Free fermionic and parafermionic quantum spin chains with multispin interactions

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, ...), 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 F-p+1(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) = p+1/N.