Parafermionization, bosonization, and critical parafermionic theories

TL;DR

This paper introduces a parafermionization scheme for 1D models, revealing new critical theories that differ from known conformal field theories, and provides explicit partition functions for these models.

Contribution

It develops a generalized parafermionization method extending Jordan-Wigner transformation, enabling analysis of novel critical parafermionic theories beyond existing conformal field theories.

Findings

Critical parafermionic theories cannot be described by existing CFTs.

Partition functions transform unconventionally under modular transformations for k>2.

Explicit partition functions for a broad class of critical parafermionic chains are obtained.

Abstract

We formulate a $\mathbb{Z}_k$-parafermionization/bosonization scheme for one-dimensional lattice models and field theories on a torus, starting from a generalized Jordan-Wigner transformation on a lattice, which extends the Majorana-Ising duality at $k=2$. The $\mathbb{Z}_k$-parafermionization enables us to investigate the critical theories of parafermionic chains whose fundamental degrees of freedom are parafermionic, and we find that their criticality cannot be described by any existing conformal field theory. The modular transformations of these parafermionic low-energy critical theories as general consistency conditions are found to be unconventional in that their partition functions on a torus transform differently from any conformal field theory when $k>2$. Explicit forms of partition functions are obtained by the developed parafermionization for a large class of critical $\mathbb{Z}_k$-parafermionic chains, whose operator contents are intrinsically distinct from any bosonic or fermionic model in terms of conformal spins and statistics. We also use the parafermionization to exhaust all the $\mathbb{Z}_k$-parafermionic minimal models, complementing earlier works on fermionic cases.