Robust extended states in Anderson model on partially disordered random regular graphs

See my report

See my report

The article discusses the recently discovered localization transition in partially disordered random regular graphs (as cited in Ref. [34]). The Anderson transition in random graphs with effective infinite dimensionality has garnered significant interest recently, partly due to its analogy with many-body localization. Recent studies have uncovered intriguing properties, particularly the potential for a non-ergodic delocalized phase. In this paper, the authors investigate partial disorder, where a fraction of the random graph sites has zero disorder. Their primary objective is to elucidate the mechanism behind the presence of a mobility edge, regardless of the disorder strength, leading to the delocalization of states near the middle of the band. The article first describes this aspect and then extends the study to related cases, such as the directed/non-Hermitian scenario and the introduction of a chemical potential for 3-cycles.

While I find the paper's subject matter interesting, I have several reservations about its current form, which prevent me from recommending it for publication in Scipost Phys. I have outlined my comments below:

1. Understanding the Mobility Edge Mechanism: The initial part of the paper appears to be the most crucial. It delves into the mechanism behind the mobility edge in the presence of partial disorder. Let me rephrase the argument to confirm my understanding: the authors explore the limit of infinite disorder and formulate Abou-Chacra-Thouless-Anderson recursion relations for the cavity Green's function in the clean (non-disordered) region. Essentially, infinite disorder excludes certain neighbors, causing fluctuating connectivity between clean sites. The authors approximate this problem by neglecting connectivity fluctuations, justified in the limit of a large connectivity. This approach leads to a self-consistent equation for the cavity Green's function, allowing the prediction of the mobility edge.

It appears to me that this problem resembles the recent rigorous solution for the localization/delocalization transition of eigenstates of the Adjacency matrix (where onsite potentials are zero) of Erdös-Rényi graphs, as detailed in arXiv:2005.14180, arXiv:2305.16294. Notably, states can be localized due to fluctuating connectivity. However, I find it unclear how this localization is described in the authors' self-consistent approach, where connectivity fluctuations are neglected.

I have additional questions concerning the numerical simulations. Figure 4 seems crucial, demonstrating the good agreement between the analytical formula for the mobility edge (mentioned only in the caption, not in the main text?) and the numerical data. However, the plotted variable is D2​ as a function of the parameters E and β. The color scale abruptly changes at D2​=0.5 (white), with mainly red for D2​>0.5 and blue for D2​<0.5. I fail to understand why D2​ is set to 0.5 at the transition. On the contrary, I would expect D2​ to tend to 0 slowly with size. My query is: since the authors possess a theoretical formula, can they precisely determine the transition numerically? The disorder does not necessarily need to be W=1000, and the section where the entire band is delocalized might not be as relevant. The focus should be on the localization transition, which is the crucial aspect of interest.

2. Sparse and Dense RRG Duality: The subsequent section briefly outlines a duality between sparse and dense Random Regular Graphs (RRG). I struggle to comprehend the authors' motivation for considering this case. Additionally, the techniques employed lack sufficient explanation for my understanding. Numerical simulations in this limit must be notably challenging, likely constrained by a smaller system size.

3. Generalizations of the Model: The other two model generalizations are quite challenging to grasp. The directed couplings case is excessively elusive, referring to a future publication. It is difficult to discern the message and the link between these results and the previous sections. The explanation of the 3-cycles case is insufficient. What exactly are 3-cycles? Why have they been included in the study?

4. Figures in the Appendix: There are numerous figures in the appendix, and their purpose is unclear to me. Could the authors provide context or explanations for these figures?

5. Language and Text Quality: Lastly, the English and overall text quality require careful editing. Several excellent tools are available to accomplish this.

See my report

1-The model under investigation is relevant and the physical motivation is compelling

1- The language and presentation of the paper is very confusing
2- The size of the numerical model with N=1024 seems a bit low, especially considering the discrepancy in Fig.3 c) d)

The article discusses the survival of delocalized states of free particles on a random regular graph where a fraction of nodes are subject to a strongly disordered potential. This is a relevant problem, since this model can serve as a toy model for the Hilbert space for certain interacting systems.
However, this paper suffers from poor presentation making it at times very hard to understand what the authors want to convey.

Question to the authors: If we take the W → ∞ limit like it seems to be done in section 3, does the model reduce to the free problem on the non-disordered subset of nodes (and some isolated disordered states)? Concretely this would be a RRG where each node has a β probability of being removed, or (at least in the case of sparse graphs) a random graph with connectivity distributed according to Eq. (7). This reduction seems rather obvious to me, but maybe I overlooked something. If this is indeed the case it should be stated clearly and not hidden in the derivation on the top of page 7. I cannot judge if this problem has been studied elsewhere before.

The abstract should be overhauled, specifically it would be good to write the meaning of (β,d) in one sentence rather then spread it out over three. The last sentence is very confusing to me.
Section 1 introduce the topic and motivate the model. The physical motivation in the introduction, namely the study of interacting models with topologically protected modes, is sound. However it could benefit from some references. Furthermore, there is an existing body of research on models comprised of clean and disordered parts (for example PHYSICAL REVIEW X 9, 041014 (2019), but there is a lot more). It would be helpful to clarify the relationship between those works and the present work.
Section 2 defines the model and demonstrate the survival of extended states numerically. Fig. 1 is convincing, however an explanation of the additional branch starting at E=20 would be welcome. Furthermore, a comment on numerical convergence with respect to all relevant parameters is missing.
Section 3 focuses on deriving a formula for the density of extended states and the percolation threshold β(d) in the W →∞. It is described that the disordered nodes do not contribute in the limit W→∞ (c.f. my question above). Eq. (17) is derived under the assumption that G is self averaging. Can you comment on why this holds?
Fig.2, why does the purple curve deviate from the blue dotted curve? can you comment on the choice of parameters? Convergence with sample and model size?
Fig. 3 c) and d) show clear deviations from the predicted curve, in particular there are a considerable number of states outside of the predicted band and the distribution looks more pointy then the predicted distribution. Why is that?
Fig 4, what does the grey color signify? If it means that there are no states with this energy, doesn't the black curve signify the band edge rather then the mobility edge? What is that extra branch in panel a)?
Section 4 discusses how RRG with connectivity d relate to RRG with connectivity N-d-1. I fail to see how this section connects to the rest of this work.
Section 5 discusses further generalizations of this model, in particular directed graphs and chemical potentials. I cannot see the relation to the previous part of the paper nor to the physical motivation mentioned in the introduction. Both sections introduce a significant amount of new concepts and terminology while adding very little to the story of the paper. Maybe it would be better to spin them off in a different work?
The captions of Fig. 6 and 7 are insufficient.
The numerical results plots look compelling, however no comment is given on convergence with respect to system size and sample count. In its current form the article is very difficult to follow due to its presentation; I can therefore not recommend publishing without major revisions to language and presentation.

1- Significantly improve the language and presentation. Here just a couple points; this list is not exhaustive:
- The abstract is hard to parse and should be reformulated, especially the last sentence.
- (d,β) and (β,d) are used inconsistently
- some concepts are not introduced like "short-cycles"
- grammatical mistakes are frequent
- Captions only state what was plotted but not what one should look at
- Sentences are at times very long and cover different only loosely related ideas
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2- add references to the physical motivation as well as compare to other works dealing with models comprising clean and disordered parts.
3- Comment on convergence of numerical computations with respect to relevant parameters like number of samples, number of nodes, etc. The number of nodes N=1024 seems very low to me. Fig 3 c) and d) shows disagreement between observed and predicted density of states, yet there is no comment on this discrepancy anywhere.
4- Explain the second branch in Figure 1
5- Either integrate section 4 and 5 better with the rest of the text or split them off in a different work.
6- Section 3 seems to be about the large disorder limit, clarify this