Some ground-state expectation values for the free parafermion Z ( N ) spin chain

We consider the calculation of ground-state expectation values for the non-Hermitian Z(N) spin chain described by free parafermions. For N = 2 the model reduces to the quantum Ising chain in a transverse field with open boundary conditions. Use is made of the Hellmann-Feynman theorem to obtain exact results for particular single site and nearest-neighbor ground-state expectation values for general N which are valid for sites deep inside the chain. These results are tested numerically for N = 3, including how they change as a function of distance from the boundary.This project has been supported by the Australian Research Council Discovery Project DP180101040 and the National Natural Science Foundation of China Grant No. 11575037