Free-parafermionic <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>Z</mml:mi><mml:mo>(</mml:mo><mml:mi>N</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math> and free-fermionic <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>X</mml:mi><mml:mi>Y</mml:mi></mml:mrow></mml:math> quantum chains
F C Alcaraz, Rodrigo A. Pimenta
Abstract
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.