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