SciPost Submission Page
Chaotic Correlation Functions with Complex Fermions
by Ritabrata Bhattacharya, Dileep P. Jatkar, Arnab Kundu
Submission summary
As Contributors:  Ritabrata Bhattacharya · Dileep Jatkar 
Preprint link:  scipost_202005_00008v2 
Date submitted:  20200921 17:50 
Submitted by:  Bhattacharya, Ritabrata 
Submitted to:  SciPost Physics 
Academic field:  Physics 
Specialties: 

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 sixpoint function and extract the corresponding sixpoint OTOC which exhibits an exponential growth. Following the program of GrossRosenhaus, 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:
Author comments upon resubmission
List of changes
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” outoftime order vs “less” outoftime order.
3. We have attempted a more selfcontained 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.
4. We have completely rewritten 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 6point 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.
Submission & Refereeing History
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Reports on this Submission
Anonymous Report 2 on 20201020 Invited Report
Report
The authors have accommodated the changes suggested in the previous reports. I find that the manuscript is a lot more streamlined than the previous version. I recommend it for publication after some rather minor corrections listed below.
Requested changes
1 Equation of curlyJ on page 4 is incorrect by a squareroot.
2 The organisation of the paper at the end of the section 1 needs an update because of the inclusion of an additional section 5.
3 there seems to be a typo above equation 2.7 “They are antisymmetric, respectively symmetric under …”
4 In figure 1, the vertex between \tau_1, \tau_5 and \tau_3 is not correctly depicted.
5 There are (6C2 x 4C2)/3! options. The division by 3! Is because interchanging the vertices among themselves doesn’t give new diagrams. Moreover, the more pressing issue is that there are index contractions between 12, 34 and 56 labeled fermions. So unless I am missing something, these additional diagrams are suppressed in N counting.
Anonymous Report 1 on 20201011 Invited Report
Report
I thank the authors for improving the manuscript and explaining their methods. Unfortunately I still cannot recommend the paper for publication without substantial improvements.
In sections 4.3 and 5 it is still unclear to me what is being done. It would help if the authors could define exactly what 6point OTOC they are computing (in Lorentzian signature). After (5.1) it is said that the six insertions are ordered in Euclidean time (t) as 0<t1<t3<t5<t2<t4<t6<2pi. This Euclidean ordering determines the order in which the operators appear on the time folded contour. It seems the authors never address the Lorentzian time insertions. Whether the analytic continuation of the 6point function with this specific Euclidean ordering gives a 6point OTOC, or a TOC, or something in between that can be represented on fewer than 3 timefolds, depends on the Lorentzian times as well. If I guess that the authors have in mind Lorentzian times (T) of the form T1=T2>T3=T4>T5=T6 with pairwise identical operators, then the Euclidean ordering described after eq. (5.1) is actually not a 6point OTOC. It can be represented on a contour with only two timefolds instead of three. I would therefore not expect this to display the features of a true 6point OTOC.
Relatedly, as explained around (5.1), the authors seem to study the 6point function only as a function of t1+t2, holding all other times as well as t1t2 fixed. Ignoring the fact that these aren't even Lorentzian times, there remains another issue: defining the 6point Lyapunov exponent as growth w.r.t. a single time difference seems unnatural (of course, depending on the configuration). But in any case I would think that a good definition of 6point Lyapunov growth should know something about all six insertion times.
As a concrete suggestion, I would recommend computing a 6point function with Euclidean times 0<t1<t3<t2<t5<t4<t6<2pi and Lorentzian times as above. This is a true 6point OTOC (i.e., it cannot be drawn on a contour with less than 3 timefolds). Comparing it to the configuration described above would be interesting. This goes hand in hand with a comparison to reference [42] which the authors have added in response to the other referee: in the updated discussion section it is claimed that results found here are consistent with the "maximally braided" 6point OTOC of [42]. I don't think this is correct for two reasons:
(i) As explained above, the OTOC that is being computed here does not seem to be truly "6point OTO", so it is quite different from the one studied in [42].
(ii) Reference [42] claims that the Lyapunov exponent in the 6point function is unchanged compared to the 4point function, which seems to contradict the main result of the present paper (and therefore should be clearly addressed by the authors).
Regarding the numerics: thank you for the additional explanations. I understand the method now. However, I am not convinced that it is correct. The authors compute a purely Euclidean 6point function (which depends on 6 times), fit its dependence on a single time coordinate to a sinusoidal and expect to read off the Lyapunov exponent from that. Several questions:
(i) It is well known that numerical analytic continuation is quite subtle (essentially because it is not unique and hence illdefined). For example, at the end of Appendix G of ref. [4] a method is described for performing such an analytic continuation in case of 2point functions: this numerical algorithm is much more subtle than what the authors do here. If my concerns are unwarranted, I apologize, but do encourage some explanation as to why none of these issues matter here.
(ii) More importantly: for the reason I explained above, knowing only the Euclidean ordering of the operators in the 6point function does not uniquely determine the realtime OTOC at all. It might not even be an OTOC, depending on what Lorentzian times one chooses. For this reason I am suspicious about obtaining the Lyapunov exponent without doing any realtime calculation at all, or even specifying the Lorentzian insertion times.
To summarize: I appreciate the more detailed explanations. However, after understanding better how the 6point OTOC was computed, I am now worried about the correctness of this calculation. The main reason for my worry is that the authors never seem to do any calculation in Lorentzian time. As explained above, having only the Euclidean 6point function is not enough to even specify what 6point OTOC one is talking about. It might be that the authors have a very specific Lorentzian configuration in mind and their simple method of analytic continuation actually computes it correctly. But if that is the case, I am not able to see it from the current manuscript.
Authors’ Responses:
Ref 2: We thank the Referee for carefully going through the resubmission and pointing out the minor corrections. We are currently updating the manuscript with the minor corrections suggested by the Referee.
We will address the Referee’s concerns paragraphwise of the corresponding Referee Report.
 In the first paragraph, the Referee has asked us to clarify the precise correlator that we calculate. We have done so in the manuscript, addressing three key aspects:
(i) The explicit Lorentzian time insertions. We have also included three figures showing the corresponding SchwingerKeldysh contour that we are considering. We have also included the precise i\epsilonprescription that we consider.
(ii) We have calculated both 2time folded and 3time folded correlators. We agree that our previous result was a 2time folded correlator. However, upon the calculation of both 2time folded and the 3time folded correlators, we do observe the same Lyapunov exponent.
(iii) We do not have a complete analytic handle on the calculations, and therefore we have indeed taken some leap of faith in taking the analytic continuation. This is the best we can do at present. However, we have demonstrated that our numerical method reproduces the wellknown Lyapunov exponent that can be extracted from a 4point OTOC, as is analytically done by MaldacenaStanford.
 In the 2nd paragraph, the Referee raises the issue that a true 6point OTOC should depend on 6 time variables. We certainly agree with this. However, it is almost impossible to have control on a function of 6 variables, which is further challenged by only a numerical access to it. Having said this, it is not just simplicity that persuades us to fix three coordinates and vary the other three equally. Such a choice is inspired by the 4point OTOC calculation of MaldacenaStanford, in which one does a similar assignment that nevertheless picks out the correct Lyapunov.
It is certainly true that the most general 6point correlator will contain more information about the system. However, we are not aspiring for a complete analyses of it in the current work.

We completely agree with the Referee’s point in the 3rd paragraph. We have corrected the relevant part of the draft. As suggested by the Referee, we have included the 3time folded calculation. We further thank the Referee for emphasising this point, since it made us look harder into our calculation and we found an error. Upon correcting this error, we now find consistency with reference [42]. This is also a reason we believe our numerics does capture the Lyapunov correctly.

In the 4th paragraph, the Referee raises concerns about the numerical analyses. Here are our responses:
(i) As mentioned earlier, we assumed our analytic continuation does capture the right Lyapunov exponent. It is beyond any doubt that this is not a rigorous statement, and it is marginally impossible to make such a rigorous statement. We have looked carefully into the literature of numerical handle on analytic continuation, and it is far from obvious that any “standard” method would actually apply here, even if one were to implement the same.
On the other hand, we believe the issues and methods of Appendix G of reference [4] are not directly relevant for us. The issue there was to numerically solve the SchwingerDyson equation and from there extract the realtime 2point correlators. We have an analytic handle on the 2point correlator, since we are working in the large q limit. Our main concern is that we cannot perform the final Euclidean timeintegration analytically. Thus, we can obtain the Euclidean correlator, numerically, and the task at hand is to analytically continue this to Lorentzian time.
The fact that our method seems to extract the correct Lyapunov can perhaps be viewed as a quick way to estimate the Lyapunov for a similar calculation in some other system, although we have no further evidence to support this claim, at present.
(ii) We have commented on this issue above.
In conclusion, we have updated the current draft with explicit details on what Lorentzian time insertions we have in mind. It is indeed true that, because of the lack of analytical control on the final Euclidean 6point correlator, we have to assume that our numerical method still extracts the correct value of the Lyapunov exponent. We have reasons to believe so, as is mentioned above and we are grateful to the Referees for the detailed scrutiny that helped us sharpen this statement, to the extent we can.
(in reply to Report 2 on 20201020)
Authors' Responses
Ref 1: We thank the Referee for a detailed report and raising very pertinent issues. There were indeed overlooks on our part. We have incorporated these changes along with changes suggested by Ref 2. We are currently updating the manuscript and we will resubmit it shortly.