Integrable quantum spin chains with free fermionic and parafermionic spectrum

We present a general study of a large family of exactly integrable quantum chains with multispin interactions. The exact integrability follows from the algebraic properties of the energy density operators defining the quantum chains. The Hamiltonians are characterized by a parameter p = 1, 2, ... related to the number of interacting spins in the multispin interaction. In the general case, the quantum spins are of infinite dimension. In special cases, characterized by the parameter N = 2, 3, ..., the quantum chains describe the dynamics of Z(N) quantum spin chains with open boundary conditions. The simplest case p = 1 corresponds to the free fermionic quantum Ising chain (N = 2) or the Z(N) free parafermionic quantum chain. The eigenenergies of the quantum chains are given in terms of the roots of special polynomials, and for general values of p the quantum chains are characterized by a free fermionic (N = 2) or free parafermionic (N > 2) eigenspectrum. The models have a special critical point when all coupling constants are equal. At this point, the ground-state energy is exactly calculated in the bulk limit, and our analytical and numerical analyses indicate that the models belong to universality classes of critical behavior with dynamical critical exponent z = (p + 1)/N and specific-heat exponent alpha = ma{10, 1 - (p + 1)/N}.