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Late time physics of holographic quantum chaos

by Alexander Altland, Julian Sonner

This Submission thread is now published as

Submission summary

Authors (as registered SciPost users): Julian Sonner
Submission information
Preprint Link:  (pdf)
Date accepted: 2021-07-23
Date submitted: 2021-07-01 10:58
Submitted by: Sonner, Julian
Submitted to: SciPost Physics
Ontological classification
Academic field: Physics
  • High-Energy Physics - Theory
Approach: Theoretical


Quantum chaotic systems are often defined via the assertion that their spectral statistics coincides with, or is well approximated by, random matrix theory. In this paper we explain how the universal content of random matrix theory emerges as the consequence of a simple symmetry-breaking principle and its associated Goldstone modes. This allows us to write down an effective-field theory (EFT) description of quantum chaotic systems, which is able to control the level statistics up to an accuracy ${\cal O} \left(e^{-S} \right)$ with $S$ the entropy. We explain how the EFT description emerges from explicit ensembles, using the example of a matrix model with arbitrary invariant potential, but also when and how it applies to individual quantum systems, without reference to an ensemble. Within AdS/CFT this gives a general framework to express correlations between "different universes" and we explicitly demonstrate the bulk realization of the EFT in minimal string theory where the Goldstone modes are bound states of strings stretching between bulk spectral branes. We discuss the construction of the EFT of quantum chaos also in higher dimensional field theories, as applicable for example for higher-dimensional AdS/CFT dual pairs.

Author comments upon resubmission

Dear Editor, we would like to thank the referees for the thorough reading of our manuscript and for their helpful suggestions. Please find below a list of responses and changes addressing these comments. We hope these address all outstanding issues and allow the paper to be published in the present form. Regards, The Authors.

List of changes

- At the bottom of page 4 in the paragraph below Eq. (1.2), we added a further sentence clarifying the fact the low-energy in the EFT treated in this paper refers to small energy differences. In particular we added the sentence "We remark that, as will become clear below, the notion of `low-energy' in the EFT we develop here, refers to small {\rm differences} of energies in the spectrum of the original system."

- We have looked at the figure on top of page 16 and do not agree with the referee's comment regarding the direction of arrows, which are both in line with the convention in the literature (on flavour-space matrix models) and correspond to the contraction flow in the corresponding formulae.

- Addressing the issue of typos and notational consistency: we have gone once again over the entire manuscript and eliminated various lingering typos. In particular (but not exclusively) we have corrected all those pointed out by Referee 1. We thank them for their attentive reading. Please find below a list of all changes that are more significant than just adding or removing a plural 's' here and there. Other changes of the more 'trivial' kind are left uncommented.

- We updated the notation of various kinds of traces occurring throughout the manuscript to be fully consistent. We now use 'Tr' for ordinary traces and 'STr' for graded traces throughout. This also applies to all figures in which such traces appear.

- "two two" changed to "to two"

- one $\epsilon$ removed from text in paragraph just below (3.23)

- "s world-sheet -> a world sheet"

- one more important typo we noticed and fixed: in Eq. (2.24) the super trace structure of formula corresponding to the diagram shown was slightly wrong and has now been corrected.

Published as SciPost Phys. 11, 034 (2021)

Reports on this Submission

Anonymous Report 1 on 2021-7-2 (Invited Report)

  • Cite as: Anonymous, Report on arXiv:2008.02271v3, delivered 2021-07-02, doi: 10.21468/SciPost.Report.3170


1- This paper very clearly reviews a formalism for understanding some universal behavior (statistics of energy levels) in chaotic quantum systems, with an emphasis on applications to quantum gravity. This formalism is not widely appreciated in the quantum gravity community, but is certainly of some interest to this community. Likewise, recent developments in this subject from the quantum gravity community may be of interest to the condensed matter/quantum chaos community, for whom the formalism described is more well known, but the applications to quantum gravity may not be. The paper does a good job of making the material accessible, and will likely be a valuable resource for researchers trying to understand progress made in the other field.

2- This paper also provides some novel applications of this formalism for 2D quantum gravity.


Typos, etc, already addressed by authors.


I think this is a very useful paper and worthy of being published in this journal.

The application of the approach to studying level statistics discussed on this paper (based on causal symmetry breaking) to holography, without recourse to ensemble-averages, is intriguing. This is a topic currently under active discussion, and this paper provides some valuable input.

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

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