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Exceptional Points in the Baxter-Fendley 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_00039v2 (pdf) |
| Date submitted: | 2023-04-21 09:32 |
| Submitted by: | Henry, Robert |
| Submitted to: | SciPost Physics |
| Ontological classification | |
|---|---|
| Academic field: | Physics |
| Specialties: |
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| 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 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
We would like to thank the reviewer for their additional comments. Below are our responses to the referee's new comments.
- The reviewer is right: Figure 3 had an incorrect scaling function applied. This has been corrected and we agree with the reviewer's numerical values. We have verified that the values shown in Figure 3 can be obtained directly from Equations 4 and 5. Following this, we also decided to remove the global rotation of the Hamiltonian given in Section 4.1, as it is unnecessary to most of the paper and has no effect on the EPs. It is still discussed in 4.1 along with the PT-antisymmetric phi=0.5 case.
- We have added a definition of Delta(epsilon) and explained why this quantity is used over another possible choice.
List of changes
+ The data in Figure 3 had been rescaled incorrectly and has been fixed.
+ The redefinition of the Hamiltonian to include a global rotation in Section 4 has been removed. These rotations are still discussed in Section 4.1.
+ The removal of the global changes the values shown in Figure 3, which have been recalculated (they are rotated by 2pi/6). Figure 3 now corresponds directly to Equations 4 and 5.
+ The removal of the global rotation does not affect the other figures as they only depend on differences of (quasi)energies.
+ A definition of Delta(epsilon) is now given in Equation 22 along with some discussion of why we use this quantity. This quantity has also been relabeled with subscript 12 instead of 01 as the quasienergies are in fact labelled from 1, not 0.
Current status:
Reports on this Submission
Report #1 by Anonymous (Referee 2) on 2023-4-21 (Invited Report)
- Cite as: Anonymous, Report on arXiv:scipost_202301_00039v2, delivered 2023-04-21, doi: 10.21468/SciPost.Report.7085
Report
I thank the authors for clarifiyng the scale issue and the ordering of the quasienergies.
I now agree with the cases $\phi=0$ and $\phi=0.1$ in Fig. 3, but unfortunately I think $\phi=0.5$ is not correct.
As in the previous version by the authors, it seems that the quasienergies do appear in complex pairs, even without a further rotation. Here are the numerical values that I found solving eq.(4,5) for $|\lambda|=1,\phi=0.5$:
k-values: $\{-2.4553+0.0826589 i,-1.82173+0.202561 i,-1.31986+0.202561 i,-0.686291+0.0826589 i\}$
quasienergy values: $\{0.634754\, +0.584635 i,0.634754\, -0.584635 i,0.990265\, +0.601812 i,0.990265\, -0.601812 i\}$
So I ask the authors to please check this point one more time.