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

TL;DR

This paper analyzes the energy spectrum and critical exponents of a simple free parafermion $Z_N$ spin chain, revealing its spectral properties and critical behavior, including specific heat and correlation length exponents.

Contribution

It provides new exact results for the ground state energy and excitation spectrum of a non-Hermitian $Z_N$ spin chain with free parafermions, highlighting its shared properties with the chiral Potts model.

Findings

Real ground state energy despite complex spectrum for $N \\ge 3$

Specific heat exponent =1-2/N

Correlation length exponents _=1 and 2/N

Abstract

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 $\nu_\perp=2/N$.