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2021

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

Alcaraz, Francisco C., Pimenta, Rodrigo A.

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Understand

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.

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

Built on

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Original
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T. Schultz, D. Mattis and E. H. Lieb, “Two-dimensional Ising model as a soluble problem of many fermions,” Rev. Mod. Phys. 36
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L.J. Slater, Generalized Hypergeometric Functions · 1966
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F. C. Alcaraz, M. N. Barber, and M. T. Batchelor, ” Conformal Invariance, the XXZ Chain and the Operator Content of Two-Dimensional Critical Systems,” Ann. Phys. (N.Y.) 182
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R. J. Baxter, “A simple solvable Z N Z_{N} Hamiltonian,” Phys. Lett
1989
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Similar

R.J. Baxter, “Superintegrable chiral Potts model: Thermodynamic properties, an inverse model and a simple associated Hamiltonian,” J. Stat. Phys
1989
Cited alongside, same era.
F.C. Alcaraz and R.A. Pimenta, “Free fermionic and parafermionic quantum spin chains with multispin interactions ,” Phys. Rev. B
Original
2005
Cited alongside, same era.
F.C. Alcaraz and R.A. Pimenta, “Integrable quantum spin chains with free fermionic and parafermionic spectrum,” Phys. Rev. B
Original
2010
Cited alongside, same era.
S. J. Elman, A. Chapman, and S. T. Flammia,”Free fermions behind the disguise”, Commun. Math. Phys
Original
2012
Cited alongside, same era.
P. Fendley, “Free parafermions,” J. Phys
Original

Then

R. J. Baxter, “The τ 2 \tau_{2} model and parafermions,” J. Phys
Original
2014
Later among the works it cites.
H. Au-Yang and J. H. H. Perk, “Parafermions in the τ 2 \tau_{2} model,” J. Phys
Original
2014
Later among the works it cites.
H. Au-Yang and J. H. H. Perk, “Parafermions in the tau-2 model II,” 2016, arXiv:1606.06319 [math-ph]
Original
2016
Later among the works it cites.
F. C. Alcaraz, M. T. Batchelor, and Z. -Z. Liu, “Energy spectrum and critical exponents of the free parafermion Z N Z_{N} spin chain,” J. Phys
Original
2017
Later among the works it cites.
F. C. Alcaraz and M. T. Batchelor, “Anomalous bulk behavior in the free parafermion Z(N) spin chain,” Phys. Rev. E

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2014

Cited alongside, same era.

B. Pozsgay, T. Gombor, and A. Hutsalyak, “Integrable hard rod deformation of the Heisenberg spin chains”, arXiv:2108:13724

Cited in the paper.

B. Pozsgay, Private communication

Cited in the paper.

2018

Later among the works it cites.