Statistics of Green's functions on a disordered Cayley tree and the validity of forward scattering approximation

We are very grateful to the Referee for her/his highly professional review of our paper. Below we present the list of amendments resulting from the comments of the Referee.

1. a) It is important to remember what is the control parameter of the transition in equations (9-14).

Reply: We thank the Referee for this clarifying question and answer to it on page 6 before Eq. (15) by the phrase: The Anderson transition “… is controlled by the energy $E$ and the disorder strength $W$ entering the characteristic function $\tilde{F}(v)$ of the distribution of on-site energies and thereby in $\epsilon_\beta$.”

b). What does $\Omega_0$ represent physically?

Reply: This is a very important question. We devoted an entire new Section 4 to answer it in a rigorous way. In addition, a short explanation about the physical meaning of this quantity was added on page 6 just after Eq. (15) and in the beginning of page 7. The discussion about the physical meaning of $\Omega_0$ was also added on page 5 around Eq. (11) and on page 6 after Eq. (13).

d) The interpretation of (18) as a renormalized disorder strength is very interesting, however without a precise description of the physical meaning of the Lyapunov exponent $\lambda_{typ}$, it is very difficult to understand. Also, it would be useful to make the link with the more often used approach of the pool method / population dynamics. How does this renormalization of disorder appear?

Reply: We added a paragraph at the end of Section 2 on page 7 explaining the physical meaning of the Lyapunov exponents in detail. In population dynamics numerics the Lyapunov exponents can be obtained by computing the two-point Green’s functions, as it was done, e.g., in Ref. [28].

e) The computation of the moments of the real Green's function is done by means of the complex Green function (with the imaginary part $\eta$ of $E$ finite, and limit $\eta \to 0$). We know that the limits $N\to \infty$ and $\eta\to 0$ do not commute. I think the authors should stress that.

Reply: We clarified this point on page 8 after Eq. (19). The limits $N\to \infty$ and $\eta\to 0$ do commute if the moment order $2\beta$ is less than $1$.

f) The authors highlight an important difference between the model they consider and that corresponding to the non-linear sigma model, where an ergodic delocalized phase exists. Does this difference also change the nature of the non-ergodic localized / delocalized phase transition?

Reply: At the end of Section 2 on page 7 of the revised manuscript we stressed that in the Anderson model with one orbital per site there is no the phase transition between non-ergodic and ergodic delocalized phases, in contrast to the non-linear sigma model. This was first demonstrated numerically by population dynamics method in Ref. [28] and shown analytically in Ref. [29] and in this paper. The reason is that the ergodic limit of the Lyapunov exponent is reached in the Anderson model with one orbital per site only in the limit of zero disorder.

g) The following calculations are more technical. Could the authors, before carrying them out, summarize the aspects which are new and those well known, and make a road map of these calculations?

Reply: This is very useful suggestion that helped us to improve the presentation of the results. We described such a roadmap in the beginning of Sec. 3 on page 7. All the results and methods of Sec. 3 are new.

c) The Cayley tree does not always have a non-ergodic delocalized phase, it depends on the boundary conditions that we consider (the imaginary part of the energy $E + i \eta$ at the boundary). In the population dynamics / pool method described in [26], it is shown that it is necessary to take $\eta\to N^{- \alpha}$, with $\alpha$ some exponent>0, to be able to observe this phase. If $\eta$ is a constant ($\alpha=0$) on the contrary, we have an ergodic delocalized phase. How does this translate into the formalism presented? There is no discussion of the considered boundary condition. Are we forced to take $Y (\Phi_n, \Phi_n^\dagger) = 1$? In the delocalized phase discussed briefly, the linearization of (9) is not always possible. What are the conditions for this linearization?

Reply: Here there is a certain misunderstanding. The point is that whether the phase is ergodic or not depends on the statistics of a single wave function. In order to distinguish between these two phases one has to have a resolution sufficient to speak about a single wave function. Such a resolution is only achievable if the broadening of an energy level $\eta$ is smaller than the level spacing $\delta=1/N$. We always imply the condition $\eta<1/N$ to be fulfilled considering the ‘real Green’s functions’. The boundary condition is indeed $Y = 1$ at the boundary which corresponds to the closed sample. However, we are mostly interested in the behavior far from the boundary considering $l<<R$. The discussion on the boundary condition and the order of limits was added on page 5 after Eq. (10).

2.a) It would be good in the introduction to recall the non-trivial successes of this approximation. For example, it makes it possible to understand that in dimension two the conductance in the Anderson localization regime has a Tracy-Widom distribution [PRL 99, 116602 (2007)] and glassy properties [PRL 122, 030401 (2019)] .

Reply: This is a correct comment. We added the reference to the above papers in the first sentence of the Introduction. We also discussed the first of these works in more detail in the Conclusion.

b) My understanding is that this approximation neglects two main effects, which are related: the effect of loops (only directed paths, thus forward) and the non-linearity of the recursion due to the self-energies that we have not taken into account. From what I understand, the deviations observed with this approximation come here, in the Cayley tree, because the considered tree is finite. Because of this finitude, the contributing paths are not simply directed paths, but can go back and forth due to the presence of an edge. Or did I misunderstand, and even for an infinite tree, we have these corrections due to non-directed paths, which attenuate the effects of resonances? In all cases, the self-energies which appear due to these non directed paths, are only real in the case considered (real Green's functions, or $\eta\to 0$ before $N\to\infty$) and therefore only shift the resonances randomly. So is it their correlation that is important?

Reply: The FSA, indeed, neglects the effect of loops considering only shortest paths. However, a different effect of loops, the existence of multiple paths between two points is generally taken into account within FSA. For the particular case of a tree, there are no loops in both senses. It is also true that the self-energies are not taken into account in FSA. However, it is not true that the effects of deviations from FSA on a tree is due to its finiteness. Even for an infinite tree and the real Green’s functions ($\eta \ll 1/N$)), the deviations will take place for a finite length r of a path between two points in the Green’s function. They are caused by the backscattering not from the boundary of a tree but rather because of the scattering by disorder. This backscattering causes correlations between the self-energies. We added a discussion on this point on pages 19 and 20.

1. The last remark I have relates to the fact that the analytical developments presented are very elaborate and that it is a little disappointing to conclude only on the forward approximation. I am convinced that other interesting predictions could be made. For example, it was recently shown that the Anderson transition in this type of graph of infinite effective dimensionality has two critical localization lengths [PR Research 2, 012020 (2020)]. Can this approach corroborate (or invalidate) these results for the Cayley tree? Can the authors give some perspectives for the future use of their analytical developments?

Reply: We share the view of the Referee that our work opens up new perspectives of research. One of them is related to the distribution of Green’s functions on realistic graphs emerging in real interaction many-body systems. This discussion is added in the Conclusion. As for the existence of two characteristic lengths in the present problem or a similar problem on random regular graph, the answer is ‘yes’, we believe that the two length with the exponents 1 and ½ do exist in those problems. The discussion on this matter is present in our recent previous work Ref. [27], but we do not consider it relevant for the current manuscript.