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Geometry of variational methods: dynamics of closed quantum systems

by Lucas Hackl, Tommaso Guaita, Tao Shi, Jutho Haegeman, Eugene Demler, Ignacio Cirac

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Submission summary

Authors (as registered SciPost users): Lucas Hackl · Jutho Haegeman
Submission information
Preprint Link:  (pdf)
Date submitted: 2020-05-04 02:00
Submitted by: Hackl, Lucas
Submitted to: SciPost Physics
Ontological classification
Academic field: Physics
  • Atomic, Molecular and Optical Physics - Theory
  • Condensed Matter Physics - Theory
  • Quantum Physics
Approach: Theoretical


We present a systematic geometric framework to study closed quantum systems based on suitably chosen variational families. For the purpose of (A) real time evolution, (B) excitation spectra, (C) spectral functions and (D) imaginary time evolution, we show how the geometric approach highlights the necessity to distinguish between two classes of manifolds: K\"ahler and non-K\"ahler. Traditional variational methods typically require the variational family to be a K\"ahler manifold, where multiplication by the imaginary unit preserves the tangent spaces. This covers the vast majority of cases studied in the literature. However, recently proposed classes of generalized Gaussian states make it necessary to also include the non-K\"ahler case, which has already been encountered occasionally. We illustrate our approach in detail with a range of concrete examples where the geometric structures of the considered manifolds are particularly relevant. These go from Gaussian states and group theoretic coherent states to generalized Gaussian states.

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Reports on this Submission

Report 1 by Thomas Klein Kvorning on 2020-7-2 (Invited Report)

  • Cite as: Thomas Klein Kvorning, Report on arXiv:2004.01015v1, delivered 2020-07-02, doi: 10.21468/SciPost.Report.1801


1. The paper presents a subject which I believe, without the article, it would be hard to access for the intended reader in a pedagogical and accessible manner. The authors have taken to use examples for newly introduced concepts.

2. The article covers a broad range of applications of variational principles and could become a standard reference manual for the subject.


1. The paper is quite long. In part, this is unavoidable for such a broad subject as the article covers, but in some aspects, you wish, as a reader, that it was more to the point. Specifically, by waiting to introduce the projective Hilbert space until page nine, roughly the same material gets presented two times, so after reading ten pages, you have gotten pretty short into the story of the paper.


The authors present the subject of phrasing quantum variational methods in a differential geometric language in a very pedagogical manner. As is evident from the article, this clarifies many subtleties, and I believe it is beneficial for a better understanding of variational methods.

Doing this is hardly new, phrasing the variational manifold in a language of differential geometry using the symplectic and metric tensors is very old indeed. Also, the different methods for approximating the spectrum, the spectral function, and the time-evolution have previously been presented elsewhere (as the authors indicate), but some have not been rewritten in the language of differential geometry, as here.

The real novelty here lies in (1) a careful treatment of the non-Kähler case. E.g., that the two different methods for time-evolution, mentioned in the article differs and that they have quite different behavior when it comes to symmetries.

(2) Using the fact that there are a maximal Kähler subspace and a direct way to get it, Preposition 1 to projecting a time-evolution to a Kähler subspace and thus keeping the Kähler properties, i.e., "Kählerization" (the preposition itself is previously known).

The article does live up to the general acceptance criteria for SciPost, and I think it lives up to one of the SciPost Physics expectations.

The novelties can not be considered a groundbreaking theoretical discovery, neither is it a breakthrough on a previously-identified and long-standing research stumbling block. But it does open up a new pathway in an existing research direction; e.g., it could open up for non-Kähler variational families who, without this article, may not have been considered.

I, therefore, recommend it for publication in SciPost Physics after some minor improvements.

Requested changes

Apart from the general comments given in the other sections of the report, I only have one real issue I think should be corrected, numbered 1) below. The other are small comments or suggestions of typo corrections.

1. There are several Propositions in the article, which I believe, at least in part, are previously known. E.g., at least one of the directions of the equivalence in Proposition 2 follows from the fact that complex projective spaces are Kähler and that complex submanifolds of Kähler spaces are in turn Kähler. Since the article is aimed at a quite broad audience, I do not think one can expect the reader to know what is "well-known facts". A reader could get the wrong impression and think there are more novelties than there are. I think this should be resolved with small comments and or references.

2. In a paper of this length, there are quite many equations, so most readers can not keep track of the equation indexes. By referencing the equations just by number, you get halted in your reading by having to go back. This could be avoided by simply reminding the reader, in words, what equation is referenced. This would make this pedagogical paper even easier to read. There are a few of these examples, but to be specific, I can mention the referencing of (33) on page 10. There one could write "using the definition of the metric and symplectic form from the Hilbert space inner product, (33)" and one would not have to go back to check whether (33) is what just the definition or something else.

3. After proposition 1 it is written "Proposition 1 is also known in the context of classifying real subspaces of complex Hilbert spaces." There should be a reference.

4. On page 10. In the sentence "The fact that the variational parameters are in general real has to be correctly taken into account when projecting time evolution...", the phrase "projecting time evolution" is confusing. I suggest writing "...when projecting the time evolution to the variational manifold..." instead. But the sentence could probably be improved further; it is a bit hard to read.

5. In (5) \psi^\prime should be a function of z_1 and z_2.

6. II C ends with a paragraph explaining that the section is a simplification, and the topic of the section will be treated in full later. As a reader, I would like to have this at the end of the section. It should come first, such that you know what you are expected to get out of the section.

7. In the second paragraph in III, should it not be "equipped with a so called Kähler structure. " and not "equipped with so called Kähler structures. "?

8. In the first paragraph, after Definition 1, it should be "forms", not "form".

9. In (42), removing the origin from C is unnecessary since the vectors are assumed to be non-zero in the above sentence.

10. Above (52) "does not span Hilbert space" should be "does not span the Hilbert space".

11. In Definition 3, it should be said that \psi represents a state in M. It is not valid for a general \psi. This might be evident to most readers but would still make the definition a bit easier to read.

12. Missing subscript on c in (99).

13. Grammar error above (121), "is" should be removed.

14. I guess footnote 15 is meant to clarify why the spectrum of \partial_\nu \chi^\mu makes sense. But I got more confused by it, why introduce a connection? My first line of thought was: As you state, one needs a connection to define a derivative of a vector field since it is a map from one tangent space to another. But even with such a connection, the spectrum of a matrix of that map in a particular coordinate system has no meaning since one can independently change the basis in the different tangent spaces. Why not write something like: Usually, defining a derivative of a vector field requires a way to relate tangent spaces at adjacent points. The derivative would have no well-defined spectrum since it is a map between two different vector spaces. However, at a stationary point, it is a map from a tangent space to itself and thus does not need a connection, and there are well-defined eigenvectors and eigenvalues.

15. What is meant by naive gradient descent on page 26? The one defined by the coordinate dependent flat metric? In any case, there should be an argument or example backing up the statement. And what is the claim? Does g approximate the energy Hessian better than a random metric?

16. \mathcal E is used for two different objects. It is pretty clear which one is meant, but maybe it is a good idea to change.

17. Proposition 1 and 2 have been renamed to proposition 2 and 3 in appendix B. It makes future referencing of them confusing.

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

Author:  Lucas Hackl  on 2020-07-13  [id 882]

(in reply to Report 1 by Thomas Klein Kvorning on 2020-07-02)

We thank the referee for the careful reading of our manuscript and his suggestions for improvement which we were very happy to implement. Please find a revised version of the manuscript attached, where we marked the respective changes in red. Let us add some comments regarding some of these changes. We will also resubmit the revised version to arXiv (without markup).

1: We agree with the referee and added respective comments/citations in front of any proposition which we consider "standard material".

3: We are not aware of a reference on the classification of real subspaces of complex Hilbert spaces, but it follows from the proof of proposition 1 that different types of real subspaces can be classified based on the spectrum of the restricted J. We adjusted the wording accordingly.

6: Section II functions as a prelude and summary of the more technical sections that follow. Readers can start with section II to get an overview and then read sections III and IV on the specific topics they are interested in. Alternatively, a more technically minded reader may skip directly to section III. Based on the referee's feedback, this was not clear in the previous version. We therefore added further remarks at the beginning of sections II and III.

7: We use the word Kähler structure as a general term referring to the three relevant structures (a metric, a symplectic form, a complex structure), which form the triangle of Kähler structures. We are aware that there exist other conventions, where one uses singular ("a manifold having a Kähler structure if it is equipped with a metric, symplectic form and complex structure"), but we decided to use plural for the "triangle of Kähler structures".

14: We agree with the referee that the covariant derivative of a vector field requires a connection. However, once such a connection is chosen we can unambiguously define $K^\mu_\nu=\nabla_\nu \mathcal{X}^\mu$, whose spectrum will be basis-independent, but will depend on the chosen connection. What we point out in the footnote is that at points $\mathcal{X}^\mu=0$, the dependence of $K^\mu_\nu$ on the connection (encoded in $\Gamma^\mu_{\nu\rho}$) drops out, so that the resulting spectrum is even independent of the connection. Of course, the reason for this effect is that $K^\mu_\nu$ is a generator of the diffeomorphism $\Phi$ at one of its fixed points. We followed the referee's suggestion to reformulate the footnote slightly.

15: The statement is that for practical calculations the convergence properties of performing gradient descent with respect to the Fubini-Study metric (equivalent to imaginary time evolution) are better than minimizing the energy expectation value directly with respect to a given parametrization without taking the geometry into account. This is what we called naive gradient descent, by which we mean to use the flat metric with respect to the given coordinates. We do not have a rigorous proof, but anecdotal evidence, i.e., even for simple systems and variational families, one finds slower convergence and may get stuck in local minima. As this was not clear from the manuscript, we clarified our statement accordingly.

16: We changed $\mathcal{E}$ to $\varepsilon$.

Finally, we added an appendix E which gives a more constructive description on how to compute the pseudo-inverse $\Omega$ in the case when $\omega$ is not invertible. We further remark that eq. (8) may be partially ill-defined if $\omega$ is non-invertible, which requires a projection as implemented by defining the pseudo-inverse in the suggested way.



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