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.

中文翻译：

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多体副费米子链中的量子临界性

我们构建了具有奇异临界点的三态 Potts 模型的局部推广。我们分析表明，这些是由 $Z_3$ parafermion 或 $u(1)_6$ 共形场论 (CFT) 乘积的非对角模不变分区函数描述的。这些对应于非平凡的置换不变量或块对角不变量，可以根据任何子凝聚来理解。在格子准费米子算子方面，构建的模型对应于具有多体项的准费米子链。我们的构建基于CFT的分配函数如何取决于对称扇区和边界条件。这使得能够将对应于一个模不变量的分区函数写成另一个在不同扇区和边界条件上的线性组合，这转化为如何写下微观模型的一般方法，调整到临界状态。我们表明该方案还可以扩展到构建 $k$-state 时钟类型模型的关键概括。