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

Submission summary

 As Contributors: Ivan Khaymovich · Pavel Nosov Arxiv Link: https://arxiv.org/abs/2108.10326v2 (pdf) Date accepted: 2021-12-16 Date submitted: 2021-11-04 20:12 Submitted by: Khaymovich, Ivan Submitted to: SciPost Physics Academic field: Physics Specialties: Condensed Matter Physics - Theory Mathematical Physics Quantum Physics Approaches: Theoretical, Computational

Abstract

The accuracy of the forward scattering approximation for two-point Green's functions of the Anderson localization model on the Cayley tree is studied. A relationship between the moments of the Green's function and the largest eigenvalue of the linearized transfer-matrix equation is proved in the framework of the supersymmetric functional-integral method. The new large-disorder approximation for this eigenvalue is derived and its accuracy is established. Using this approximation the probability distribution of the two-point Green's function is found and compared with that in the forward scattering approximation (FSA). It is shown that FSA overestimates the role of resonances and thus the probability for the Green's function to be significantly larger than its typical value. The error of FSA increases with increasing the distance between points in a two-point Green's function.

Current status:
Publication decision taken: accept

Editorial decision: For Journal SciPost Physics: Publish
(status: Editorial decision fixed and (if required) accepted by authors)

Dear Editor,

We are grateful to the referee for the high evaluation of our paper and for thorough careful reading of the manuscript. His/her critical remarks have allowed us to significantly improve the presentation of our work.

In the revised version of our manuscript, we address all the points mentioned by the referee.
The point-to-point reply to the referee is given below the report, while the list of changes is placed below.

Sincerely yours, the authors.

List of changes

1. On page 6 before Eq. (15) we have clarified the control parameter of the Anderson transition in Eqs. (9-14).
2. We have developed and described a physical meaning of the parameter $\Omega_0(v)$ on page 5 around Eq. (11), on page 6 after Eq. (13), on page 6 after Eq. (15), and in the beginning of page 7.
3. We have added a paragraph at the end of Section 2 on page 7 explaining the physical meaning of the Lyapunov exponents in the problem and the difference of a single-orbital model on the Cayley tree with respect to the non-linear sigma model.
4. We have described the roadmap of calculations in the beginning of Sec. 3 on page 7.
5. We have clarified the non-commutativity of limits of $\eta\to 0$ and $N\to\infty$ on page 8 after Eq. (19).
6. The discussion on the boundary condition and the order of limits has been added on page 5 after Eq. (10).
7. The discussion on the deviations of the forward scattering approximation from the exact analytical results of our paper has been added on pages 19 and 20.
8. The new perspective of research opened by our paper has been discussed in the Conclusion.
9. We have added the clarification of the meaning of the arguments t and v of $\Omega_r$ on page 5 before Eq. (11).
10. In order to avoid repeating notations, we have changed them in Eqs. (80-81).
11. Several relevant references (including the ones mentioned by the Referee) have been added to the Introduction and the Conclusion sections.
12. Several clarifications, minor amendments, and corrections of typos have been done throughout the manuscript.

Submission & Refereeing History

Resubmission 2108.10326v2 on 4 November 2021
Submission 2108.10326v1 on 25 August 2021

Reports on this Submission

Strengths

1-New exact and approximate results on localisation physics in the Cayley tree, with potential applications to understanding both Anderson localisation and versions of many-body localization.

2-New results on the strongly disorder one-dimensional limit of Anderson localisation.

Weaknesses

Figures could be of higher quality

Report

Having read the paper and the reports from the first referee, I agree that this is a nice work with nontrivial results that deserve publication in SciPost Physics. I do not have much to add to the reports already written, though I have a couple of minor suggestions for improvements in the presentation that the authors can consider.

Requested changes

1-in the general discussion in the introduction, which considers all dimensions, the definition of r = |i-j| is not clear.
2-In Eq. (2) there is a P(y) on the left hand side, but no y on the right hand side.
3-All figures have rather clumsy looking labels that are placed in strange places. Since there are also legends put on the plots in the same style, it is not always obvious what is being plotted against what. I would suggest making the axis label in a more normal way.
4-The details of footnote 1, and the notation in the equations, could be more easily understood if the authors provided a figure defining the tree and the parameters r and R etc. This is maybe not needed for an expert, but it would be useful for those that are not experts on the method and trees but still interested in the results.
5-There are a couple of places where the wording could be improved. On page 6 between Eq. (14) and (15), they write "it appears to be at beta = 1/2". This "appears to be" makes the reader thing that it only appears to be at beta = 1/2 but is actually not there, but somewhere different. Since no explanation follows, I assume they mean to simply say that it is at beta = 1/2. After Eq. (19) the authors write "Upon averaging over disorder [...] the levels pass through the energy E just causing ...". This sentence is unclear.
6. $\theta$, $J_0$ and $K_{1-2\beta}$ are not defined when first introduced. One can guess these are Heaviside step function and Bessel functions, but it would be better to define them.
7. Eq. (42) is not an equation.

• validity: top
• significance: high
• originality: high
• clarity: good
• formatting: reasonable
• grammar: good

Author:  Ivan Khaymovich  on 2021-12-09  [id 2018]

(in reply to Report 2 on 2021-12-08)

The revised version is attached

Author:  Ivan Khaymovich  on 2021-12-08  [id 2016]

(in reply to Report 2 on 2021-12-08)

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 - We thank the Referee for pointing this out. In the revised version we corrected $r = |i-j|$ to $r_{ij}$ throughout the text.

2 - We have corrected Eq. (2) and its discussion right beforehand by using the variable $z = \ln G + r \ln(W/2)$.

3 - We have modified the figures accordingly.

4 - We have added the corresponding figure and add some discussion into the text moved from the footnote.

5 - We have corrected the corresponding phrases.

6 - In the revised version of the manuscript, we have added the above definitions.

7 - As Sec. 3.3 is a technical derivation of the integration over variables $s_k$ and $\tilde s_k$, we take some freedom to use expressions, but not equations, like (39) and (42) where it is clear from reading the manuscript.

Report

In the new version of their article, the authors have greatly clarified the points I had raised. I find the given physical interpretations very interesting, especially the new section 4.

I am not sure I agree with the statement in the conclusion that the agreement of numerical simulations (Refs. [6,7]) with the forward scattering approximation, in dimension two and in the strongly localized regime, is specific to this dimension. On the contrary, I believe that this agreement should persist in higher dimensions. My interpretation is that in the strongly localized regime, it is the competition between the paths that dominates (contrary to the weakly localized regime, where it is the interference between the paths which dominates). This competition is done by a global optimization, in a similar way to the physics of directed polymers. But I recognize that this remains an open problem which could be addressed in 3D or on random regular graphs for example. The question therefore arises as to whether the effects described by the authors are specific to 1D and to the Cayley tree where a single path connects two points of the network, or if these effects are important even in generic graphs where many paths contribute.

This discussion shows that the authors' interesting approach not only answers but also opens up interesting questions, in addition to being a technical  tour de force ''. So I can only recommend the publication of the manuscript in SciPost.

• validity: -
• significance: -
• originality: -
• clarity: -
• formatting: -
• grammar: -

Author:  Ivan Khaymovich  on 2021-11-12  [id 1935]

(in reply to Report 1 on 2021-11-11)
Category:
remark

We thank the referee for his/her thorough reading of our revised manuscript and for high evaluation of changes.

We agree that the issue with FSA in higher dimensions in an open question, especially analytically.
We have only meant that in numerical simulations it seems that a specific Tracy-Widom form works only for 2D.