Criticality in Translation-Invariant Parafermion Chains

TL;DR

This paper investigates critical phases in translation-invariant $ ext{Z}_N$ parafermion chains, identifying multiple conformal field theories and phase transitions, including continuous and Kosterlitz-Thouless types, through numerical analysis.

Contribution

It provides a detailed numerical study of critical phases in $ ext{Z}_N$ parafermion chains with both nearest and next-nearest neighbor couplings, mapping to self-dual $ ext{Z}_N$ clock models and identifying multiple critical phases.

Findings

Identified six critical phases with central charges 4/5, 1, and 2.

Found continuous phase transitions between $c=1$ and $c=2$ phases.

Conjectured a Kosterlitz-Thouless transition between $c=4/5$ and $c=1$ phases.

Abstract

In this work we numerically study critical phases in translation-invariant $\mathbb{Z}_N$ parafermion chains with both nearest- and next-nearest-neighbor hopping terms. The model can be mapped to a $\mathbb{Z}_N$ spin model with nearest-neighbor couplings via a generalized Jordan-Wigner transformation and translation invariance ensures that the spin model is always self-dual. We first study the low-energy spectrum of chains with only nearest-neighbor coupling, which are mapped onto standard self-dual $\mathbb{Z}_N$ clock models. For $3\leq N\leq 6$ we match the numerical results to the known conformal field theory(CFT) identification. We then analyze in detail the phase diagram of a $N=3$ chain with both nearest and next-nearest neighbor hopping and six critical phases with central charges being $4/5$, 1 or 2 are found. We find continuous phase transitions between $c=1$ and $c=2$ phases, while the phase transition between $c=4/5$ and $c=1$ is conjectured to be of Kosterlitz-Thouless type.