# Chaotic Correlation Functions with Complex Fermions

### Submission summary

 As Contributors: Ritabrata Bhattacharya · Dileep Jatkar Preprint link: scipost_202005_00008v1 Date submitted: 2020-05-19 02:00 Submitted by: Bhattacharya, Ritabrata Submitted to: SciPost Physics Academic field: Physics Specialties: Condensed Matter Physics - Theory High-Energy Physics - Theory Approach: Theoretical

### Abstract

We study correlation functions in the complex fermion SYK model. We focus, specifically, on the h = 2 mode which explicitly breaks conformal invariance and exhibits the chaotic behaviour. We explicitly compute fermion six-point function and extract the corresponding six-point OTOC which exhibits an exponential growth. Following the program of Gross-Rosenhaus, we estimate the triple short time limit of the six point function. Unlike the conformal modes with high values of h, the h = 2 mode has contact interaction dominating over the planar in the large q limit.

###### Current status:
Has been resubmitted

### Submission & Refereeing History

Resubmission 1810.13217v3 on 8 December 2020
Resubmission scipost_202005_00008v2 on 21 September 2020

Submission scipost_202005_00008v1 on 19 May 2020

## Reports on this Submission

### Strengths

1- A very good Introduction section clearly motivates the problem, describes the known results and summarizes the new results of the paper.

2- References are quite appropriately quoted.

### Weaknesses

1- I find that the assumptions as well as motivations are confusingly stated in parts of the paper.

2- Details of the computations are presented within the main text of the paper which distracts the reader from the main message.

### Report

In this paper, higher point (6-point) correlation functions are studied in complex SYK model. This is quite interesting because such computations are usually difficult to perform in general. A study of out-of-time ordered correlation functions performed in the paper is even more instructive in improving our understanding of fine-grained observables in a chaotic holographic theory. I would recommend this paper for publication after the following concerns have been addressed.

### Requested changes

1- Connection and comparison to the work of arXiv:1712.04963 is highly recommended because the theory under discussion is one of the rare examples where explicit computations are possible.

2- I feel it would be instructive to demonstrate some of the 90 possible independent configurations discussed after equation (3.3) to help the reader visualize them.

3- In equation (3.11), I feel that in describing all the steps that lead to the simplification of answer, a reader might be distracted from the fact that the main aim of the simplification was to obtain a conformal 3-point function. Explicitly stating so after the presentation of the equation might be useful. (This is an example of weakness -2 I described above)

4- Below equation 4.15, the authors state that the sum over n is complicated to perform, but this sum has already been performed in the reference [4] of the paper.

5- While the expressions of the 6-point computed in section 4.3 might be 'messy' as quoted by the authors, I strongly believe that it should be presented in the paper, at least in an appendix, as it might be useful for future studies in this direction.

6- In first point of the summary of results for the six-point function, it needs to be stated more clearly whether the 6-point function decreases as a function of time differences or $\mu \beta$. It is not clear from the text.

7- In the summary of results for the six-point function, the difference between the first and the second case is confusing. It is claimed that the first case corresponds to taking the contribution of soft modes first followed by the short time limit; and, the second case exchanges this order of limits. However, equation 4.26 is written in terms of the four-point functions with the contribution of the soft modes already accounted for. A clear explanation here would be greatly appreciated.

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

### Report

The authors seem to know the subject well and their calculations seem correct. Sometimes the written text is a bit imprecise (see below) and there are many typographical issues and typos (which I won't list below). The structure of the paper is a bit unfortunate: most of the paper deals with details of technical calculations (some of which might be unnecessary since the authors in the end consider large $q$, which would presumably simplify the analysis if applied from the beginning), and the interesting section 4.4 containing new physics is extremely short/vague and pushed to the very end of the text. While I would recommend to improve these structural issues, below is a list of some more specific points that the authors might want to consider. After addressing these points appropriately, I will recommend publication since the result is relevant and new.

### Requested changes

1- After eq. 1.2: the authors say that the two-point function $\langle [V,W]\rangle$ might in principle exhibit exponential growth. I think it is clear that this can't happen, since this is known to measure linear response.

2- At the very bottom of p. 2: the 6-point OTOC can indeed be simplified using the KMS condition. However, the individual pieces thus obtained do not take exactly the form that the authors claim here. Application of the KMS condition yields imaginary time shifts in some of the operator insertions. This might be important for the regularization of the OTOC (which the authors might want to address somewhere). It might also be worth pointing out that the out of time order pieces identified here are not all of the same type: for example, a correlation function such as $\langle VWVWVW \rangle$ is "more'' out of time order than one of the form $\langle VWVVWW \rangle$: the former requires an additional timefold, while the latter is qualitatively more similar to the 4-point OTOC. Do the authors expect this distinction to play a role?

3- Sections 2 and 3: without detailed knowledge of previous work (in particular reference [41]), the reader might be quite lost here (e.g., equations 2.4, 2.5, 3.1, 3.2). Various quantities are explained in words briefly, but not to a degree that really defines them. It would be nice to be a little more self-contained.

4- Section 4.4: this is the most important section of the paper as it contains the novel results that go beyond small modifications of previous work. Unfortunately the section is extremely short. As a result it is unclear to me what is being done and I am not quite able to tell whether the analysis is correct or not. I will try to break down some of my confusions:

The first paragraph describes figure 1. It is not clear what figure 1 is showing. It is said that all times in the 6-point OTOC are kept different and the plot shows the behavior as a function of $\tau_1$ as $\tau_1 \rightarrow \tau_2$. If the 6-point OTOC still depends on 4 other times, then what does this plot capture? Have the other 4 times been fixed? To what values (relative to $\tau_2$)? What is the physical significance of this short time limit (and how is the OTOC regularized in such a limit)?

In the third paragraph it is described how the authors compute the Lyapunov exponent without having an analytical expression that could be analytically continued to real time. From the very brief explanation I am not able to tell how and why this procedure works. I would recommend that the authors provide more details.

Figure 2 is also confusing to me. Again, it is not entirely clear what is being plotted here. (I assume it is the OTOC with all times fixed except for $\tau_1$, which is varied. If that's correct, how are the other times fixed relative to each other?). Why do we expect to see oscillatory behavior (I thought OTOCs grow exponentially and then saturate)?

The authors end up extracting a Lyapunov exponent $\lambda_L \approx 1.4$. This value is clearly an interesting result and deserves more physical explanation: $(i)$ Have the authors confirmed (numerically) that taking the chemical potential to smaller and smaller values, the Lyapunov exponent approaches $3/2$? $(ii)$ What can be said about the scrambling time? (The Lyapunov exponent alone doesn't contain all the information about higher-point chaos. If the six-point scrambling time increases by the same factor as the Lyapunov exponent, then the exponential growth cannot be distinguished from the 4-point OTOC raised to some power, which would be a bit disappointing. It is therefore important to compare the two quantities.) $(iii)$ How does this value of the exponent compare with other studies of six-point OTOCs in similar systems? For instance, would it be easy to adapt the numerics to compare various different out of time order configurations and compare their Lyapunov exponents and scrambling times? The Lyapunov exponent/scrambling time for six-point OTOCs with more generic insertion times might be different.

• validity: good
• significance: ok
• originality: ok
• clarity: low
• formatting: acceptable
• grammar: good

Author Ritabrata Bhattacharya on 2020-08-30
(in reply to Report 1 on 2020-06-27)

We thank both the Referees for carefully going through our manuscript and raising extremely constructive comments and criticisms. We apologise for late resubmission, which was partly due to Covid19 related lockdown and poor internet connectivity.

In this revised version, we have attempted our best to address these issues. Below, we enlist the specific changes that have been incorporated in the revised version.

List of changes:

Referee 1:

1. We agree with the Referee’s comment and we have removed the corresponding remark.

2. We agree completely with the Referee on this point. Our aim was to provide an impression of the structure. Therefore, we have added a footnote clarifying further the notational abuse that we have made use of, for simplicity. In the explicit calculation, we have taken care of this shift inside the calculations.

Also, we have further added comments regarding the Referee’s comment on “more” out-of-time order vs “less” out-of-time order.

1. We have attempted a more self-contained description in section 2 and 3. The updates include further explanation following eqns (2.5) and (2.6), (3.1), (3.2) and diagram of contact and planar contributions.

2. We have completely re-written the earlier subsection 4.4 into a new section 5. In this section, we have provided many explicit details of the calculations, as well as improved on the figures and related details. We hope this better explains the results and addresses the questions raised by the Referee.

Referee 2:

1. We have added a short paragraph in the Conclusion, adding a comment in connection of the ref 1712.04963. We thank the Referee for pointing out this reference to us.

2. We have added a footnote demonstrating explicitly how the combinatoric factor of 90 arises in the eqn (3.3).

3. We have stated the goal clearly, at the end of eqn (3.3).

4. The paragraph following eqn (4.15) is edited to clarify the issues. We have also explicitly included the reference [4] here.

5. Added Appendix B with the relevant and explicit details of the various functions.

6. Clarified that the 6-point correlator decreases with increasing \mu\beta.

7. We have clarified first two points summarising results of the computation of six point correlation function. Hopefully this addresses the referee’s concern.

### Attachment:

Author Ritabrata Bhattacharya on 2020-08-30
(in reply to Report 1 on 2020-06-27)

We thank both the Referees for carefully going through our manuscript and raising extremely constructive comments and criticisms. We apologise for late resubmission, which was partly due to Covid19 related lockdown and poor internet connectivity.

In this revised version, we have attempted our best to address these issues. Below, we enlist the specific changes that have been incorporated in the revised version.

List of changes regarding the referee report:

1. We agree with the Referee’s comment and we have removed the corresponding remark.

2. We agree completely with the Referee on this point. Our aim was to provide an impression of the structure. Therefore, we have added a footnote clarifying further the notational abuse that we have made use of, for simplicity. In the explicit calculation, we have taken care of this shift inside the calculations.

Also, we have further added comments regarding the Referee’s comment on “more” out-of-time order vs “less” out-of-time order.

1. We have attempted a more self-contained description in section 2 and 3. The updates include further explanation following eqns (2.5) and (2.6), (3.1), (3.2) and diagram of contact and planar contributions.

2. We have completely re-written the earlier subsection 4.4 into a new section 5. In this section, we have provided many explicit details of the calculations, as well as improved on the figures and related details. We hope this better explains the results and addresses the questions raised by the Referee.

Author Ritabrata Bhattacharya on 2020-08-30
(in reply to Report 1 on 2020-06-27)

We thank both the Referees for carefully going through our manuscript and raising extremely constructive comments and criticisms. We apologise for late resubmission, which was partly due to Covid19 related lockdown and poor internet connectivity.

In this revised version, we have attempted our best to address these issues. Below, we enlist the specific changes that have been incorporated in the revised version.

List of changes:

Referee 1:

1. We agree with the Referee’s comment and we have removed the corresponding remark.

2. We agree completely with the Referee on this point. Our aim was to provide an impression of the structure. Therefore, we have added a footnote clarifying further the notational abuse that we have made use of, for simplicity. In the explicit calculation, we have taken care of this shift inside the calculations.

Also, we have further added comments regarding the Referee’s comment on “more” out-of-time order vs “less” out-of-time order.

1. We have attempted a more self-contained description in section 2 and 3. The updates include further explanation following eqns (2.5) and (2.6), (3.1), (3.2) and diagram of contact and planar contributions.

2. We have completely re-written the earlier subsection 4.4 into a new section 5. In this section, we have provided many explicit details of the calculations, as well as improved on the figures and related details. We hope this better explains the results and addresses the questions raised by the Referee.

Referee 2:

1. We have added a short paragraph in the Conclusion, adding a comment in connection of the ref 1712.04963. We thank the Referee for pointing out this reference to us.

2. We have added a footnote demonstrating explicitly how the combinatoric factor of 90 arises in the eqn (3.3).

3. We have stated the goal clearly, at the end of eqn (3.3).

4. The paragraph following eqn (4.15) is edited to clarify the issues. We have also explicitly included the reference [4] here.

5. Added Appendix B with the relevant and explicit details of the various functions.

6. Clarified that the 6-point correlator decreases with increasing \mu\beta.

7. We have clarified first two points summarising results of the computation of six point correlation function. Hopefully this addresses the referee’s concern.