QUROPE Quantum Information Processing and Communication in Europe

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