Energy spectrum and critical exponents of the free parafermion $Z_N$ spin chain

Results are given for the ground state energy and excitation spectrum of a simple $N$-state $Z_N$ spin chain described by free parafermions. The model is non-Hermitian for $N\ge 3$ with a real ground state energy and a complex excitation spectrum. Although having a simpler Hamiltonian than the superintegrable chiral Potts model, the model is seen to share some properties with it, e.g., the specific heat exponent $\alpha =1-2\mathrm{/}N$ and the anisotropic correlation length exponents ${\nu }_{\parallel }=1$ and ${\nu }_{\perp }=2\mathrm{/}N$.