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

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.

Figures (15)

• Figure 1: Representations of the product $W^{(\ell)}W^{(\ell')}$. The crosses and the circles are the generators in $W^{(\ell)}$ and $W^{(\ell')}$, respectively. The links connect the generators that do not commute. The arrows give the directions of the multiplication rule in the algebra (\ref{['2.19']}).
• Figure 2: Representation of the product $W^{(\ell)} W^{(\ell')}$. The links connect generators that do not commute. This configuration is not allowed, since there is three commuting operators in $W^{(\ell)}$ (crosses) that do not commute with a single operator in $W^{(\ell')}$ (circles).
• Figure 3: The existence of a word $W^{(\ell)}$ with four commuting operators that does not commute with a single operator imply the existence of a related word (see the right of the figure), that is already forbidden (see Fig. \ref{['Q2']}).
• Figure 4: The clusters of non-commuting operators in the product $W^{(\ell)} W^{(\ell')}$. In the figure we have three clusters with ($n_c=4,n_{c'}=3$), ($n_c=2,n_{c'}=1$) and ($n_c=1,n_{c'}=1$) generators in $W^{(\ell)}$ and $W^{(\ell')}$.
• Figure 5: Examples of clusters formed in product $W^{(\ell)}W^{(\ell')}$. The clusters in (a) and (b) are connected even clusters and (c) and (d) are connected odd clusters. The clusters (e), (f) and (g) are unconnected ones.