Criticality in Translation-Invariant Parafermion Chains

Parafermionic zero modes have been recently proposed to emerge at certain topological defects in Abelian fractional quantum Hall systems. In this work, we investigate the phase diagram of a translationally invariant $Z_3$ parafermion chain, with nearest- and next-nearest-neighbor hopping terms. The model can be mapped to a $Z_3$ Potts model with nearest-neighbor couplings via a generalized Jordan-Wigner transformation. The phase diagram is obtained numerically via accurate density matrix renormalization group method, and six gapless phases with central charges being 4/5, 1 or 2 are found. By checking the energy derivatives, we observe 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.