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

## TL;DR

This paper derives exact ground-state expectation values for the non-Hermitian Z(N) spin chain using the Hellmann-Feynman theorem, extending known results for the quantum Ising model to general N and analyzing boundary effects.

## Contribution

It provides the first exact analytical expressions for certain ground-state expectation values in free parafermion Z(N) chains for arbitrary N.

## Key findings

- Exact expectation values for sites deep inside the chain.
- Numerical validation for N=3.
- Analysis of boundary effects on expectation values.

## Abstract

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-neighbour ground-state expectation values for general N which are valid for sites deep inside the chain. These results are tested numerically for N=3, along with how they change as a function of distance from the boundary.