# Phases of scrambling in eigenstates

### Submission summary

 As Contributors: Tarek Anous · Julian Sonner Arxiv Link: https://arxiv.org/abs/1903.03143v2 (pdf) Date submitted: 2019-05-03 02:00 Submitted by: Anous, Tarek Submitted to: SciPost Physics Academic field: Physics Specialties: Condensed Matter Physics - Theory High-Energy Physics - Theory Approach: Theoretical

### Abstract

We use the monodromy method to compute expectation values of an arbitrary number of light operators in finitely excited ("heavy") eigenstates of holographic 2D CFT. For eigenstates with scaling dimensions above the BTZ threshold, these behave thermally up to small corrections, with an effective temperature determined by the heavy state. Below the threshold we find oscillatory and not decaying behavior. As an application of these results we compute the expectation of the out-of-time order arrangement of four light operators in a heavy eigenstate, i.e. a six-point function. Above the threshold we find maximally scrambling behavior with Lyapunov exponent $2\pi T_{\rm eff}$. Below threshold we find that the eigenstate OTOC shows persistent harmonic oscillations.

### Ontology / Topics

See full Ontology or Topics database.

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

### Submission & Refereeing History

Resubmission 1903.03143v3 on 13 June 2019

Submission 1903.03143v2 on 3 May 2019

## Reports on this Submission

### Anonymous Report 1 on 2019-6-4 (Invited Report)

• Cite as: Anonymous, Report on arXiv:1903.03143v2, delivered 2019-06-04, doi: 10.21468/SciPost.Report.992

### Strengths

1- Well written, transparent structure
2- Necessary background is explained in a useful fashion
3- Presentation of new results is intuitive, but at the same time precise
4- Content is interesting and timely
5- Referencing is good

### Weaknesses

1- The content is interesting and useful, though perhaps a little bit limited in scope

### Report

This paper investigates a version of the eigenstate thermalisation hypothesis in the context of 2d CFTs with large central charge and a sparse spectrum. The authors show that correlation functions of any even number of (simple) probe operators in heavy primary states look approximately like thermal expectation values. The relevant temperature is the ETH temperature associated with the heavy state. As an application, the authors compute the OTOC of four probe operators in a heavy primary state and show that under the above assumptions it exhibits maximal Lyapunov growth with the Lyapunov exponent given by the same microcanonical temperature associated with the heavy state. Interestingly, there is a sharp transition between exponential Lyapunov growth and oscillatory behaviour when the heavy primary creating the state approaches the BTZ threshold.

The paper is very well written, clearly structured, and shows appropriate awareness of previous literature. The authors explain all the necessary background in a pedagogical fashion. The paper definitively adds valuable information to the current state of the field. I will recommend the paper for publication.

One comment though, regarding identity block dominance: I understand that the full implications/justifications for this assumption are probably not entirely understood even for the HHLL block. But I wonder if the authors have some more to say about this assumption in the case of intermediate channels between light operators in the HHLL...L blocks. In particular, does the method presented here require a hierarchy of conformal weights of the string of light operators ($h_{Q_1} \gg h_{Q_2} \gg \ldots$)? If so, would one hope for the results about OTOCs to hold outside this regime?

typos:
- Caption of figure 2.
- First sentence on page 14.

• validity: high
• significance: high
• originality: good
• clarity: top
• formatting: perfect
• grammar: perfect