Quantum criticality in many-body parafermion chains

We construct local generalizations of 3-state Potts models with exotic critical points. analytically show that these are described by non-diagonal modular invariant partition functions products $Z_3$ parafermion or $u(1)_6$ conformal field theories (CFTs). These correspond either to non-trivial permutation invariants block diagonal invariants, one can understand in terms anyon condensation. In lattice operators, the constructed chains many-body terms. Our construction is based on how function a CFT depends symmetry sectors and boundary conditions. This enables write corresponding as linear combination another over different conditions, which translates general recipe down microscopic model, tuned criticality. scheme also be extended $k$-state clock type models.