Criticality in Translation-Invariant Parafermion Chains


In this work, we numerically study critical phases in translation-invariant ZN parafermion chains with both nearest- and next-nearest-neighbor hopping terms. The model can be mapped to a ZN 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 ZN clock models. For 3$łeq$N$łeq$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.

Phys. Rev. B