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Free fermionic and parafermionic quantum spin chains with multispin interactions

We introduce a new a family of $Z(N)$ multispins quantum chains with a free-fermionic ($N=2$) or free-parafermionic ($N>2$) eigenspectrum. The models have $(p+1)$ interacting spins ($p=1,2,\dots$), being 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 dynamical critical exponent $z_c=\frac{p+1}{N}$.