Free-parafermionic $Z(N)$ and free-fermionic $XY$ quantum chains

The relationship between the eigenspectrum of Ising and XY quantum chains is well known. Although the Ising model has a $Z(2)$ symmetry and the XY model a $U(1)$ symmetry, both models are described in terms of free-fermionic quasi-particles. The fermionic quasi-energies are obtained by means of a Jordan-Wigner transformation. On the other hand, there exist in the literature a huge family of $Z(N)$ quantum chains whose eigenspectra, for $N>2$, are given in terms of free parafermions and they are not derived from the standard Jordan-Wigner transformation. The first members of this family are the $Z(N)$ free-parafermionic Baxter quantum chains. In this paper we introduce a family of XY models that beyond two-body also have $N$-multispin interactions. Similarly to the standard XY model they have a $U(1)$ symmetry and are also solved by the Jordan-Wigner transformation. We show that with appropriate choices of the $N$-multispin couplings, the eigenspectra of these XY models are given in terms of combinations of $Z(N)$ free-parafermionic quasi-energies. In particular all the eigenenergies of the $Z(N)$ free-parafermionic models are also present in the related free-fermionic XY models. The correspondence is established via the identification of the characteristic polynomial which fixes the eigenspectrum. In the $Z(N)$ free-parafermionic models the quasi-energies obey an exclusion circle principle that is not present in the related $N$-multispin XY models.