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Local Density of States Correlations in the Lévy-Rosenzweig-Porter random matrix ensemble
by Aleksey Lunkin, Konstantin Tikhonov
Submission summary
| Authors (as registered SciPost users): | Lunkin Aleksey |
| Submission information | |
|---|---|
| Preprint Link: | https://arxiv.org/abs/2410.14437v1 (pdf) |
| Date submitted: | 2024-10-29 13:52 |
| Submitted by: | Aleksey, Lunkin |
| Submitted to: | SciPost Physics |
| Ontological classification | |
|---|---|
| Academic field: | Physics |
| Specialties: |
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| Approach: | Theoretical |
Abstract
We present an analytical calculation of the local density of states correlation function $ \beta(\omega) $ in the L\'evy-Rosenzweig-Porter random matrix ensemble at energy scales larger than the level spacing but smaller than the bandwidth. The only relevant energy scale in this limit is the typical level width $\Gamma_0$. We show that $\beta(\omega \ll \Gamma_0) \sim W/\Gamma_0$ (here $W$ is width of the band) whereas $\beta(\omega \gg \Gamma_0) \sim (W/\Gamma_0) (\omega/\Gamma_0)^{-\mu} $ where $\mu$ is an index characterising the distribution of the matrix elements. We also provide an expression for the average return probability at long times: $\ln [R(t\gg\Gamma_0^{-1})] \sim -(\Gamma_0 t)^{\mu/2}$. Numerical results based on the pool method and exact diagonalization are also provided and are in agreement with the analytical theory.
Author indications on fulfilling journal expectations
- Provide a novel and synergetic link between different research areas.
- Open a new pathway in an existing or a new research direction, with clear potential for multi-pronged follow-up work
- Detail a groundbreaking theoretical/experimental/computational discovery
- Present a breakthrough on a previously-identified and long-standing research stumbling block
Current status:
Reports on this Submission
Report #1 by Carlo Vanoni (Referee 2) on 2024-12-30 (Invited Report)
Strengths
1- The results presented in the paper are solid
2- The analytical calculations are presented with enough details
3- The analytical results are verified numerically and they agree
Weaknesses
1- There are some minor mistakes in two equations
2- There are some typos in the text
3- A comment in the conclusions of the paper can be improved
Report
This paper presents an analytical calculation of the finite-frequency correlation function of the local density of states of Lévy-Rosenzweig-Porter random matrices, using a "semi-classical" approximation in the computation of the imaginary part of the self-energy.
The results presented in the paper are compared with numerical results and the authors find good agreement with the analytical predictions in the desired range of validity of their approximations.
Despite being valid for this specific class of random matrices and probably not directly applicable to more complicated models, I find the results interesting and worthy of publication in SciPost Physics.
I report here some comments about the manuscript.
1- Eqs. (4), (5) and (6) are inconsistent, there must be some signs that are not correct. I believe that the definition of self-energy reported as the second part of Eq. (4) is wrong, and should be $\Sigma_j(z) = z - H_{jj}- 1/G_{jj}(z)$. This way the usual textbook definition is met and the rest of the paper is consistent with it.
2- I think that a factor 2 is missing when going from Eq. (11) to Eq. (12) when the authors define an auxiliary imaginary frequency. I think the right definition should be $ix = \omega + 2 i \delta$. Or instead, $ix = \omega/2 + i \delta$, removing the factor of 2 in ($ix / 2$) in the following equations. I think this is not affecting the validity of the results, just a matter of consistency in the definitions.
3- In the conclusions, the authors discuss their results with the findings of a previous paper discussing the Jacobi algorithm applied to MBL. The connection they find is very interesting, but there are a couple of questions I would like to ask. According to Fig. 3, the authors find that $\mu/2 = 0.7$, so that $\mu = 1.4$, while the other paper argues that the same curve should have slope $\mu - 1 = 0.4$. The difference between 0.7 and 0.4 is not small, but not even huge, and might correspond to the behavior before the long-time asymptote. Could the authors add a curve to Fig. 3 corresponding to the slope $\mu - 1$ and verify if it is consistent with this observation? Also, in the reference the authors compare to, the behavior with slope $\mu - 1$ is not always valid, but according to Eq. (15) in that reference, that behavior is valid in an intermediate time regime. Could the authors comment more on this and check if their results are valid in the same regime?
Requested changes
Some minor changes to the text.
1- I would remove "the" in "the section #" at the end of the Introduction, it doesn't sound correct. Also, at the very beginning of the Introduction "The Anderson localization" -> "Anderson localization".
2- In the first paragraph of the Introduction "neglect or simplification" -> "neglect or simplify"
3- Below Eq. (3), "...with the result is..." is not correct, maybe use "...with the result being..."?
4- Below Eq. (8), "This means that the integral is coming from $\xi \approx \epsilon$". Do the authors mean that the main contribution to the integral comes from $\xi = \epsilon$ and thus the result can be obtained by replacing the integral with the value at $\epsilon$?
5- Below Eq.(20) "The similar equation" -> "A similar equation".
6- Below Eq.(36) it is not clear what "expands over the full zone" means. Can the authors be more clear?
7- In Eq. (37) there is an unmatched $\langle \rangle$ pair.
8- In Sec. 5.2, "chosen in such a way such that" -> "chosen in a way such that".
Recommendation
Publish (easily meets expectations and criteria for this Journal; among top 50%)