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Exceptional Points in the Baxter-Fendley Free Parafermion Model

by Robert A. Henry, Murray T. Batchelor

This Submission thread is now published as

Submission summary

Authors (as registered SciPost users): Robert Henry
Submission information
Preprint Link: scipost_202301_00039v3  (pdf)
Date accepted: 2023-05-22
Date submitted: 2023-04-24 00:33
Submitted by: Henry, Robert
Submitted to: SciPost Physics
Ontological classification
Academic field: Physics
  • Condensed Matter Physics - Theory
  • Condensed Matter Physics - Computational
  • Mathematical Physics
  • Quantum Physics
Approaches: Theoretical, Computational


Certain spin chains, such as the quantum Ising chain, have free fermion spectra which can be expressed as the sum of decoupled two-level fermionic systems. Free parafermions are a simple generalisation of this idea to $Z(N)$-symmetric clock models. In 1989 Baxter discovered a non-Hermitian but $PT$-symmetric model directly generalising the Ising chain, which was much later recognised by Fendley to be a free parafermion spectrum. By extending the model's magnetic field parameter to the complex plane, it is shown that a series of exceptional points emerges, where the quasienergies defining the free spectrum become degenerate. An analytic expression for the locations of these points is derived, and various numerical investigations are performed. These exceptional points also exist in the Ising chain with a complex transverse field. Although the model is not in general $PT$-symmetric at these exceptional points, their proximity can have a profound impact on the model on the $PT$-symmetric real line. Furthermore, in certain cases of the model an exceptional point may appear on the real line (with negative field).

Author comments upon resubmission

Dear editor and referees,

We apologise for the confusion on this point, and we appreciate your patience.

The referee is indeed right, if a convention is applied to make the quasienergies have the smallest complex argument possible, or equivalently to have arguments in the range (-pi/N, pi/N]. There are N equivalent choices for each quasienergy corresponding to the N roots of Equation 4. We think this is a good convention to apply because it reveals that the quasienergies are conjugate pairs for phi=0.5 as the referee observed. The quasienergies previously shown in Figure 3 also satisfy Equations 4 and 5 but we have switched to this convention as it's a clear improvement.

List of changes

+ Figure 3 has been changed to use the convention that the quasienergies have arguments in the range (-pi/N, pi/N].
+ This convention is explained in the text in Section 4.
+ The discussion of the antisymmetric point is corrected in light of this convention.
+ Applying this convention does not affect any of the other material in the paper.
+ An acknowlegement of the referees' contributions has been added.

Published as SciPost Phys. 15, 016 (2023)

Reports on this Submission

Anonymous Report 1 on 2023-4-24 (Invited Report)


I thank the authors for explaining in detail the conventions used. In my opinion the paper is ready to be published in SciPost.

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