Destruction of Localization by Thermal Inclusions: Anomalous Transport and Griffiths Effects in the Anderson and André-Aubry-Harper Models

This manuscript comprises a detailed numerical analysis of two toy models conceived to study the breakdown of localization in disordered non-interacting chains. Both of these models introduce locally dissipative effects designed to mimic the effect of delocalised regions. The authors provide an exhaustive comparison of the two models, providing convincing evidence that they may both be understood in terms of Griffiths physics. Specifically, that their transport properties are dictated by rare but large localised regions in which there are no dephasing effects. This is shown through a detailed analysis of both typical quantities and their distributions.

The manuscript is well written, and, after accounting for the requested changes below, I would recommend it be published in Sci-post, for which it is suitable.

--- Recent work has developed the idea (see Refs [53], [54]) that the thermodynamic MBL transition and the finite size MBL “transition” may be driven by distinctly different physics. With [54] noting indeed that “the MBL phase transition actually occurs … far from the numerically-accessible crossover between the finite-size MBL regime and thermalization, reinforc[ing] the idea that the physics of this crossover is likely quite different from that of the ultimate phase transition. This suggests that this crossover should probably be studied as a distinct phenomenon from the MBL phase transition.” I would encourage the present authors to make clearer in their introduction which regime the authors are focussing on: the physics of the thermodynamics transition, or the physics of the numerically accessible transition (I assume the former, as it is in this regime in which spatial variation and Griffiths effects are believed to be important), and to make sure their citations are consistent with this.

--- In Eq 3 I think $k = 1 – c$, however I think $k$ was not introduced in the text.

--- In Figs 5 and 9 the procedure used to make the density plot was unclear. Some explanation and a color bar indicating precisely how I should interpret intermediate colors would seem necessary.

--- Regarding the comment: “As such this result seems to point out that the emergence of subdiffusion and broad dissipation statistics is a robust feature of localized phases coupled to thermal inclusions, irrespectively of the nature of the on-site potential.” It seems important to qualify this with a comment that it applies in the case where the dephasing dynamics is *assumed* to apply to a randomly distributed set of sites. I am aware that the authors make a comment that configurational disorder may potentially lead to this situation in quasiperiodically modulated many body system, however this later comment is somewhat conjectural, whereas the comment above is much more sweeping.

1-Clearly written with good quality figures.
2-Rather complete study of two models of thermal inclusion in localised systems, comparing random disorder with quasi-periodic potentials

1-Some of the appendices could be more self-contained.

This is a careful study of two models of thermal inclusions in localised systems. One where the thermal regions are modelled by local dephasing and second where they are modelled by coupling to random regular graphs. Both models have been studied in detail in the case of random disorder (Refs. 58 and 59). The main contribution of this work is to extend these studies to quasi-periodic potentials, namely the André-Aubry-Harper chain, and compare and contrast the two models of thermal inclusion in the quasi-periodic case to the random case. The main conclusion of the work is that these two cases behave qualitatively the same (apart from the fact that the André-Aubry-Harper chain has a localisation transition in the absence of dephasing).

As both models have been studied in detail, this work may read as slightly incremental. There is new data provided for the random case, complementing the already published data, but not really providing any new insight. However, even if the behaviour of the quasi-periodic case is qualitatively the same, I do think the paper raises some questions and can likely inspire some follow-up work, and therefore I believe it likely satisfies the third expected acceptance criteria of SciPost Physics. The authors may need to comment on wether their work satisfies the 5th general acceptance criteria, requesting that all reproducing-enabling resources be provided.

I have some comments on the manuscript below. My main question after reading the paper is whether one could have expected anything different for the results. I find it interesting to compare potential Griffiths effects in quasi-periodic potentials but to what extend is that really being done here, since the thermal inclusions are still being modelled by randomly located dephasers or random regular graphs. The authors briefly mention this question but I did not quite feel that question was answered. It would be useful to have a bit more extended discussion, perhaps backed up by some data if possible, on this point, if something more can be said about it.

1-In the main text, when discussing the energy dependence in the GOE case, it would be useful to explain the mode of the leads and what energy they are talking about here.
2-After Eq. (14): "Thanks to the translational invariance within the transverse planes." Not sure what they mean here. Aren't these random regular graphs.
3-Figure 4. Can the other provide some more details of the fitting to obtain $\beta(p)$. What are the error bars? Does $\beta(p)$ flow with system size $L$? The authors mention that the super-diffusion can be, or is, only apparent but will go to diffusion at large system sizes. Can they rule out that similar flow of the exponent does not happen for the sub-diffusive case and that it will flow to diffusion at very large systems? I am here interested mainly in wether something can be said with the available data, not some hypothetical even larger system sizes than they can reach.
4-Is there a reason to not include the curves for $p=1$ in, for example, Fig. 4. It could be useful to have that limit as well.
5-Figure 5. What is being plotted in the color scale and what is the scale. The caption seems to suggest that something binary is being plotted, but the white region would suggest otherwise.
6-In the text explaining Figure 7, the other discuss what happens at small $p$, but no data is provided in the case of small $p$. Would it be useful to include that data? Also the caption to the figure simply says "Comparison" but not comparison of what. Finally: the authors discuss how the two models are qualitatively the same after introducing a $p_\text{eff}$, yet in some cases a peak in one model gives a valley in the other. Not sure how seriously one should take this comparison.
7-Maybe add also $\gamma_i$ to Figure 7?
8-Since the first two authors are co-first authors, it would maybe make sense to mark that somehow also in the author list , not only in the acknowledgment.
9-Appendix A: The authors fix $J=1$, likely this should be $t=1$.
10-Notation in Eq. (19). Doesn't $P$ depend on $C$?
11-Appendix B could be made a bit more self contained by a few more sentences of introduction. What are the boson variables, what is the starting point, etc. The could explain more clearly what loops they are referring to when they neglect all loops. And $G_i^{v}$ is not defined. This notation is used in their earlier work but not defined here.
12-There are several references that are not complete. This seems to be mainly arXiv references that are missing the arXiv numbers.


Typos and minor things
1-Throughout analysis should be thorough analysis.
2-Figure 1: Sum in $\mathcal{D}_d$ should be over $i$ instead of $I$.
3-Couples of sites $\rightarrow$ pairs of sites
4-In Eq. (8). $k$ is not defined. Maybe it was supposed to be $c-1$?
5-Just before section 5.1, $t$t=1 $\rightarrow$ $t=1$
6-Page 17. A sentence starts with "On the other,".
7-Eq. (22): should $\psi \rightarrow \phi$
8-Eq. (28): should $\delta_{i,j}(t) \rightarrow \delta_{i,j}\delta(t)$.
9-Is there an extra minus sign in Eq. (48)?