Free fermionic and parafermionic quantum spin chains with multispin interactions

Abstract

We introduce a family of $Z(N)$ multispin quantum chains with a free fermionic ($N=2$) or free parafermionic ($N>2$) eigenspectrum. The models have $(p+1)$ interacting spins ($p=1,2,\ensuremath{\cdots}$), which are Hermitian in the $Z(2)$ (Ising) case and non-Hermitian for $N>2$. We construct a set of mutually commuting charges that allows us to derive the eigenenergies in terms of the roots of polynomials generated by a recurrence relation of order $(p+1)$. In the critical limit we identify these polynomials with certain hypergeometric polynomials ${}_{p+1}{F}_{p}$. Also in the critical regime, we calculate the ground-state energy in the bulk limit and verify that they are given in terms of the Lauricella hypergeometric series. The models with special couplings are self-dual and at the self-dual point show a critical behavior with a dynamical critical exponent ${z}_{c}=\frac{p+1}{N}$.