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

TL;DR

This paper introduces new multispin quantum chain models with non-uniform interaction ranges that maintain free-particle spectra, extending previous models with homogeneous interactions and analyzing their critical properties.

Contribution

It generalizes existing $Z(N)$ exchange algebra models by allowing site-dependent interaction ranges while preserving free-particle spectra.

Findings

Derived conditions for site-dependent interaction ranges to ensure free spectra.

Introduced simple examples of non-homogeneous interaction models.

Analyzed critical properties and dynamical exponents for specific cases.

Abstract

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.