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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, Nathan Goldman

This is not the latest submitted version.

This Submission thread is now published as SciPost Phys. 12, 018 (2022)

Submission summary

As Contributors: Nathan Goldman · Bruno Mera
Arxiv Link: (pdf)
Date submitted: 2021-06-23 13:41
Submitted by: Goldman, Nathan
Submitted to: SciPost Physics
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 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.

Current status:
Has been resubmitted

Submission & Refereeing History

Published as SciPost Phys. 12, 018 (2022)

Reports on this Submission

Anonymous Report 3 on 2021-8-30 (Invited Report)

  • Cite as: Anonymous, Report on arXiv:2106.00800v3, delivered 2021-08-30, doi: 10.21468/SciPost.Report.3467


1-generally comprehensive and readable write-up.
2-a mathematically interesting idea.
3-rather straightforward derivation.


1- the idea, as far as presented in the paper, is only applicable to degenerate and unphysical class of Hamiltonians.
2-a great number of buzz words not directly related to the paper content in the abstract, introduction and conclusion confuse a potential reader and undermine the significance of the contribution.
3-no attempt to access the generatily of conclusions, which, in my opinion renders the paper a purely mathematical contribution.


Dear all,
it's a first time I write a referee report for SciPost. Among other things, I was asked to check if this manuscript satisfies the acceptance criteria of SciPost. On a personal note, I find these criteria childish and silly. Moreover, their strict, literal and overall implementation would result - God forbids - in a quick and painfull death of physics as a branch of science. However, I do what I asked for.

This is a fine paper that address the Dirac Hamiltonians defined as linear superpositions of the Clifford algebra generators. It adresses in uniform manner arbitrariy space dimension as well as arbitrary dimension of the generators. While the members of Dirac Hamiltonian ensembles have been used as minimal models for various topological solids, an actual solid can never be described by Dirac Hamiltonian.

The main and, as far as a non-specialist can see, only achievement of the authors are two upper bonds on topological numbers in terms of "quantum"volume of the Brillion zone, expressed by Eqs. 1 and 2. This is an interesting idea that brings together two global characteristics of the geometry defined on periodical families of quantum states. It's a pity that it cannot be extended to any more physical situations.

I admit that the manuscript provides the comprehensive derivation of the main result for anybody who wishes to follow their exteremly formal outline, this derivation being decorated with a number of known examples.

This manuscript presents no discovery, breaktrhough, pathway with clear potential, neither a link between two research fields. As such, it does not satisfy the publication criteria of SciPost.

As mentioned, it is a fine paper with a solid scientific content. It may be published, for instance, in Physics Core, or any other decent journal. In my opinion, the best way to prepare the manuscript to this publication is to implement the changes I list under Requested Changes. By no means would I intrude the authors with an actual request: I just follow the formatting suggested by this particlular journal.

Requested changes

1- In the abstract: please remove 3 first sentences. They do not reflect the content of this paper yet may produce an unintended impression that those are results of this mere paper!
2- The whole 3-paragraph introduction seems generic and could suit almost any paper in the field. Strictly speaking, it is redundant and serves as a placeholder for citations. The real stuff begins in the subchapter "Scope..." More specific introduction would improve the presentation.
3- The concluding part consists of 7 sentences. From those, 2nd and 3rd is not directly related to the content of the work. The 6th and 7th might make sence but would require much more detailed explanations that, again, are not directly related to the research presented.

  • validity: good
  • significance: ok
  • originality: good
  • clarity: high
  • formatting: excellent
  • grammar: perfect

Anonymous Report 1 on 2021-7-27 (Invited Report)

  • Cite as: Anonymous, Report on arXiv:2106.00800v3, delivered 2021-07-27, doi: 10.21468/SciPost.Report.3306


1- Original and greatly interesting idea for the quantum engineering and topological community
2- Very well written and highly comprehensible
3- Many different examples provided, showing that their construction applied to a variety of systems


1-Eq. (1) and (2) in the introduction may look a bit complex for non experts, perhaps the authors can add a short summary in the intro about the meaning of Eq. (1) and (2) for a specific system


The authors study the relation between two important concepts describing the geometry of electronic wavefunctions, the quantum metric and topological invariants given by the Berry curvature. These two objects has been at the center of a variety of novel effects in condensed matter systems, yet the relation between them has not been well established so far. The authors tackle this important problem, demonstrating that in a certain class of Dirac Hamiltonians of arbitrary dimension, a one-to-one relation between the quantum geometry and the topological invariants can be made. Interestingly, the findings of the authors suggest that topological invariants can be probed by dynamic susceptibility measurements that allow probing the quantum metric. Therefore, beyond the theoretical value of their work, this manuscript can be greatly relevant for future experiments in quantum matter.

The manuscript is well written and highly comprehensive, and the results presented are fully compatible with well-known results in the field. In particular, the authors explicitly demonstrate their idea for a variety of well-known topological states, including chiral phases and Chern insulators in 1,2,3,4 dimension. The manuscript provides a highly accessible demonstration of their original idea, and I believe that it would be of great interest to the theory community on topological and engineered quantum matter. For the reasons stated above, I strongly recommend the publication of their manuscript in Scipost Physics.

Requested changes

1- I would suggest that the authors can a short summary in the introduction of the meaning of Eq. (1) and (2), as they may look a bit complex for non expert readers

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

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