Free fermionic and parafermionic multispin quantum chains with non-homogeneous interacting ranges

A large family of multispin interacting one-dimensional quantum spin models with $Z(N)$ symmetry and a free-particle eigenspectra are known in the literature. They are free-fermionic ($N=2$) and free-parafermionic ($N\geq 2$) quantum chains. The essential ingredient that implies the free-particle spectra is the fact that these Hamiltonians are expressed in terms of generators of a $Z(N)$ exchange algebra. In all these known quantum chains the number of spins in all the multispin interactions (range of interactions) is the same and therefore, the models have homogeneous interacting range. In this paper we extend the $Z(N)$ exchange algebra, by introducing new models with a free-particle spectra, where the interaction ranges of the multispin interactions are not uniform anymore and depends on the lattice sites (non-homogeneous interacting range). We obtain the general conditions that the site-dependent ranges of the multispin interactions have to satisfy to ensure a free-particle spectra. Several simple examples are introduced. We study in detail the critical properties in the case where the range of interactions of the even (odd) sites are constant. The dynamical critical exponent is evaluated in several cases.