SciPost Submission Page
Hopf Exceptional Points
by Tsuneya Yoshida, Emil J. Bergholtz, Tomáš Bzdušek
Submission summary
| Authors (as registered SciPost users): | Emil Bergholtz · Tsuneya Yoshida |
| Submission information | |
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| Preprint Link: | https://arxiv.org/abs/2504.13012v3 (pdf) |
| Date submitted: | Oct. 29, 2025, 10:56 a.m. |
| Submitted by: | Tsuneya Yoshida |
| Submitted to: | SciPost Physics |
| Ontological classification | |
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| Academic field: | Physics |
| Specialties: |
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| Approach: | Theoretical |
Abstract
Exceptional points at which eigenvalues and eigenvectors of non-Hermitian matrices coalesce are ubiquitous in the description of a wide range of platforms from photonic or mechanical metamaterials to open quantum systems. Here, we introduce a class of Hopf exceptional points (HEPs) that are protected by the Hopf invariants (including the higher-dimensional generalizations) and which exhibit phenomenology sharply distinct from conventional exceptional points. Saliently, owing to their $\mathbb{Z}_2$ topological invariant related to the Witten anomaly, three-fold HEPs and symmetry-protected five-fold HEPs act as their own ``antiparticles". Furthermore, based on higher homotopy groups of spheres, we predict the existence of multifold HEPs and symmetry-protected HEPs with non-Hermitian topology captured by a range of finite groups (such as $\mathbb{Z}_3$, $\mathbb{Z}_{12}$, or $\mathbb{Z}_{24}$) beyond the periodic table of Bernard-LeClair symmetry classes.
Author indications on fulfilling journal expectations
- Provide a novel and synergetic link between different research areas.
- Open a new pathway in an existing or a new research direction, with clear potential for multi-pronged follow-up work
- Detail a groundbreaking theoretical/experimental/computational discovery
- Present a breakthrough on a previously-identified and long-standing research stumbling block
Author comments upon resubmission
Dear Dr. Carlo Beenakker,
Dear Referees,
Re: scipost_202507_00091v1
Hopf Exceptional Points
by
Tsuneya Yoshida, Emil J. Bergholtz, and Tomas Bzdusek
We would like to thank you very much for your communication on August 4th and on September 18th concerning our manuscript (scipost_202507_00091v1). After carefully studying the referee's reports and substantially revising the manuscript accordingly, we are resubmitting our manuscript titled "Hopf Exceptional Points". We are grateful to all referees for their careful reading of our manuscript and their insightful comments, all of which we found helpful in improving our manuscript.
We have carefully studied all the comments by both referees and revised the manuscript accordingly. We appreciate the recommendation of for the publication of our manuscript in SciPost Physics made by Referee 2, stating
"I find this work highly pertinent and timely. It represents a very interesting extension of the existing topological classification of exceptional points. I therefore recommend it for publication."
We are also thankful for Referee 1's positive comment:
"It is a timely contribution to a topical field"
Besides the above positive comments, both referees raised questions pointing out our insufficient explanations in the previous manuscript. By taking into account all of the comments by both referees, we have revised the manuscript.
We believe that in this revised version, we have appropriately addressed all the points raised by all referees and that the revised manuscript is now suited for publication in SciPost Physics. Your kind consideration of our manuscript is greatly appreciated.
We also attach our revised manuscript with the changes highlighted (see reply to Report 1 on 2025-08-04). We do hope that our revised manuscript with our reply will meet with your approval.
Yours sincerely,
Tsuneya Yoshida$^{1, 2}$, Emil J. Bergholtz$^{3}$, and Tomas Bzdusek$^{4}$
$^{1}$ Department of Physics, Kyoto University, Kyoto 606-8502, Japan
$^{2}$ Institute for Theoretical Physics, ETH Zurich, 8093 Zurich, Switzerland
$^{3}$ Department of Physics, Stockholm University, AlbaNova University Center, 10691 Stockholm, Sweden
$^{4}$ Department of Physics, University of Zurich, Winterthurerstrasse 190, 8057 Zurich, Switzerland
E-mail:
yoshida.tsuneya.2z@kyoto-u.ac.jp, emil.bergholtz@fysik.su.se, tomas.bzdusek@uzh.ch
Dear Referees,
Re: scipost_202507_00091v1
Hopf Exceptional Points
by
Tsuneya Yoshida, Emil J. Bergholtz, and Tomas Bzdusek
We would like to thank you very much for your communication on August 4th and on September 18th concerning our manuscript (scipost_202507_00091v1). After carefully studying the referee's reports and substantially revising the manuscript accordingly, we are resubmitting our manuscript titled "Hopf Exceptional Points". We are grateful to all referees for their careful reading of our manuscript and their insightful comments, all of which we found helpful in improving our manuscript.
We have carefully studied all the comments by both referees and revised the manuscript accordingly. We appreciate the recommendation of for the publication of our manuscript in SciPost Physics made by Referee 2, stating
"I find this work highly pertinent and timely. It represents a very interesting extension of the existing topological classification of exceptional points. I therefore recommend it for publication."
We are also thankful for Referee 1's positive comment:
"It is a timely contribution to a topical field"
Besides the above positive comments, both referees raised questions pointing out our insufficient explanations in the previous manuscript. By taking into account all of the comments by both referees, we have revised the manuscript.
We believe that in this revised version, we have appropriately addressed all the points raised by all referees and that the revised manuscript is now suited for publication in SciPost Physics. Your kind consideration of our manuscript is greatly appreciated.
We also attach our revised manuscript with the changes highlighted (see reply to Report 1 on 2025-08-04). We do hope that our revised manuscript with our reply will meet with your approval.
Yours sincerely,
Tsuneya Yoshida$^{1, 2}$, Emil J. Bergholtz$^{3}$, and Tomas Bzdusek$^{4}$
$^{1}$ Department of Physics, Kyoto University, Kyoto 606-8502, Japan
$^{2}$ Institute for Theoretical Physics, ETH Zurich, 8093 Zurich, Switzerland
$^{3}$ Department of Physics, Stockholm University, AlbaNova University Center, 10691 Stockholm, Sweden
$^{4}$ Department of Physics, University of Zurich, Winterthurerstrasse 190, 8057 Zurich, Switzerland
E-mail:
yoshida.tsuneya.2z@kyoto-u.ac.jp, emil.bergholtz@fysik.su.se, tomas.bzdusek@uzh.ch
List of changes
(1) In response to Referee 1's comment 1-[1], we have added footnote 1.
(2) In response to Referee 2's comment 2-[1], we have added footnote 2.
(3) In response to Referee 1's comment 1-[6], we have revised a sentence in the third paragraph of Sec. 1.
(4) In response to Referee 1's comment 1-[2], we have cited Ref. [88]
(5) In response to Referee 1's comment 1-[2], we have revised sentences in the third paragraph of Sec. 1.
(6) In response to Referee 1's comment 1-[6], we have cited Refs. [86,87,108]
(7) In response to Referee 1's comment 1-[8], we have cited Ref. [36].
(8) In response to Referee 1's comment 1-[3(b)], we have revised sentences in the last paragraph of Sec. 1
(9) In response to Referee 1's comment 1-[5], we have added footnote 3.
(10) In response to Referee 1's comment 1-[3(b)], we have added footnote 4.
(11) In response to Referee 1's comment 1-[5], we have added sentences below Eq. (2) and Eq. (A.8).
(12) In response to Referee 1's comment 1-[4], we have revised a sentence below Eq. (4).
(13) In response to Referee 1's comment 1-[3(c)], we have revised a sentence below Eq. (4).
(14) In response to Referee 1's comment 1-[3(a)], we have added footnote 5.
(15) In response to Referee 1's comment 1-[3(d)], we have added footnote 6.
(16) In response to Referee 1's comment 1-[3(a)], we have added footnote 7.
(17) In response to Referee 2's comment 2-[1], we have added a sentence at the end of Sec. 5.3.
(2) In response to Referee 2's comment 2-[1], we have added footnote 2.
(3) In response to Referee 1's comment 1-[6], we have revised a sentence in the third paragraph of Sec. 1.
(4) In response to Referee 1's comment 1-[2], we have cited Ref. [88]
(5) In response to Referee 1's comment 1-[2], we have revised sentences in the third paragraph of Sec. 1.
(6) In response to Referee 1's comment 1-[6], we have cited Refs. [86,87,108]
(7) In response to Referee 1's comment 1-[8], we have cited Ref. [36].
(8) In response to Referee 1's comment 1-[3(b)], we have revised sentences in the last paragraph of Sec. 1
(9) In response to Referee 1's comment 1-[5], we have added footnote 3.
(10) In response to Referee 1's comment 1-[3(b)], we have added footnote 4.
(11) In response to Referee 1's comment 1-[5], we have added sentences below Eq. (2) and Eq. (A.8).
(12) In response to Referee 1's comment 1-[4], we have revised a sentence below Eq. (4).
(13) In response to Referee 1's comment 1-[3(c)], we have revised a sentence below Eq. (4).
(14) In response to Referee 1's comment 1-[3(a)], we have added footnote 5.
(15) In response to Referee 1's comment 1-[3(d)], we have added footnote 6.
(16) In response to Referee 1's comment 1-[3(a)], we have added footnote 7.
(17) In response to Referee 2's comment 2-[1], we have added a sentence at the end of Sec. 5.3.
Current status:
Voting in preparation
Reports on this Submission
Strengths
The clarity of the paper has improved. Only a few small presentational issues remain, as outlined below.
Weaknesses
Comment 1-[1]:
The authors now qualify the use of "quasiparticle" in a footnote. In the paragraph with the new footnote, however, there is also no issue. The issue arises shortly later when they extend this notion to the band singularities themselves, stating "... exceptional point cannot be its own “antiparticle”," along the further examples "meaning that they act as their own antiparticle...", "Notably, the HEP acts as its own antiparticle..." already mentioned in the first report.
Non of the three provided references support this extension. This is easily checked by searching these references for the keyword "quasiparticle".
Refs [1] and [2] refer to quasiparticles exclusively in the context of Bogoliubov quasiparticles, fermionic quasiparticles, and charge e / 2 quasiparticles moving along with a soliton. Ref [11] explicitly constructs generalizations of fermionic quasiparticles, such as a fermionic spin-1 generalization of an ordinary Weyl fermion. Annihilation and creation properties are then encoded in the matrix structure of these - not in the existence of nonexistence of the band singularity itself.
In all cases, changes of the band singularity are discussed as *topological phase transitions*. Maybe the confusion arises because we can associate topological charges to band singularities. That does not make these singularities into particles that can dynamically annihilate; they are structures in parameter space, in contrast to the quasiparticles listed in the opening paragraph, and the topological charges distinguish different phases.
Given that this very solid terminology, I suggest that the authors maintain the distinction between (parametric) topological phase transitions in the band structure and the notion of (dynamical) quasiparticles populating the bands.
Comment 1-[3(d)] For clarification, the EP2 example was given as an indication of what level of abstraction could qualify as a "simple" explanation. This would of course have to be adapted to the HEP case. I therefore re-raise this issue as accessibility of the manuscript would be increased if the authors could include a simple picture, just in case that such an explanation is readily available.
The authors now qualify the use of "quasiparticle" in a footnote. In the paragraph with the new footnote, however, there is also no issue. The issue arises shortly later when they extend this notion to the band singularities themselves, stating "... exceptional point cannot be its own “antiparticle”," along the further examples "meaning that they act as their own antiparticle...", "Notably, the HEP acts as its own antiparticle..." already mentioned in the first report.
Non of the three provided references support this extension. This is easily checked by searching these references for the keyword "quasiparticle".
Refs [1] and [2] refer to quasiparticles exclusively in the context of Bogoliubov quasiparticles, fermionic quasiparticles, and charge e / 2 quasiparticles moving along with a soliton. Ref [11] explicitly constructs generalizations of fermionic quasiparticles, such as a fermionic spin-1 generalization of an ordinary Weyl fermion. Annihilation and creation properties are then encoded in the matrix structure of these - not in the existence of nonexistence of the band singularity itself.
In all cases, changes of the band singularity are discussed as *topological phase transitions*. Maybe the confusion arises because we can associate topological charges to band singularities. That does not make these singularities into particles that can dynamically annihilate; they are structures in parameter space, in contrast to the quasiparticles listed in the opening paragraph, and the topological charges distinguish different phases.
Given that this very solid terminology, I suggest that the authors maintain the distinction between (parametric) topological phase transitions in the band structure and the notion of (dynamical) quasiparticles populating the bands.
Comment 1-[3(d)] For clarification, the EP2 example was given as an indication of what level of abstraction could qualify as a "simple" explanation. This would of course have to be adapted to the HEP case. I therefore re-raise this issue as accessibility of the manuscript would be increased if the authors could include a simple picture, just in case that such an explanation is readily available.
Report
My general assessment that this is a high-quality paper in a topical field remains, and the remaining presentational issues are very minor.
Requested changes
The authors should reconsider to address comments 1-[1] and 1-[3(d)] as outlines above.
Recommendation
Publish (easily meets expectations and criteria for this Journal; among top 50%)
Report
The authors have clarified, in a footnote, the terminology related to Hopf that is used to characterize their exceptional points. After this minor revision of their manuscript, I have no further substantial criticisms with regard to its publication. I still think that this manuscript is timely and brings some new results to the hot topic of non-Hermitian topology, and would thus recommend its publication.
Recommendation
Publish (meets expectations and criteria for this Journal)