Disorder in AdS$_3$/CFT$_2$

Dear Editor,
we thank both the referees for carefully reading our manuscript and for the detailed comments that we address below. A detailed list of all changes is given below. First however, given that we have answered all of the referees questions in detail, we kindly ask to reconsider our submission for publication in SciPost Physics.
We first address the comments of report 1 dated 29.05.2022 and afterwards the comments of report 2 dated 02.06.2022.

*** report dated to 29.05.2022

2.1) Indeed, the metric can be obtained without involving perturbation theory from the gauge fields by using eq.~(14) in our paper. The resulting components are functions of $\mu$, $\bar{\mu}$, and their derivatives. For clarity, we have now included the non-perturbative expressions for the metric components in Appendix A, eq.~(66). The charges $\cal L$ and $\bar{\cal L}$ appearing in these equations are given by eq.~(12) and eq.~(13), respectively. Averaging these metric components analytically involves integrals for which closed expressions could not be found. Therefore, as the referee expected, we expand in $\epsilon$ and $\bar{\epsilon}$ to be able to analytically compute the averages. We now state this more clearly below eq.~(21). Regarding the suggestion about using numerics, we refer to the next question where we made use of numerical analysis to solidify some of our statements.

2.2) The metric components given in eq.~(67) [former eq.~(62)] are expanded in the disorder strengths $\epsilon,\bar{\epsilon}$, not in the radial coordinate $z$. Going to higher orders in $\epsilon$ does not lead to a qualitative change of the $z^2$ terms, and hence the infrared divergence persists. We confirm this now, once by calculating the average of the fourth-order expansion of the metric components (eq.~(79)) in Appendix C and once by a numerical analysis described in Appendix D. In both cases, the IR divergence can be found, see eq.~80 and fig.~2. Hence, we confirm the referee's expectation that the marginally relevant disorder gives rise to this IR singularity.

3.1) Before addressing the actual questions, we clarify a potential misunderstanding. The energy-momentum tensor eq.~(32) resulting from the averaged metric does not satisfy QNEC for all values of the disorder strengths $\epsilon$ and $\bar{\epsilon}$. We analysed this in sec.~3.2, cf.~Fig.~1 in the main text. However, as discussed in sec.~4, after resummation, the resulting energy-momentum tensor fulfills QNEC. We furthermore point out that the fulfillment of QNEC after resummation is not a requirement for fixing the constants in the resummation procedure, but directly follows from it, cf.~eqs.~(61) and (62).

Both the energy-momentum tensors in eq.~(32) (before resummation) and in eqs.~(47)/(56) imply that the disorder sources mass and angular momentum. We state this below eq.~(32) for the energy momentum-tensor before resummation.

The two energy-momentum tensors compare in the following way. The state in eq.~(32) does not satisfy the trace anomaly equation relating the trace to the boundary Ricci scalar. In addition, it does not satisfy QNEC in general. Its dual bulk geometry contains curvature singularities in the IR region (eq.~(26)), is not an AdS spacetime everywhere (eq.~(27)), and has a singularity in the causal structure (eq.~(28)). All of these properties are cured by the resummation procedure performed in sec. 4.

3.2) We agree that in hindsight, the result for the trace anomaly might not be surprising. However, to the best of our knowledge, it is not a priori clear that for a metric not solving the vacuum Einstein equations, the trace anomaly equation must be violated. A counterexample of a non-vacuum solution where the trace anomaly equation holds can be found e.g. in ``Marc Henneaux, Cristian Martinez, Ricardo Troncoso, and Jorge Zanelli.
Black holes and asymptotics of 2+1 gravity coupled to a scalar field.
Phys. Rev. D, 65:104007, 2002''.

Regarding QNEC, there is no direct relation between a metric not satisfying the vacuum Einstein equations and the possible violation of QNEC. Rather, there exist counter examples which can be found in reference [49] in the main text and references therein. Generically, the QNEC inequality may not be saturated in a state with bulk matter switched on, but that does not mean it has to be violated.

In order to further analyse the origin of violation of the trace anomaly equation and QNEC, we would need knowledge about the effective theory to which our averaged metric is a solution to. Such an effective theory is, at the time of writing, however not available to us; we comment on this in more detail in the conclusion section of the main text. The results for the trace anomaly and QNEC as we explain them in the paper provide help in constraining the search for an effective theory.

4.1) To define our resummation, we proceed in a similar way as the references pointed out in the text. We modify the metric and demand the resummation to cure the divergent behaviour. The fact that we do so in eq.~(41) after averaging is only due to convenience in the calculation. Since, by definition, the resummation functions only depend on $z$, and in particular not on the random phases $\gamma_n$, the modification commutes with the averaging procedure. Concerning the relation to the CS formulation, so far, staying in radial gauge, we could not find a good ansatz for the resummation on the level of the Chern--Simons (CS) fields. However, since on the classical level the Einstein--Hilbert (EH) formulation is equivalent to the CS formulation, there has to be a way to implement the resummation also in the CS formalism, most probably by going beyond radial gauge. As we comment in the conclusion, we leave this task for future work.

Furthermore, there is a related question at which level the average is to be imposed. In our work, we choose to impose it after the metric components are obtained. It might also be interesting to directly average the CS connection. Since the metric is bilinear in the connection, the result will be different, implying that the classical equivalence of EH and CS interferes with introducing disorder. We have added some comments regarding this issue below eq.~(20).

Regarding the relation between the resummed metric and the one originally obtained from the disorder, we point out that the resummation is not performed arbitrarily. Throughout the resummation process described in sec.~4, various reasonable physical conditions are imposed on the metric. These conditions lead to constraints on the resummation parameters fixing the parameters uniquely, up to freedom of choice for $b_{12},b_{22},b_{32}$, whose choice does not violate QNEC. We discuss the determination of $b_{12},b_{22},b_{32}$ in the last step of the resummation explained on p.13-15 in the main text.

To say more about the relation of the resummed and the averaged metric, we inserted the resummed metric with general open coefficients eq. (41) in the Einstein equations. Expanding to second order in $\epsilon,\bar{\epsilon}$, where we assume again that the open coefficients can be thought of as a power series in $\epsilon,\bar{\epsilon}$ and demanding that the Einstein equations are satisfied yields differential equations fixing the $z$ dependence of the coefficients $\alpha(z)$ and $\beta(z)$. Up to an integration constant, solving the differential equations directly yields the resummed solution eq. (45) with $b_{12}=b_{22}=b_{32}=0$. The integration constant then again has to be set to zero using the BPS condition as described in section 4. So we find that the resummed metric is not a generic new solution to the Einstein equations, but is fixed by solving differential equations following from the averaged metric including the resummation parameters. We have added these explanations in the main text after discussing the BPS bound at the bottom of page 14 and the top of page 15, in the paragraph containing the new equations (57)-(60).

4.2a) We thank the referee for asking this question. Indeed, the position $z_h$ where $g_{tt}(z_h)=0$ may be interpreted as a horizon. Denoting the induced mass and angular momentum as $M=\frac{\epsilon^2+\bar{\epsilon}^2}{24}$ and $J=\frac{\bar{\epsilon}^2-\epsilon^2}{24}$, the $tt$-component of the resummed metric may suggestively be written as
\begin{align}
\d s^2=-\frac{1}{z^2}+M+\mathcal{O}(\epsilon^4)\stackrel{?}{=}-\frac{1}{z^2}+M+\frac{M^2-J^2}{4}z^2.
\end{align}
To see if this really is the case, the fourth-order analysis now included in Appendix C will have to undergo the resummation procedure. Accompanying this, an effective description in the Chern--Simons formulation has to reproduce these fourth-order considerations using an appropriate effective gauge field. In order for this to describe a black hole solution, the gauge field has to satisfy the holonomy condition. We intend to work this out in more detail to see whether disorder can source black hole solutions.

4.2b) The resummation procedure we perform yields a reasonable physical system connected to the disorder we initially introduce. This is now made more clear also by a slightly different resummation procedure that we describe in the paragraph around the new equations (57)-(60), namely by fixing the resummation functions via solving differential equations resulting from the Einstein equations. As we now show by the analysis in Appendix C, going to fourth-order in perturbation theory does neither resolve the curvature divergence in the IR region nor the violation of the trace anomaly equation. Moreover, in Appendix D we show numerically that the curvature divergence is also present non-perturbatively.

We set these resummation parameters to zero in order to minimise the influence of the resummation on the result. In more detail, the parameters receive bounds by the BPS analysis performed in section 4. By these bounds, we determine the minimal amount of extra mass that we have to include via the resummation procedure in order to obtain a geometry satisfying the BPS bound. In particular, the minimal amount we have to include yields a resummed metric saturating the bound for generic $\epsilon$, $\bar{\epsilon}$. Since we do not want to include more mass by the resummation as necessary, we set the corresponding parameters to the lowest possible value. Following the above lines, we have enlarged the discussion about this point in the paragraph between eqs. (52) and (53) on page 14.

*** report dated to 02.06.2022

(1) The marginal relevance of our choice of disorder implies that we should expect drastic changes in the IR bulk region. Therefore, as we point out in the paragraph below eq. (29), we are not surprised to encounter divergent quantities. In the infrared, where the divergent behaviour appears, the perturbative treatment breaks down since the perturbative corrections grow unbounded and eventually become much larger than the zeroth order of the quantity in question. Therefore, the marginal relevance of our disorder leads to a breakdown of the perturbative treatment in certain limits.

In our approach, we make use of the fact that the three-dimensional equations of motion of gravity in a negatively curved spacetime can be solved exactly. This yields, for every choice of (static) chemical potentials $\mu(\phi),\bar{\mu}(\phi)$, the solutions for the charges ${\cal L},\bar{\cal L}$ (eq. (12) and eq. (13) in the main text). For every choice of the chemical potentials, which in particular includes our disordered chemical potentials (eq. (16) and eq. (17) in the main text), these charges are fixed such that the spacetime has constant negative curvature, i.e. is locally AdS$_3$. However, averaging the metric components results in a geometry which, to $\mathcal{O}(\epsilon^2)$, does not have constant negative curvature. In particular, the curvature diverges for $z\to\infty$ (see eq. (26) in the main test). Furthermore, this geometry leads to an expression for the entanglement entropy that can be negative for sufficiently large length $l$ of the RT geodesic (see eq. (40) in the main text). Hence we conclude that these features, which are clearly unphysical, result operationally from the disorder average.

To summarise, we see that using marginally relevant disorder leads to a breakdown of perturbation theory in certain limits. In our approach, this breakdown of perturbation theory is indicated by quantities showing unphysical behaviour. This unphysical behaviour is due to computing quantities from the disorder averaged metric without the resummation being implemented. Along these lines, we have enlarged the explanations in the second paragraph on p. 4 and the first and second paragraphs on p. 16 to make the connection between the unphysical behaviour, the breakdown of perturbation theory, and the marginal relevance, more clear. Moreover, we added clarifying comments below eq. (24), in the paragraph below eq. (29) and eq. (40).

(2) The resummation functions, as written in eq.~(42) in the main text, are defined to only depend on the radial coordinate $z$. This is in line with the resummation functions of the references cited; there as well, the resummation functions depended only on the radial coordinates and in particular not on the random phases. Therefore, the resummation functions commute with the averaging procedure eq.~(18). More generally, due to the definition in eq.~(18), for any function $f$ that does not depend on the random phases it holds that
\begin{align}
\expval{f}=\lim\limits_{N\to\infty}\int_0^{2\pi}\prod_{n=1}^N\frac{\d\gamma_n}{2\pi}\,f=f\lim\limits_{N\to\infty}1=f.
\end{align}
In this sense, we confirm the statement of the referee that the average over the resummation functions is trivial. For the specific case of the metric components, the commutation can be seen as follows. Consider resumming the metric components before disorder, that is
\begin{align}
g_{tt}\to\frac{g_{tt}}{\alpha(z)}\quad\text{and}\quad g_{\phi\phi}\to\frac{g_{\phi\phi}}{\beta(z)}.
\end{align}
The average of the resummed metric components is then given by
\begin{align}
\expval{\frac{g_{tt}}{\alpha(z)}}=\frac{\expval{g_{tt}}}{\alpha(z)},\quad\expval{\frac{g_{\phi\phi}}{\beta(z)}}=\frac{\expval{g_{\phi\phi}}}{\beta(z)}.
\end{align}
For an appropriate choice of the coefficients within $\alpha$ and $\beta$, the resummed geometry is well-behaved. From the above two equations it is clear that the resummation commutes with the average. In the main text, in line with the approaches of the cited references using a resummation, we are ultimately interested in the averaged geometry. In particular, we do not demand that the geometry should be regular in each disorder realisation, so before performing the average. Therefore, we are free to include the resummation functions before or after averaging. Following this reasoning, we have improved the discussion on p.~12 and p.~13 below the list where we discussed the commutation of the resummation with the averaging procedure.

(3) To our knowledge, it is not known whether the Poincaré-Lindstedt method applied in holographic systems affects physical observables. Being inspired by this method, the same is true for our resummation procedure. Considering the energy-momentum tensor obtained before averaging in eq.~(32), it is clear that the resummation has to have some effect on the components since the trace does not vanish and the trace anomaly in curved backgrounds is not satisfied. To restore these results, it is clear that the components of the energy-momentum tensor have to be modified.

To fully answer the question about the effect of the resummation procedure on physical observables, knowledge about the effective theory is required. We briefly touch on this question in our conclusion in the second paragraph of ``Poincar\'e-Lindstedt inspired resummation''. The effect of the PL or our resummation on physical observables is an interesting subject to study, however, due to the lack of an effective theory, we have to postpone studying this in detail to future work.

(4) Originally, the plot was generated without using colours. Therefore, the line was included to mark the border between QNEC being satisfied and violated. In the current version with colours, this line is no longer necessary. We, therefore, follow the recommendation of the referee and removed the line.

(5) It is true that the disorder causes a breaking of the translational symmetry in $\phi$ direction. However, all of our analysis we perform only after disorder averaging the metric. The averaged metric components, displayed in eq.~(25), do not depend on $\phi$, indicating that the translational invariance is restored by the averaging procedure. We therefore think that a mass for the graviton would only be visible before performing the average. To our understanding, the fact that the Einstein vacuum equations are not satisfied is linked to the marginally relevant nature of the disorder.

(6) Indeed, using eq.~(14) it can be obtained straightforwardly to which metric components the disorder contributes. We now include the metric components before expanding in the disorder strength in appendix A, eqs.~66. As can be seen there, the disorder influences $g_{tt}$, $g_{tz}$, $g_{t\phi}$ and $g_{\phi\phi}$. The disorder is such that upon averaging, $\expval{g_{tz}}=0$, i.e., the averaged Poincar\'e patch metric is again in Fefferman--Graham form.

(7) We thank the referee for pointing out these references. We included them in our revised version of the introduction in the first paragraph located entirely on page 3, together with some brief comments on the systems studied therein.

(8) Again, we thank the referee for pointing out these references. We included them in our revised version in the introduction in the first paragraph on page 3 (references [18,19]). When discussing the divergences we find in our analysis, we again comment on these two references on page 9 in the second two last paragraph before section 3 along the following lines. While the non-perturbative analysis explained in these references is very interesting in interpreting the results originally obtained by Hartnoll and Santos (reference [14] in our paper), the setup is different to our approach. We do not consider additional matter fields like the scalar field to introduce disorder, but use the chemical potentials within the metric to source the disorder. Due to this conceptual difference, we do not think that the IR divergences found in our approach can be related to the analysis of 2004.06543 and 2110.11978. It will however be interesting to study how their approach could be adapted to our setup.

(9) Following the suggestions by the referee, we have enlarged the discussion about Anderson localisation in the introduction (second to last paragraph on page 3), also discussing the references pointed out in the report.

Regarding the necessity of quantum effects, we comment the following. Anderson localisation as a phenomenon is not tied to quantumness in the system, but is rather a generic result if waves are considered. While the original paper of Anderson dealt with a tight-binding model where the wave function has the probability interpretation of quantum mechanics, the mathematics behind do not require this interpretation. The notion of localisation can also be found for systems of electromagnetic ([33-35] in the main text) or acoustic waves ([36-38] in the main text) with disorder. This is due to the fact that Anderson localisation results from waves interfering after taking different scattering paths in the disordered potential. Assuming the strong scattering limit, the interference becomes completely destructive, yielding the localisation interpretation. Therefore, the lack of quantum effects does not prohibit Anderson localisation, and correspondingly large $N$ limits may help in shedding light on this phenomenon. We have included this discussion in the introduction in the last paragraph on page 3.

- added the references 1509.02547, 1401.7993, 1505.05171, 1802.08650, 1409.6875, 1511.05970 on prior studies of disorder and commented on their results in the first paragraph located entirely on p. 3 (starting with: ``Einstein-Maxwell theory in D=4 with a disordered ...'')

- enlarged the discussion on Anderson localisation in the introduction in the second to last paragraph on p. 3 (starting with ``The possibility of Anderson localisation in holographic systems ...''). Added and commented on the results regarding Anderson localisation of 1507.00003, 1601.07897, 1602.01067, 1711.10953

- added discussion on the necessity of quantum effects for Anderson localisation in the last paragraph of p. 3, ending on the top of p .4. In this discussion, added the references [33-38]

- added a sentence in the second paragraph on p. 4: ``As we will discuss in more detail in section 2.2, ...''

- added comment on the non-compatibility of the classical equivalence of gravity in the second and first order formulation and the averaging procedure at the end of the paragraph below eq. (20) (p. 7)

- clarified why we expand the exact metric components in \epsilon and \bar\epsilon in the paragraph below eq. (21) (p. 7)

- added a sentence below eq. (24) ``This will be indicated ...''

- added a sentence in the paragraph below eq. (29) ``Since before averaging, the solutions for the charges ...''

- added the references 2004.06543, 2110.11978 and commented on their results on p. 3 in the first paragraph (references [18,19]). Also, added a brief discussion on the relation between their and our approach on p. 9 in the second to last paragraph before section 3

- added a half-sentence below eq. (40) ``..., resulting from the disorder averaging''

- removed the light blue line in figure 1

- enlarged the discussion regarding the commutation of our resummation and the averaging in the paragraph after the list on p. 12

- refined the discussion how to fix the resummation parameters in the paragraph between eqs. (52) and (53) on p. 14

- added an alternative derivation of the resummed metric using the Einstein equations at the bottom of p. 14 and the top of p. 15, in the paragraph containing the new eqs. (57)-(60)

- added a sentence in the second paragraph on p. 16 ``The marginally relevant nature of our disorder ...''

- added a sentence in the third paragraph on p. 16 ``Again, this is due to the marginally ...''

- added exact results for the metric components, before expanding in \epsilon and \bar\epsilon, in Appendix A (eqs. (66a)-(66f))

- added average of the fourth-order expansion of the metric components and the resulting curvature and boundary EM tensor in the new Appendix C

- added numerical evaluation (i.e., without expanding in \epsilon and \bar\epsilon) of the Ricci scalar curvature in the new Appendix D