Free parafermionic quantum spin systems
Since the solution of the Ising model in two dimensions free fermionic models played an important rule in the understanding of the critical properties of Statistical Mechanics models. In this talk we discuss the physical properties of a N-state Z(N) quantum spin chain described in terms of free parafermions. Differently from the fermionic case (N = 2) for N > 2 the related Hamiltonian is non-Hermitian. The eigenspectrum although complex has a real ground-state. The finite-size behavior of the eigenspectrum enable us to calculate some of the critical exponents for arbitrary N. We are going to show that although having a quite simple eigenspectra the model has the same critical exponents as the super integrable chiral Potts model, with specific heat exponent α = 1 − 2/N and the anisotropic correlation lenght exponents νk = 1 and ν⊥ = 2/N. We also show that the model show quite distinct physical properties for the case of open and periodic boundary conditions. The critical exponents governing the finite-size correction of the model are sensitive to the boundary conditions. Surprisingly if the bulk properties seems to depend if the model has free end or periodic boundary conditions. Several surprises appear in the eigenpectrum of the model as we change its boundary condition The model with open boundary conditions are exact integrable (free fermion spectrum), hoerver for the case of periodic boundary the model is probably not exact integrable.