SciPost Submission Page
Geometry of variational methods: dynamics of closed quantum systems
by Lucas Hackl, Tommaso Guaita, Tao Shi, Jutho Haegeman, Eugene Demler, Ignacio Cirac
This is not the current version.
|As Contributors:||Lucas Hackl · Jutho Haegeman|
|Arxiv Link:||https://arxiv.org/abs/2004.01015v1 (pdf)|
|Date submitted:||2020-05-04 02:00|
|Submitted by:||Hackl, Lucas|
|Submitted to:||SciPost Physics|
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.
Submission & Refereeing History
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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.
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.