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

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 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.

9 pages, 5 figures, 1 table. v2: eq(31) corrected. v3: published version

References in corpus (4)

• Free fermionic and parafermionic quantum spin chains with multispin interactions
• Free fermions behind the disguise
• Integrable quantum spin chains with free fermionic and parafermionic spectrum
• Energy spectrum and critical exponents of the free parafermion $Z_N$ spin chain

Cited by in corpus (9)

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• Exceptional Points in the Baxter-Fendley Free Parafermion Model
• A Brief History of Free Parafermions
• Random free-fermion quantum spin chain with multi-spin interactions
• Quantum circuits with free fermions in disguise
• Exact real time dynamics with free fermions in disguise
• A Graph-Theoretic Framework for Free-Parafermion Solvability
• Exceptional point rings and $PT$-symmetry in the non-Hermitian XY model
• Noninteracting tight-binding models for Fock parafermions