Quantum criticality in many-body parafermion chains

TL;DR

This paper constructs and analyzes exotic critical points in 3-state Potts and clock models using conformal field theory, revealing new non-trivial modular invariants and their lattice realizations with parafermion chains.

Contribution

It introduces a method to construct lattice models with exotic critical points based on CFT modular invariants, including permutation and block diagonal types.

Findings

Identified non-diagonal modular invariants for $Z_3$ parafermion and $u(1)_6$ CFTs.

Developed a recipe to realize these invariants in lattice models with many-body parafermion terms.

Extended the construction to $k$-state clock models at criticality.

Abstract

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.