Investigating finite-size effects in random matrices by counting resonances

Dear Editor,

Thank you very much for forwarding us the referee reports.

First of all, we would like to express our sincere gratitude to the Reviewers for their
positive assessment of our work but even more for their constructive comments, questions,
and remarks which allowed us to substantially improve the manuscript.
Below we provide an itemized reply to the referee reports and highlight the changes in the
text that we made.

We anticipate that, in revising the manuscript, we extended Appendix B to clarify the
connection between some of our results and previous works, we modified Fig. 7 so that it
is easier for the reader to understand the significance of the plots, more clearly connected
Sec. 6 and the rest of the paper and extended the range of parameters in Fig. 3, discussing
their physical content. In addition, we have added some new data in Fig. 3 in the localized
phase and provided an analytical explanation in Eq. (19).

We refer to the replies to the Referees for the complete list of changes in the manuscript.
For your and the Referee’s convenience, we have highlighted in red the changes we have
made while preparing this revised version, which we kindly ask you to consider for publi-
cation in SciPost.

Yours faithfully,

Anton Kutlin and Carlo Vanoni

Report of the referee #1, supplemented with our comments

Referee: In the manuscript, the authors present a self-consistent approach to resonance
counting in disordered quantum systems. They propose a resonance criterion that connects
this to a phenomenological ansatz for the wave functions, allowing for the calculation
of quantities such as participation entropy, support set dimension, and related beta function.
They provide further microscopic support for their phenomenological theory and test
its predictions against exact diagonalization results for various Rosenzweig-Porter (RP)
models.
The manuscript reports on original and interesting work, providing insights into resonance
counting and finite-size scaling of related quantities in disordered quantum systems. I also
find the manuscript well written and, therefore, support its publication after the authors
address the following questions and comments.
Authors: We are grateful to the Referee for their careful reading of the manuscript and
their positive assessment towards our work. We now provide a reply to their comments
and questions, and correspondingly improve the text when necessary, marking in red the
changed text.

1. Referee: A curious result in my opinion is the scaling property of the weight of the
wave functions head. The authors show that in the thermodynamic limit the latter
approaches unity in the localized and ergodic phases, implying that all weight is
in the head and tails do not contribute. While this is intuitive, I find it somehow
surprising that the weight takes a single fixed value in the entire intermediate fractal
regime (rather than a γ-dependent value). Do they have any explanation for (i) why
it’s a single value in the entire fractal regime, and (ii) why this is 1/2? This should
be related to the change of wave function statistics from Porter Thomas distribution
to the modified distribution, discussed in Appendix B? Can they comment more on
this?
Authors: We thank the Referee for their question. As the Referee suggested, the
motivation is in the discussion presented in Appendix B. In order to be fully consistent
between the main text and the appendix and omit any confusion, we renamed
C from the appendix as \tilde{C} and then established a link between these two quantities
using the Porter-Thomas distribution, backing up and explaining the numerical
results from the main text.

2. Referee: I have two basic questions regarding a statement made about the resonance
counting based on dressed hopping elements. The authors comment that the esti-
mate for the Gaussian RP model “severely underestimates the support set volume
(...)”. This also applies for conventional RMT Hamiltonian (where a similar estimate
N1/2 also severely underestimating the number of resonances)? They continue “(...)
unable to correctly locate the ergodic fractal transition”. I think some more explanation
of this point would be helpful, since the change in scaling from usual RMT can
be identified with the ergodic fractal transition? (In the end, the Thouless inspired
criterium gives the “square” of the condition derived from dressed resonances).
Authors: We thank the Referee for their questions, as they allow us to clarify this
point. As the Referee pointed out, the criterion M∼min{N^(1−γ/2),N^(1/2)}would also
underestimate the number of resonances for RMT, where it is natural (and correct) to
expect M∼N. As we now mention in the new footnote, it is not a surprise given the
already mentioned in the paper relation between the naive resonance condition (1)
and the Anderson criterion of localization, while the ergodic transition is commonly
studied by the Mott’s criterion of ergodicity.

3. Referee: In criterium (7), the authors carefully distinguish contributions from direct
and indirect resonances, but in criterium Eq. (10) such distinction does not play a
role? Can they comment on this?
Authors: We thank the Referee for their comment. As discussed in Section 6,
the connection between the direct and indirect resonance condition and the self-
consistent criterion can be seen through the exact secular equation for the size-
increment. From the analysis of the secular equation, we have arrived to Eq. (33),
which is very similar to Eq. (7) (just the definition of one of its components is a
bit different) and allows to obtain Eq. (10). The difference between Eq. (7) and
Eq. (33) is what makes the self-consistent criterion more precise and predictive than
Eq. (7), in that Eq. (10) effectively takes into account resonances to all orders, as
explained in Section 6, while Eq. (7) only up to second order in perturbation theory.
To improve the clarity of the text, we have added a reference to Section 6 near to
Eq. (10).

4. Referee: I would find it helpful to add a brief discussion on which of the considered
quantities are basis independent and which not.
Authors: We thank the Referee for their suggestion. In fact, all quantities considered
in the paper are basis-dependent: on-site energies (as the notion of ‘site’ is basis-
dependent), inter-site hopping (for the same reason), fractal dimension (which is
a function of the eigenstates’ components and, hence, of basis), entropy, even the
eigenvalues’ shifts due to the new site’s addition because ‘the notion of site is basis-
dependent’. However, if we link the notion to the physical site in a coordinate basis,
the basis becomes the only reasonable choice to work with.

5. Referee: If possible, a more detailed discussion on the deviations between analytical
predictions and exact diagonalization for the log-normal and Bernoulli RP models
would be beneficial. For example, they briefly mention a relation to matrix sparsity,
could they provide further commentary on this?
Authors: We thank Referee for the request. As we mentioned in the conclusion
section as well as at the end of section 5.3, we think the deviations between the
analytics and numerics is due to the quality of the ansatz for the distribution of
ψ^2_{head}, which is unlikely to be as simple as in the Gaussian RP; to emphasize this
once again, we added the corresponding remarks to section 5.2 between the figures
6 and 7 and also to the conclusion section. We did not consider how this ansatz can
be improved but we are pretty sure it is possible using a model-specific information
such as, e.g., the matrix sparsity (or anything else).

6. Referee: A more general question (out of curiosity and not the focus of their work)
is which of their findings they expect to transfer to “true” disordered many-body
systems with long range interactions. These systems share similarities with the
Gaussian RP model but are more sparse and exhibit correlated disorder. How would
this affect their results?
Authors: We thank Referee for bringing up this intriguing problem. We believe that
the self-consistent condition should also work in the “true” many-body system, but
the analytical progress will be hindered by the correlations between the eigenenergies
and the dressed hopping and the difficulties in choosing the correct ansatz for the
head distribution. Still, it would be interesting to check the self-consistent resonance
counting approach numerically in this case, and we will probably do it later if time
allows.

Report of the referee #2, supplemented with our comments

Referee: Despite a long history and a very large research effort, Anderson localization
remains a difficult subject. The reason is that even though the propagation of waves in
a random medium, a problem that includes single-body quantum evolution in a random
potential, is qualitatively well-understood, it is hard to make precise and sharp statements,
and a lot of the understanding is based on heuristic ideas and arguments.
Resonances between sites with nearby orbital energies is a transport mechanism concept
that has played a central role in the studies of localization for a long time, but it is difficult
to assign a sharp meaning to this concept. The goal of the present paper is to use this
idea to study the energy levels and eigenfunctions in a random medium, in particular as
a tool to capture finite size effect.
The main result is the self-consistent resonance condition, a mean-field like approximation,
that is then applied to several random-matrix models of localization. Special attention
is given to the Gaussian Rosenzweig-Porter model, where the theory approximates quite
closely the exact statistics; this is unsurprising, given that the GRP is a mean-field model
of localization. Even though the predictive power of the theory is smaller for the other
random matrix models, qualitative behavior in them is captured by the theory, in particular
the behavior in the thermodynamic limit. It is of interest to examine the validity of
the theory and its possible improvements for finite-dimensional lattices, but this question
is clearly outside the scope of the current paper.
While not a breakthrough, the resonance condition proposed in this work is a significant
step in the effort to better understand quantum energy levels and wavefunctions in random
environments, so I recommend publication of the paper.
Authors: We are grateful to the Referee for their careful reading of the manuscript and
their positive assessment towards our work. We now provide a reply to their comments
and questions, and correspondingly improve the text when necessary, marking in red the
changed text.

1. Referee: The paper assigns particular importance to the log-normal Rosenzweig-
Porter model, but the discussion of the analysis of this model at the top of page
14 is hard to follow, and in particular it is not clear which parts of the discussion
are based on the self-consistent resonance condition and which parts on numerical
diagonalization. This is also a problem in Fig 7 which presents results for this model,
where in particular the relation between the two panels is not clarified.
Authors: We thank the Referee for pointing out this weak point of our presentation.
In the revised version of the manuscript, we clarified Fig. 7 by adding titles to the
plots, so that their content is clear even without reading the caption. Moreover, we
represented with a shaded area the region of the plots that are speculative and part
of our discussion, while the part at smaller system sizes represent the data produced
via exact diagonalization and reported in the right panel of Fig. 6.

2. Referee: Can the authors provide some insight regarding the source of the discrepancy
between the self-consistent resonance theory and exact results for the LNRP
and Bernoulli RP models, and how one may improve the accuracy of the approximation?
Authors: We thank the Referee for the request. As we mentioned in the conclusion
section as well as at the end of section 5.3, we think the deviations between the
analytics and numerics is due to the quality of the ansatz for the distribution of
ψ^2_{head}, which is unlikely to be as simple as in the Gaussian RP; to emphasize this
once again, we added the corresponding remarks to section 5.2 between the figures
6 and 7 and also to the conclusion section. We did not consider how this ansatz can
be improved but we are pretty sure it is possible using model-specific information
such as, e.g., the matrix sparsity (or anything else).

3. Referee: Section 6 on the microscopic approach, while clearly related to the other
sections, seems to stand apart in terms of ideas and methods. It would be better
to either more strongly tie it to the rest of the paper or relegate it to another
publication.
Authors: We thank the Referee for their observation. We agree that the methods
used in Sec. 6 are different from the rest of the paper, but we believe that the
content of that section is important, in that it provides an independent derivation of
the resonance conditions we used in the rest of the paper, with a more solid ground
derivation. To comply with the Referee’s comment, we emphasized at the beginning of
Sec. 6 the connection with the rest of the paper, as well as adding references to
Sec. 6 on pag. 8, remarking that the resonance criterion derived in Eq. (10) will be
derived in an independent way in Sec. 6.

4. Referee: While the graphs themselves are clear and polished, the captions are not
always self-contained, sometimes not very clearly labeled, and color choices are sometimes
not ideal. Note a mistake in the legends of Fig 6.
Authors: We thank the Referee for their comment. We fixed the mistake in the
legend of Fig. 6 and improved Fig. 7 as described in a previous reply, to improve
the readability of the plots. Regarding the color chosen for the plots, we tried to use
a color map allowing to easily distinguish different curves and, at the same time,
avoiding soft colors, very often difficult to see. We are happy to change our choices
if the Referee has some more specific indications that might further improve the
presentation.

5. Referee: The sharp transition from head to tail statistics in the wavefunction ansatz
(as depicted in Fig 2) seems too crude. Could it be a significant source of error?
The authors address this question in section 6, concluding that replacing it by a
smoothed transition does not improve the approximation. Nevertheless, this a possible
weakness of the theory which merits further scrutiny.
Authors: We thank the Referee for their comment. As mentioned in the reply to
question 2., the ansatz we provided can be for sure improved. We have tested a
“smoothed” version of the ansatz for the wavefunction, but we concluded that this
change does not bring significant benefits to the results. We believe, instead, that
a different choice of the distribution for the wavefunction head can bring to more
consistent improvements, but it is likely that such a choice is model-dependent (our
simple ansatz works very well for the Gaussian RP), and thus we leave its study to
other, model-oriented, works.