Disorder in AdS$_3$/CFT$_2$

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.

- 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 $N=15$''.