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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: | https://arxiv.org/abs/2301.11031v3 (pdf) |
| Date submitted: | 2023-04-05 14:22 |
| 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
Dear editors and referees,
Thank you for considering our manuscript for SciPost Physics. We wish to thank the referees for their comments which have resulted in significant improvements to the paper. In particular, we have added an argument for why the quasienergy degeneracies must in general be (the only) EPs of the Hamiltonian, based on Fendley's work. We have responded to each of the referees' points below, in addition to the list of changes.
Sincerely, Robert Henry and Murray Batchelor
Responses to referee 1 ~~~~~~~~~~~~~~~~~~~~~~ + 1. An EP is indeed the appearance of a nontrivial Jordan structure at an isolated point. We have made this clear in the text. + 2. The misprint in equation (25) has been corrected.
Responses to referee 2 ~~~~~~~~~~~~~~~~~~~~~~ + 1. Following the referee's suggestion, we further investigated the coalescence of the eigenvectors analytically. Further examination revealed that Fendley's solution (Free Parafermions 2014) implies that a quasienergy degeneracy of the kind we have demonstrated also leads to coalescence of the corresponding eigenvectors. This is explained in a new section (2) in the revised paper. The fact that the eigenvectors can be shown to coalesce analytically rendered some of our analysis throughout the paper to confirm such coalescence redundant. Therefore we have removed Figure 4, as this was its only purpose and it was perhaps confusing anyway. Figure 6 in the revised paper still includes some numerical confirmation of the coalescence. + 2. The referee pointed out that our motivation in terms of diverging correlation functions was not well explained. In fact, the divergent behaviour of the fidelity susceptibility on the real line previously shown in Figure 7 is equivalent to diverging correlations, and provides a more concrete motivation since the behaviour of the susceptibility near an EP is well-understood. Therefore we have moved the content of Section 6 to the motivation section and focused on the fidelity susceptibility rather than diverging correlations. + 3. phi=0 is correct for the top right panel in Fig. 2 + 4. The referee is right - Fig. 3 should be L=5. We have also carefully verified that the correct values of L are stated elsewhere in the paper. + 5. The definitions of Delta_epsilon and Delta_E are now given in the text. + 6. Figure 4 has been removed as detailed above, as it is no longer necessary. + 7. The definition of gamma in Fig.7 was indeed wrong and has been corrected. + 8. The number of EPs should indeed be N(L-1) with the trivial EP excluded. We have made this more clear in a number of places in the text.
List of changes
Some of these changes are further discussed in the responses to the referees.
+ A new section (Section 2) arguing that the eigenvectors must coalesce at quasienergy degeneracies, based on Fendley's work (Free Parafermions 2014) has been added. Later parts of the paper which focused on testing this numerically have been somewhat shortened and revised. Figure 4 has been removed as it is not necessary given this addition.
+ The motivation section has been re-expressed in terms of the fidelity, which has been moved from Section 6 to the new Section 3.
+ The "trivial" EPs have been removed from figures 3, 5 and 6 as they are not in fact EPs. They are still discussed in the text in 4.3.
+ In section 1.1 a more precise description of EPs in terms of the Jordan structure is given.
+ The sign of the expansion coefficient 'a' in Eq. (24) has been changed to match Alcaraz et al., although 'a' is still used instead of kappa for clarity.
+ Other small corrections suggested by the referees have been implemented, detailed in the responses.
+ Brief discussion of the case phi=0.5, which appears to be PT antisymmetric, has been added to Section 4.
+ The conclusion has been restructured and some additional details are given, but the content is mostly unchanged.
+ Various small changes for style and clarity have been made throughout the paper, which do not affect the content.
Current status:
Reports on this Submission
Report #1 by Anonymous (Referee 4) on 2023-4-19 (Invited Report)
- Cite as: Anonymous, Report on arXiv:2301.11031v3, delivered 2023-04-19, doi: 10.21468/SciPost.Report.7056
Report
The authors have adequately responded to most of my earlier remarks. In particular, I think the argument about the coalescence of the eigenvectors is correct, as supported by numerical evidence. The new version is better and more clear.
However, the following small details still needs clarification:
1) Regarding the data in Fig. 3, I still cannot reproduce it. I must be missing some detail, but here is what I get:
For $N=3$, $L=4$ and $|\lambda|=1,\phi=0$, the nontrivial solutions of eq.(18) are:
$k\in \{8\pi/9,6\pi/9,4\pi/9,2\pi/9\}$
whereas the quasienergy is,
$\epsilon(k)=(4\cos^2(k/2))^{1/3}$.
Pluging in the numerical values, we get the quasienergies
$\{0.494083, 1, 1.32899, 1.52292\}$
which does not match right upper panel of Fig. 3.
I cannot reproduce the other cases ($\phi\neq 0$) as well.
Can the authors clarify this point?
2) Around eq.(22), the assumed ordering of the quasienergies must be specified. That is, is it $|\epsilon_0|<|\epsilon_1|<...$ or $Re(\epsilon_0)<Re(\epsilon_1)<...$ or something else?
I believe that the paper will be ready for publication after the authors have clarified the two points above.
Requested changes
see report