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The relationship between the eigenspectrum of Ising and XY quantum chains iswell 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-fermionicquasi-particles. The fermionic quasi-energies are obtained by means of aJordan-Wigner transformation. On the other hand, there exist in the literaturea huge family of $Z(N)$ quantum chains whose eigenspectra, for $N>2$, are givenin terms of free parafermions and they are not derived from the standardJordan-Wigner transformation. The first members of this family are the $Z(N)$free-parafermionic Baxter quantum chains. In this paper we introduce a familyof 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 alsosolved by the Jordan-Wigner transformation. We show that with appropriatechoices of the $N$-multispin couplings, the eigenspectra of these XY models aregiven in terms of combinations of $Z(N)$ free-parafermionic quasi-energies. Inparticular all the eigenenergies of the $Z(N)$ free-parafermionic models arealso present in the related free-fermionic XY models. The correspondence isestablished via the identification of the characteristic polynomial which fixesthe eigenspectrum. In the $Z(N)$ free-parafermionic models the quasi-energiesobey an exclusion circle principle that is not present in the related$N$-multispin XY models.

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