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Disorder in AdS$_3$/CFT$_2$

by Moritz Dorband, Daniel Grumiller, René Meyer, Suting Zhao

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Authors (as registered SciPost users): Moritz Dorband · Daniel Grumiller · Rene Meyer
Submission information
Preprint Link:  (pdf)
Date submitted: 2023-01-13 10:03
Submitted by: Dorband, Moritz
Submitted to: SciPost Physics
Ontological classification
Academic field: Physics
  • Gravitation, Cosmology and Astroparticle Physics
  • High-Energy Physics - Theory
Approach: Theoretical


We perturbatively study marginally relevant quenched disorder in AdS$_3$/CFT$_2$ to second order in the disorder strength. Using the Chern-Simons formulation of AdS$_3$ gravity for the Poincar\'e patch, we introduce disorder via the chemical potentials. We discuss the bulk and boundary properties resulting from the disorder averaged metric. The disorder generates a small mass and angular momentum. In the bulk and the boundary, we find unphysical features due to the disorder average. Motivated by these features, we propose a Poincar\'e-Lindstedt-inspired resummation method. We discuss how this method enables us to remove all of the unphysical features.

Current status:
Has been resubmitted

Author comments upon resubmission

Dear Editor, we thank the referees for accepting and carefully reading our manuscript as well as for the detailed comments that we address below. A list of all changes to our manuscript is given after our answers to the referees comments.

*** report dated to 29.11.2022

1) We apologise for our somewhat ambiguous formulation, which we clarify in the following. The metric resulting from averaging over realisations in the numerical computation is indeed a function both of $z$ and $\phi$. Of this metric, we compute the Ricci scalar by its definition in terms of the metric and derivatives of the Christoffel symbols, again as a function of $z$ and $\phi$. We state this in the forth sentence of the last paragraph on page 22. However, we do not invoke a numerical package, so both the metric and the resulting Ricci scalar are given by analytical expressions. We chose to do so since this treatment works well at least up to $N=15$, which is sufficient for the arguments we give in the paper. For $N=100$, while the code still works in principle, since we are using analytical expressions, plotting the output for the Ricci scalar becomes extremely lengthy on the hardware available to us. Since the comment about averaging over $N=100$ realisations is not essential to what is discussed in appendix D, we chose to remove the corresponding sentences on the bottom of page 22. We now only state that our code works well at least up to $N=15$, which is sufficient for our analysis.

2) We confirm the observation of the referee: the resummed solution does not agree with the numerical solution. Such a matching is expected in the setup of e.g. ref. [14] of our paper since they are using the Poincaré--Lindstedt method. In our case, while our resummation is inspired by this method, it works fundamentally different. In particular, our resummation is defined such that the singular behaviour in the IR, which as we show by our numerical analysis is clearly present beyond perturbation theory before resummation, is removed. Therefore in the IR, we expect the resummed solution to always be different from the numerical solution. This is the reason why we call our method ``Poincaré--Lindstedt inspired''. In the first paragraph entirely on page 16 we elaborate on this difference. To further indicate this difference, we have added a few words in brackets in page 4 in the beginning of the second to last paragraph, stating that our method is different from the original Poincaré--Lindstedt method of e.g. ref. [14]. We do however emphasise that the relationship between the resummed metric and the geometry induced by the disordered chemical potentials is not arbitrary. By our resummation, we fix all the open resummation parameters uniquely by imposing physically reasonable conditions, or alternatively by also invoking the vacuum Einstein equations. Therefore, the imprint of the original sources of disorder is still present.

2.1) We agree with the referee that finding the resummed metric from solving the vacuum Einstein equations is surprising, at least without using further input. However, the result for the dual energy momentum tensor after resummation (eq. (56) of our paper) is in a form which is compatible with the vacuum Einstein equations. So by knowing this result, we do not expect having to introduce an additional matter field.

3.1) Indeed, computing the energy momentum tensor for each realisation and averaging afterwards yields a healthy result, which in particular also satisfies the trace anomaly equation. We give the corresponding results in the following. Computing the energy momentum tensor for each realisation and averaging results in \begin{align} \langle\langle T_{ij}^{\text{ren}}\rangle\rangle=\frac{c}{288\pi}\begin{pmatrix}-\frac{3}{2}(\bar{\epsilon}-\epsilon)^2 & (\bar{\epsilon}-\epsilon)^2 \ (\bar{\epsilon}-\epsilon)^2 & \frac{1}{2}(\epsilon^2+10\epsilon\bar{\epsilon}+3\bar{\epsilon}^2\end{pmatrix}, \end{align} where $\langle\langle\cdot\rangle\rangle$ denotes the disorder average. Since before averaging, the boundary metric $\gamma^{(0)}$ is still $\phi$-dependent, its Ricci scalar does not vanish. After averaging, it is given by \begin{align} \langle\langle R[\gamma^{(0)}]\rangle\rangle=-\frac{(\epsilon-\bar{\epsilon})^2}{12}. \end{align} Computing the trace of $T_{ij}^{\text{ren}}$ using $\gamma^{(0)}$ and averaging the result yields \begin{align} \langle\langle\tr(T_{ij}^{\text{ren}})\rangle\rangle=-\frac{c(\epsilon-\bar{\epsilon})^2}{288\pi}, \end{align} satisfying \begin{align} \langle\langle\tr(T_{ij}^{\text{ren}})\rangle\rangle=\frac{c}{24\pi}\langle\langle R[\gamma^{(0)}]\rangle\rangle. \end{align} Moreover, $T_{ij}^{\text{ren}}$ is covariantly conserved w.r.t. the boundary metric $\gamma^{(0)}$. In all of these computations, the average was performed at the very end. In particular, the trace of the energy momentum tensor was computed using the energy momentum tensor before averaging. Averaging the determinant of the energy momentum tensor shows that the BPS condition is satisfied. Comparing the above to the energy momentum tensor computed for the averaged metric (eq. (32) in our paper), the trace anomaly equation for curved backgrounds does hold in the above case (eq. (33) in our paper). The value of the trace of the energy momentum tensor is the same in both cases (compare eq. (33) in our paper), but since the boundary metric still depends on $\phi$ in the above approach, the corresponding Ricci scalar does not vanish. Since in our paper, we focus on studying the properties of the averaged geometry, we did not include the above results in the paper.

To the summary: As we discuss in our answer to 2), our resummation, while being inspired by the Poincaré--Lindstedt method of e.g. ref. [14], works fundamentally different. In particular, it renders curvature quantities finite. Therefore, the numerical solution is expected to be different from the resummed solution in the IR. Nevertheless, by our resummation method all resummation parameters are uniquely determined.

List of changes

- added a comment in brackets on page 4 in the beginning of the second to last paragraph ``(but different from)''
- slightly changed the formulation in the paragraph above eq. (57) on page 15; ``approaches''->``approach'' and ``however''->``remarkably''
- removed the comment about $N=100$ in the last paragraph on page 22, ``This code works well at least up to $N=100$, however computing with the corresponding averaged metric becomes lengthy. In particular, plotting the resulting Ricci scalar unfortunately is restricted by the hardware available to us to $N=15$''.

Reports on this Submission

Anonymous Report 1 on 2023-2-13 (Invited Report)

  • Cite as: Anonymous, Report on arXiv:2204.00596v5, delivered 2023-02-13, doi: 10.21468/SciPost.Report.6740


Dear Authors,

Let me first and mostly address item number 2) of your last reply since it deals with the point that I cannot find convincing about this work.

I would say my main objection to the 'resummation' procedure and therefore to one of the main results of the present work is not yet addressed.
Let me describe the process followed in the manuscript as I understand it, and then state my main objection.
The authors plug in a disordered chemical potential which I understand implies, via the CS gravity formalism, that a disordered source is introduced through the metric (is this correct?). Then they solve the gravity system, first analytically in a perturbative expansion in the disorder strength, and then numerically, for a very realistic disorder (including a lot of modes). Both solutions agree in showing that the IR metric is divergent. This finding is not altogether surprising given that the disorder they introduce is Harris-relevant and thus expected to grow towards the IR.
The authors then take the disordered-average metric, at second order in disorder strength, and look for a somewhat minimalistic modification that will make it finite. They also show that this finite metric solves the vacuum Einstein's equations of motion.
(That is why I asked if the geometry corresponding to the introduction of a source for the metric is generically expected to solve the vacuum Einstein's equations, otherwise I guess one could try a much more generic form for the Einstein's equations).

But, crucially, they cannot show (or I have failed to find it) a direct connection between this solution of the vacuum Einstein's equations and the original problem in the Chern-Simons formalism where the disordered chemical potential is defined. Therefore, to me, the connection between the disordered chemical potential and the smooth solution of the vacuum Einstein's equations that they define as the resummed metric is not proven.

In my view more evidence is needed to establish that the 'resummed' metric describes the correct IR physics corresponding to the introduction of a disordered chemical potential in this system. I would say that Gauge/Gravity tells us that the IR of the dual system is governed by the metric resulting from solving the equations of motion after one has fixed the sources corresponding to the physical problem one wants to study. In this case, after setting a disordered chemical potential, via the Chern-Simons formalism of AdS3 gravity, the authors arrive at an IR-divergent metric, which is not totally surprising (see results like the infinitely disordered IR fixed points). I would expect a very strong evidence would be needed to show that this IR divergent metric is not the physical answer and that instead the IR smooth metric they propose is instead the correct (and of course unique) description of the low energy physics of the dual system.
[An analogy that comes to mind when I consider the situation the authors find when solving the system -an IR divergent metric as a result of switching on some particular sources- is what it is usually done when looking for a dual of a QCD-like theory. There one actually constructs duals where the space ends at a singularity and do not try to 'resumm' it away -since said singularity or end of space is expected to reflect the IR physics of QCD]

As the authors admit, they cannot implement their 'resummation' method in the CS formulation of the problem. But it is in this formulation where it is clear that they are introducing a disordered source via a chemical potential. Thus, the connection between the 'resummed metric' (41) and the original problem seems lacking.
As I said before, and the authors admitted, this is very different to the Poincaré-Lindstedt method they quote, cause in that case it is crystal clear that the resummed metric does result from solving the problem corresponding to the introduction of the original disordered source.

Finally, the authors say that
"By our resummation, we fix all the open resummation parameters uniquely by imposing physically reasonable conditions, or alternatively by also invoking the vacuum Einstein equations. Therefore, the imprint of the original sources of disorder is still present."
I see some problems with this reasoning:
a)As I said, it is not clear to me that a smooth metric has to be found for this problem.
b)Is it clear that the metric corresponding to the disordered problem they introduce via the CS formalism has to solve the vacuum Einstein's equations?
c)Is it the ansatz they take for the resummation the most generic possible and is it obvious why the solution should be obeying a BPS bound (if the 'resummed' metric were to be a solution corresponding to some matter content, should a BPS bound apply at all?)?
d) I think more is needed to prove that 'the imprint of the original sources of disorder is still present'.

3.1) I asked about the energy-momentum tensor averaged over realisations cause indeed it seems that from that point of view, as it should, everything is fine and QNEC conditions and anomaly eqs are fulfilled, right?
Then, one could say that from the point of view of the dual holographic state, as far as the energy-momentum tensor is concerned, there is no need to do a resummation that brings us to a different T_{ij} whose relation to the one corresponding to the original disordered source is not clearly proven.
It might be interesting to compare the T_{ij} the authors report in 3.1) of their reply to the energy momentum tensor after the resummation process.
By the way, is there some metric \gamma missing in the equation for T_{ij}^{ren} in the reply?

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