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 translational 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 2 phases, while the phase transition between $c=4/5$ and 1 is conjectured to be of Kosterlitz-Thouless type.

• Received 24 July 2014
• Revised 25 February 2015

DOI:https://doi.org/10.1103/PhysRevB.91.115133