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From integrability to chaos in quantum Liouvillians
by Álvaro Rubio-García, Rafael A. Molina, Jorge Dukelsky
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Submission summary
| Authors (as registered SciPost users): | Rafael Molina · Alvaro Rubio-García |
| Submission information | |
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| Preprint Link: | https://arxiv.org/abs/2102.13452v1 (pdf) |
| Date submitted: | 2021-03-03 11:59 |
| Submitted by: | Rubio-García, Alvaro |
| Submitted to: | SciPost Physics |
| Ontological classification | |
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| Academic field: | Physics |
| Specialties: |
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| Approaches: | Theoretical, Computational |
Abstract
The dynamics of open quantum systems can be described by a Liouvillian, which in the Markovian approximation fulfills the Lindblad master equation. We present a family of integrable many-body Liouvillians based on Richardson-Gaudin models with a complex structure of the jump operators. Making use of this new region of integrability, we study the transition to chaos in terms of a two-parameter Liouvillian. The transition is characterized by the spectral statistics of the complex eigenvalues of the Liouvillian operators using the nearest neighbor spacing distribution and by the ratios between eigenvalue distances.
Current status:
Reports on this Submission
Report #2 by Anonymous (Referee 2) on 2021-8-1 (Contributed Report)
- Cite as: Anonymous, Report on arXiv:2102.13452v1, delivered 2021-08-01, doi: 10.21468/SciPost.Report.3325
Strengths
1- the paper is very well written
2- the results are relevant and the models used to illustrate them are of actual interest
3- it exploits nicely the machinery of quantum integrable systems as starting point for the analysis of a transition to chaos
4- it provides explicit physical examples of the connection between chaos/integrability and Poisson/RMT statistics of the Liouvillian spectrum in quantum open systems
Weaknesses
1- the use of the term "quantum chaos" is very restrictive and the important aspect of the lack of a classical limit in the systems studied was not addressed.
2- the semiclassical ideas behind the intermediate statistics (for example Berry-Robnik) were not mentioned
3- there is no mention to the high-spin situation where an actual semiclassical limit can be use to unambiguously define quantum chaotic regimes
4- a very obvious aspect that appears to be missing is the dependence of the fitting parameter \gamma with the strength \beta of the chaotic term in the Liouvillian in eq. 17.
Report
Concerning the weak points, as in many papers addressing the "renewed" interest in quantum chaos, the authors seem to assume that a consistent definition of quantum chaos is the compliance to RMT. It is conceptually inappropriate to say that the lack of understanding of several aspects of the integrable/chaotic transition for the systems considered in the paper is due to the fact that the are open. The real reason is that the systems considered lack a proper classical limit where the semiclassical techniques that allow for a precise definition of chaos and the study of their impact on the quantum properties. In this sense, I consider the first paragraphs of the introduction as slightly misleading. In the same spirit, a discussion about the semiclassical understanding of the integrable/chaotic transition would be important, perhaps just mentioning the main idea behind the celebrated Berry-Robnik approach.
Requested changes
1- A careful delimitation of the different kind of systems where BGS can or cannot be properly claimed.
2-including a study of the depenedence of \gamma with \beta.
Report #1 by Anonymous (Referee 1) on 2021-7-19 (Invited Report)
- Cite as: Anonymous, Report on arXiv:2102.13452v1, delivered 2021-07-19, doi: 10.21468/SciPost.Report.3259
Strengths
1- The work is sound, interesting and contributes to current efforts to understand and characterise chaos in open quantum systems.
2- It provides a new family of models displaying a transition from integrability to chaos and shows/reinforces the relevance of certain chaos markers in the context of dissipative quantum systems.
Weaknesses
In my opinion, the transition from integrability to chaos in the proposed models should be studied and characterized further to complete the work. In particular, the following questions should be addressed:
1- At a fixed value of $\alpha$, how do the variations with $\beta$ in the different chaos observables change with the system size ? It would be helpful to show 2d plots of chaos observables as a function of $\beta$ for different values of $N$. Can some finite-size scaling be performed on these observables ?
2- What is the type of transition observed between integrability and chaos ? Is it a cross-over or a dissipative phase transition in the limit where $N$ tends to infinity ? Does it depend on the value of $\alpha$ ?
Report
In this work, the authors present and study a new family of integrable Liouvillians for spin systems and show how to interpolate between the integrability of these models and chaos by adding some integrability breaking term to the Liouvillian. Their integrable models are based on Richardson-Gaudin models. The work is sound, interesting and contributes to current efforts to understand and characterise chaos in open quantum systems. It provides a new family of models displaying a transition from integrability to chaos and shows/reinforces the relevance of certain chaos markers in the context of dissipative quantum systems. For this work to make a meaningful and useful contribution to the community, I believe that the transition from integrability to chaos in the proposed models needs to be further studied and characterised.
Requested changes
In addition to a more thorough study of the transition from integrability to chaos, a number of points should be clarified in the manuscript and some inaccuracies and typos should be corrected. Here are some that I have identified:
1- In the introduction, it should be mentioned that $P(s)\propto s^\beta$ is the expected behaviour for small values of $s$
2- in the second column of the Introduction, “… an important important step …” should read “… an important step …”
3- the discussion following Eq. (1) on the Liouvillian spectrum and the steady state (which may not be unique) is not accurate when the zero Liouvillian eigenvalue is degenerate
4- In Eq. (3), the numbers $z_i^a$ and $x_i$ are not defined in the text
5- In the last line of the paragraph following Eq. (6), "... at site $k_i$ can ..." should read "... at site $i$ can ..."
6- In Eq. (16), my impression is that the first term of the right-hand side should correspond to Eq. (12), which is not the case in the manuscript
7- The authors should justify the choice of orthonormal vectors for the factors entering the chaotic and integrable parts [as described after Eq. (15)]
8- In Eq. (17), $\mathcal{L}_{\mathrm{int}}(\alpha)$ is not defined, although it should correspond to Eq. (12)
9- In Sec. III, the text describing the unfolding procedure of the spectrum is too short and vague to be clear
10- The caption of Fig. 1 does not provide sufficient details. How are the random values of the parameters $w_i$ in Eq. (15) drawn ? Is there a link with the $\omega_i$ used for the $\eta_i$ parameters ?
11- In Fig. 2, colorbars for the $\gamma$, $\langle r \rangle$ and $-\langle \cos\theta\rangle$ values are missing
Author: Alvaro Rubio-García on 2021-09-20 [id 1768]
(in reply to Report 1 on 2021-07-19)
We thank both referees for the careful reading of the manuscript and for their constructive criticism. We have made the best effort to comply with the requested changes. We believe this new version might be ready for publication. We respond to the specific questions and requests of the referees below.
Following the advice of the referee (which was similar to a request of the second referee) we have further analyzed the transition from integrability to chaos in more detail. Specifically, we have studied the transition for different values of the parameter alpha as a function of beta and the size of the system. Unfortunately, the matrix dimensions grow too fast (as usual in many-body quantum systems) to be able to study a span of different sizes large enough to make a proper size scaling. We have concentrated on the extreme values of alpha and added a figure of the increase in gamma, the parameter measuring the transition to chaos, as beta is increased for alpha=0 and alpha=1. We have made smaller calculations for other values of alpha that point to similar results in those cases. The results point to a sharper transition to chaos as the system size is increased. This may indicate that an actual phase transition occurs but the reached system sizes are not large enough to make a definite claim.
We respond below to the other specific changes requested item by item:
1- In the introduction, it should be mentioned that P(s)∝sβ is the expected behaviour for small values of s
We have added the sentence "for small s" after the mention of "P(s)∝sβ" in the first paragraph.
2- in the second column of the Introduction, “… an important important step …” should read “… an important step …”
This typo has been corrected in the new manuscript version.
3- the discussion following Eq. (1) on the Liouvillian spectrum and the steady state (which may not be unique) is not accurate when the zero Liouvillian eigenvalue is degenerate
We have added a sentence at the end of the first paragraph of Section 2 mentioning the possibility of having degenerate steady states and what happens in this case.
4- In Eq. (3), the numbers zai and xi are not defined in the text
These parameters are now defined after Eq. (3).
5- In the last line of the paragraph following Eq. (6), "... at site ki can ..." should read "... at site i can ..."
This error is now corrected in the new version of the manuscript.
6- In Eq. (16), my impression is that the first term of the right-hand side should correspond to Eq. (12), which is not the case in the manuscript
While both equations by themselves are right, it would be clearer for the reader if the first terms of Eqs. (12) and (16) are the same. We have modified the first term of Eq. (16) so that they now coincide.
7- The authors should justify the choice of orthonormal vectors for the factors entering the chaotic and integrable parts [as described after Eq. (15)]
This choice has been made in light of our numerical results using both orthonormal and non orthonormal vectors. We observed that orthonormal vectors showed stronger signatures of chaos than non orthonormal ones. A clarifying sentence has been added in the paragraph after Eq. (15).
8- In Eq. (17), Lint(α) is not defined, although it should correspond to Eq. (12)
We now define L_int(\alpha) at the left hand side of Eq. (12).
9- In Sec. III, the text describing the unfolding procedure of the spectrum is too short and vague to be clear
We have expanded the description of the unfolding procedure around Eq. (18). We think it now gives a clear description of the unfolding procedure used in this work.
10- The caption of Fig. 1 does not provide sufficient details. How are the random values of the parameters wi in Eq. (15) drawn ? Is there a link with the ωi used for the ηi parameters ?
We have added a sentence in the second paragraph of Section 3 detailing that the ωi parameters are drawn from a uniform distribution in the range [-1/2, 1/2] and that we then orthonormalize the vectors using a Grand-Schmidt procedure. The ωi parameters are not related to the ηi ones and we have changed the definition of the ηi parameters so that there is no room for confusion.
11- In Fig. 2, colorbars for the γ, ⟨r⟩ and −⟨cosθ⟩ values are missing
The colorbars have been added to the Figure.
Submission
We submit a pdf file where every change made to the manuscript has been highlighted.
Attachment:
Anonymous on 2021-10-18 [id 1861]
(in reply to Alvaro Rubio-García on 2021-09-20 [id 1768])
I thank the authors for the additional results presented in Fig. 3 of the revised manuscript.
I also recall my main criticism, which was that the transition from integrability to chaos in the proposed models should be further studied and characterised to complete the work.
The authors now present results on the transition parameter $\gamma$ for different system sizes and mention in the text and conclusion that their results "indicate a sharp transition in the thermodynamic limit". I do not agree with this claim. The curves shown in Fig. 3 for the same value of $\alpha$ and different system sizes $L$ do not intersect. In view of this, it is difficult to imagine where the transition is supposed to take place, and even to know if there is a transition in the limit $L\to\infty$. It might perfectly be that $\gamma \to 1$ for any value of $\beta$ (except perhaps zero) as $L\to\infty$. It is of utmost importance to clarify the nature of the so-called transition. Therefore, I cannot presently recommend the publication of this revised manuscript.
Author: Alvaro Rubio-García on 2021-11-17 [id 1950]
(in reply to Anonymous Comment on 2021-10-18 [id 1861])
We agree with the Reviewer in the relevance of completely characterizing the transition to chaos in the thermodynamical limit (TL). For a correct characterization of the spectral statistics and the transition to chaos we need the full spectra of a non-Hermitian matrix. However, the sizes of the Liouvillian matrix in the sector <Sz>=1 grow exponentially with the system's size as:
L = 9, dH = 2907
L = 10, dH = 8350
L = 11, dH = 24068
L = 12, dH = 69576
L = 13, dH = 201643
L = 14, dH = 585690
While sizes like L=12,13,14 could be computed with the help of a supercomputer or a big cluster (to which we do not currently have access to), we think these sizes would probably still not provide enough information to infer the behavior of the transition in the TL. In light of this, we remark the sentence that we have added to the new version of the manuscript when describing Figure 3, which points to the impossibility of characterizing the transition in the TL. We think, however, that the position of this sentence should be modified, leaving the end of the paragraph describing Figure 3 as:
"[...] As L increases the system reaches the chaotic limit for lower values of β, pointing to a sharp transition from integrability to chaos in the thermodynamical limit as random Lindblad jumps are added to the model. However, the system sizes we are able to reach are not large enough to make a definite claim in this regard. The transition seems to occur earlier for α = 0 probably because there is a wider distribution of Hamiltonian parameters in the integrable part of the model with α = 1."
Therefore, it cannot cause the reader to think that the impossibility of a definite claim is referred to the dependence of the transition with α.
Anonymous on 2021-11-18 [id 1954]
(in reply to Alvaro Rubio-García on 2021-11-17 [id 1950])To be clear, I am not asking the authors to push numerical calculations to larger and larger sizes. What I am asking for is clear evidence for or against a transition in the thermodynamic limit. This could be achieved by a thorough finite-size scaling approach. If it turns out that there is no transition, as the data seem to suggest, then the results will probably be less useful for the community. That is why it is important to know what this transition is. In conclusion, I still think that in its current form, the work is not ready for publication.
Author: Alvaro Rubio-García on 2021-12-13 [id 2028]
(in reply to Anonymous Comment on 2021-11-18 [id 1954])
We thank for the Referee's insistence on the subject of the finite size scaling of the transition, which has lead us to new results pointing to a sharp transition to chaos as soon as random Lindblad jumps are added to the system in the thermodynamic limit. In particular, we show in the new version of the manuscript (Fig. 3) how our current computations suggest that the critical point $\beta_c$ at which the system reaches the fully chaotic regime $\gamma = 1$ scales with the system size as $1/L^2$. This gives strong support to the conclusion that the critical point in the thermodynamic limit is $\beta_c=0$.
We attach to this comment a new version of the manuscript.
List of changes: - Fig.3 has been modified to show the scaling of the $\gamma$ parameter with $L^2$. - The description of Fig.3 has been modified in the body of the manuscript to reflect the new conclusion that we derive from the figure, i.e. that the transition to chaos happens at $\beta_c=0$ in the thermodynamic limit. - A new sentence has been added to the last paragraph of the conclusions reflecting the above comments.
Attachment:
Anonymous on 2022-01-31 [id 2139]
(in reply to Alvaro Rubio-García on 2021-12-13 [id 2028])I thank the authors for the new analysis of their numerical data. However, in my opinion, the new version of Fig. 3 brings little added value. In its current form, the collapse of the curves for different system sizes is not evident and might simply be an artifact of the horizontal scale used for the plot. More seriously, the analysis made by the authors shows that there is no transition (existence of a transition point at $\beta\ne 0$) from integrability to chaos in the thermodynamic limit. The smooth transition from Poisson to RMT level statistics is probably simply due to finite size effects. Only at $\beta=0$ does the system display Poissonian level statistics, which follows from the fact that the model is integrable by construction at this point. In conclusion, the proposed model does not seem to exhibit the expected transition from integrability to chaos in the thermodynamic limit, and the original goal of this work seems to me to be compromised. I do not see what significant practical or conceptual value the model proposed by the authors could have and therefore, I cannot recommend this work for publication.
Author: Alvaro Rubio-García on 2021-09-20 [id 1769]
(in reply to Report 2 on 2021-08-01)Concerning the weakness and the requested changes we agree that the many-body character of the Liouvillian is the novelty of the work and that this is the main problem for a theoretical understanding as the relationship between the statistical measures and quantum chaos is only well founded in semiclassical theory. This was not stressed enough in the original version of the introduction. We have added several sentences in the first paragraphs of the introduction regarding this issue and we have also added an explanation of the semiclassical theory of Berry and Robnik for the intermediate statistics.
We have also studied in more detail the dependence of the fitting parameter \gamma with the strength \beta of the chaotic term. We have added a new Fig. 3 in the results section, and a new paragraph the conclusions. This point has been also the main request of the first report and a more detailed answer in this regard can be found there.
We have submitted a version with highlighted changes to the reply to Referee 1