Criticality in Translation-Invariant Parafermion Chains

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 $3\le N\le 6$ 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.

published version

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• A Brief History of Free Parafermions
• Dynamical topological excitations in parafermion chains
• Twofold twist defect chains at criticality