  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 $\alpha=1-2/N$ and the anisotropic
correlation length exponents $\nu_\parallel =1$ and $\nu_\perp=2/N$.
