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Exceptional Points in the BaxterFendley Free Parafermion Model
by Robert A. Henry, Murray T. Batchelor
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Submission summary
Authors (as registered SciPost users):  Robert Henry 
Submission information  

Preprint Link:  scipost_202301_00039v3 (pdf) 
Date accepted:  20230522 
Date submitted:  20230424 00:33 
Submitted by:  Henry, Robert 
Submitted to:  SciPost Physics 
Ontological classification  

Academic field:  Physics 
Specialties: 

Approaches:  Theoretical, Computational 
Abstract
Certain spin chains, such as the quantum Ising chain, have free fermion spectra which can be expressed as the sum of decoupled twolevel fermionic systems. Free parafermions are a simple generalisation of this idea to $Z(N)$symmetric clock models. In 1989 Baxter discovered a nonHermitian 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
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)