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Relating the topology of Dirac Hamiltonians to quantum geometry: When the quantum metric dictates Chern numbers and winding numbers

by Bruno Mera, Anwei Zhang, Nathan Goldman

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Submission summary

Authors (as registered SciPost users): Nathan Goldman · Bruno Mera
Submission information
Preprint Link:  (pdf)
Date accepted: 2021-11-10
Date submitted: 2021-10-04 13:37
Submitted by: Goldman, Nathan
Submitted to: SciPost Physics
Ontological classification
Academic field: Physics
  • Condensed Matter Physics - Theory
  • Quantum Physics
Approach: Theoretical


Quantum geometry has emerged as a central and ubiquitous concept in quantum sciences, with direct consequences on quantum metrology and many-body quantum physics. In this context, two fundamental geometric quantities are known to play complementary roles: the Fubini-Study metric, which introduces a notion of distance between quantum states defined over a parameter space, and the Berry curvature associated with Berry-phase effects and topological band structures. In fact, recent studies have revealed direct relations between these two important quantities, suggesting that topological properties can, in special cases, be deduced from the quantum metric. In this work, we establish general and exact relations between the quantum metric and the topological invariants of generic Dirac Hamiltonians. In particular, we demonstrate that topological indices (Chern numbers or winding numbers) are bounded by the quantum volume determined by the quantum metric. Our theoretical framework, which builds on the Clifford algebra of Dirac matrices, is applicable to topological insulators and semimetals of arbitrary spatial dimensions, with or without chiral symmetry. This work clarifies the role of the Fubini-Study metric in topological states of matter, suggesting unexplored topological responses and metrological applications in a broad class of quantum-engineered systems.

Author comments upon resubmission

Dear Editors, Dear Referees,

We are glad to resubmit our manuscript entitled "Relating the topology of Dirac Hamiltonians to quantum geometry: When the quantum metric dictates Chern numbers and winding numbers" for your consideration as an Article in SciPost Physics.

First of all, we would like to thank both Referees for their reading of our work, and for their appreciations.

We have analyzed the two invited reports (#1 and #3) thoughtfully. The first report, which "strongly recommend(s) the publication in SciPost Physics" highlights the importance of our findings "for the quantum engineering and topological communities". This first report concludes by stating that our "manuscript can be greatly relevant for future experiments in quantum matter". In contrast, the second report (#3) contradicts those very same statements, by indicating that our idea "is only applicable to unphysical class of Hamiltonians", that our work is "a purely mathematical contribution" and that it is "a pity that it cannot be extended to any more physical situations". That same report (#3) nevertheless concludes that our manuscript is "a fine paper with a solid scientific content", which may be published in a "decent journal".

Above all, we strongly support the opinion of Referee #1 according to which the scope and results of our work are indeed relevant to experiments in quantum matter. Lattice systems described by a Dirac Hamiltonian (such as the emblematic 1D SSH model, the 2D Haldane model, the ideal 3D Weyl model, ... ) can be finely engineered in a broad class of physical settings (e.g. ultracold gases in optical lattices, photonics devices, electric circuits, ... ). Furthermore, these systems have been shown to be very well suited to measure quantum geometry [see, for instance, N. R. Cooper et al., Reviews of Modern Physics 91, 015005 (2019); T. Ozawa et al., Reviews of Modern Physics 91, 015006 (2019)]. In this sense, we strongly refute the statements of Referee #3 according to which our results are "only applicable to unphysical class of Hamiltonians" and that our work is "a purely mathematical contribution".

We hereby submit a revised version of our work, which takes the remarks and suggestions of the two Referees into account; see List of Changes below.

We hope that the Referees will be pleased by these revisions and that they will recommend publication of our work in the journal.

Yours sincerely,

The authors.

List of changes

Our changes are summarized below:

(1) Following the main criticism of Referee #3, the revised text better emphasizes the relevance of our work for ongoing experimental efforts. In particular, the text now explicitly refers to emblematic implementations of Dirac Hamiltonians in synthetic lattice systems, as well as to quantum-geometry measurements that have been recently performed in these systems.

(2) Following a suggestion of Referee #1, we now provide a "short summary in the introduction of the meaning of Eq. (1) and (2)". This reads [see below Eq. (2)]: "The relations presented in Eqs. (1) and (2) show that the volume of the Brillouin zone, as measured by the quantum metric, provides an upper bound to the topological invariants of generic Dirac Hamiltonians, in all dimensions. "

(3) Following a remark of Referee #3, we have included a whole new Section 6 dedicated to the implications of our metric-curvature relations for quantum metrology (an idea which was only briefly formulated in the original manuscript, and which, according to Referee #3 "require(d) much more detailed explanations").

(4) Following a remark of Referee #3, we have also clarified several statements in our concluding section, which the Referee #3 found "not directly related to the content of the work".

(5) We note that Referee #3 suggested to remove the three first sentences of our abstract because they "may produce an unintended impression that those are results of this mere paper". We have slightly revised the abstract so as to remove all possible ambiguity regarding the actual contributions of our work.

(6) Regarding the criticism of Referee #3 "The whole 3-paragraph introduction seems generic and could suit almost any paper in the field": We have decided to keep our introductory paragraphs in Section 1, because we believe that such an opening section can indeed be general before diving into more specific aspects; in particular, we believe that a general reader might appreciate this pluridisciplinary view on the quantum metric. We hope that the Referee #3 (who wrote "By no means would I intrude the authors with an actual request") will accept our choice of style.

Published as SciPost Phys. 12, 018 (2022)

Reports on this Submission

Anonymous Report 1 on 2021-10-13 (Invited Report)

  • Cite as: Anonymous, Report on arXiv:2106.00800v5, delivered 2021-10-13, doi: 10.21468/SciPost.Report.3666


- The paper puts forward an interesting relation between Dirac Hamiltonians and quantum geometry, a topic of great interest for the condensed matter community


- All my previous comments have been address by the authors. No remaining weaknesses


The authors have addressed the comments of the previous report and modified their manuscript accordingly. As elaborated in my previous report, I believe that their results are of great interest to the condensed matter community. I have also read in detail the report of Referee #3 and the response of the authors, and I believe that the authors have successfully addressed all the comments raised. Therefore, given all the points above, I strongly recommend the publication of their manuscript in Scipost Physics.

Requested changes

1- No changes requested

  • validity: top
  • significance: high
  • originality: high
  • clarity: high
  • formatting: perfect
  • grammar: perfect

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