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 quasiparticles. The fermionic quasienergies are obtained by means of a Jordan-Wigner transformation. On the other hand, there exists 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. Similar 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 quasienergies. 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 quasienergies obey an exclusion circle principle that is not present in the related N-multispin XY models.