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From integrability to chaos in quantum Liouvillians
by Álvaro Rubio-García, Rafael A. Molina, Jorge Dukelsky
|As Contributors:||Rafael Molina · Alvaro Rubio-García|
|Date submitted:||2021-10-05 16:35|
|Submitted by:||Rubio-García, Alvaro|
|Submitted to:||SciPost Physics|
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.
Author comments upon resubmission
Anonymous Report 1
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 have 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.
Anonymous Report 2
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 sentence in 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.
List of changes
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.
Following the comments made by both Referees we have added a new Fig. 3 in the results section with the growth of the parameter \gamma as the transition parameter \beta is increased. We have also added a new sentence in the last paragraph of the conclusions commenting these results.
Submission & Refereeing History
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- Report 2 submitted on 2021-08-01 22:46 by Anonymous
- Report 1 submitted on 2021-07-19 14:46 by Anonymous