series are derived. The $\mathrm{SU}(3)_k$ chiral spin liquids are lattice analogs of non-Abelian spin-singlet fractional quantum Hall states, and the $k=2$ member hosts Fibonacci anyons.