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Traveling/non-traveling phase transition and non-ergodic properties in the random transverse-field Ising model on the Cayley tree
by Ankita Chakrabarti, Cyril Martins, Nicolas Laflorencie, Bertrand Georgeot, Éric Brunet and Gabriel Lemarié
This Submission thread is now published as
Submission summary
| Authors (as registered SciPost users): | Ankita Chakrabarti · Nicolas Laflorencie |
| Submission information | |
|---|---|
| Preprint Link: | scipost_202302_00036v2 (pdf) |
| Date accepted: | 2023-10-09 |
| Date submitted: | 2023-09-22 01:40 |
| Submitted by: | Chakrabarti, Ankita |
| Submitted to: | SciPost Physics |
| Ontological classification | |
|---|---|
| Academic field: | Physics |
| Specialties: |
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| Approaches: | Theoretical, Computational |
Abstract
We study the random transverse field Ising model on a finite Cayley tree. This enables us to probe key questions arising in other important disordered quantum systems, in particular the Anderson transition and the problem of dirty bosons on the Cayley tree, or the emergence of non-ergodic properties in such systems. We numerically investigate this problem building on the cavity mean-field method complemented by state-of-the art finite-size scaling analysis. Our numerics agree very well with analytical results based on an analogy with the traveling wave problem of a branching random walk in the presence of an absorbing wall. Critical properties and finite-size corrections for the zero-temperature paramagnetic-ferromagnetic transition are studied both for constant (independent of the system volume) and algebraically vanishing (scaling as an inverse power law with the system volume) boundary conditions. In the later case, we reveal a regime which is reminiscent of the non-ergodic delocalized phase observed in other systems, thus shedding some light on critical issues in the context of disordered quantum systems, such as Anderson transitions, the many-body localization or disordered bosons in infinite dimensions.
Author comments upon resubmission
We are resubmitting our revised manuscript, titled "Traveling/non-traveling phase transition
and non-ergodic properties in the random transverse-field Ising model on the Cayley tree," for your
consideration for publication in SciPost Physics. We appreciate the time taken by you and the
reviewers in evaluating our work.
We are delighted that both reviewers found our manuscript interesting and valuable for the
community. Referee 1’s comments were particularly positive, strongly recommending its publication
in SciPost Physics.
Referee 2 provided a very detailed report, raising important and interesting questions regarding the analogy between the random transverse field Ising model we consider and the Anderson transition, both on the Cayley tree. In response, we have provided detailed explanations and made revisions to the manuscript to clarify these points. We have also followed their advice to enhance
the overall presentation of our manuscript. Furthermore, we have meticulously reviewed the bibliography.
To facilitate your review process, we have highlighted all changes made to the main text in
magenta within the manuscript.
We greatly appreciate your consideration of this revised manuscript and eagerly await your response.
Sincerely, the Authors.
List of changes
1. We have changed the abstract clarifying what "constant and algebraically
vanishing boundary conditions" mean.
2. We have added a sentence to the second paragraph of the introduction, discussing the substantial body of work describing non-ergodic delocalized/multifractal phases.
3. We have included a paragraph in the introduction that discusses the Jordan-Wigner fermionization of the spin model on the Cayley tree in comparison to the 1D case.
3. We have included two new paragraphs in the introduction that discuss the analogy with the traveling wave problem in the context of similar behaviors between the random transverse field Ising model and the Anderson transition on the Cayley tree.
4. We have combined the previous sections 4 and 5 into a single section 4, where numerical results are discussed immediately after the corresponding theoretical predictions.
5. We have reorganized the placement of the figures, positioning them as closely as possible to the corresponding discussions in the text.
6. We have included a comment in the current Figure 5 (formerly Figure 3) about the bimodal shape of the distribution.
7. We have inserted a paragraph at the end of Section 5, ’Non-ergodic phase,’ to address how our approach enables us to handle the intricate finite-size effects.
8. In Section 6, we have replaced the term ’inverse participation ratio’ with ’second moment’ for I 2 in Eq. (31) and Fig. 12.
9. In the conclusion, we have included two paragraphs to discuss the analogy between our cavity mean field approach and Anderson localization, devoting a paragraph to emphasize the limitations of this analogy .
10. In the conclusion, we have added a paragraph discussing the differentiation between the Bethe lattice and the finite Cayley tree, as well as the intriguing perspective of Random Regular Graphs and small-world networks. We also discuss about the intricate
finite-size effects in the context of the the non-ergodic ordered phase.
11. In the conclusion, we have added a paragraph discussing comparison of the outcomes of our cavity mean field approach with other approaches.
12. We have meticulously reviewed the bibliography. We have removed the citations to Ref. [30] when exclusively discussing
the Cayley tree case.
To facilitate the review process, we have highlighted all changes made to the main text in
magenta within the manuscript.
Published as SciPost Phys. 15, 211 (2023)
Reports on this Submission
Report 1 by Ivan Khaymovich on 2023-9-22 (Invited Report)
- Cite as: Ivan Khaymovich, Report on arXiv:scipost_202302_00036v2, delivered 2023-09-22, doi: 10.21468/SciPost.Report.7852
Report
The authors have addressed all of my questions thoroughly and improved the manuscript.
I will comment only on a couple of more things:
Question 1. Yes, indeed, it is my fault in mentioning the Jordan-Wigner transformation for the loop-less graphs. It is the 1d structure which is important for that mapping.
Questions 2-4 were related to the difference between the Rosenzweig-Porter model, where the Thouless energy scales down as a power of the system size ($N$), and the short-range models (on the Cayley tree or RRG), where the Thouless energy remains finite or scales as a (power of the) diameter of the graph ($\sim \ln N$). But, as the authors focus on the ground-state physics, most of my questions seem to be irrelevant.
I have only the following minor reference issues:
a. [23, 27, 75, 97, 110] page numbers are missing
b. [30, 57, 68, 76, 78, 79, 85, 86, 93, 101, 108, 117] (doi) link is missing
c. Links to some arXiv are missing and the format is different: compare [81] to [25, 32, 99]
d. The format of references is heterogeneous in terms of the presence/absence of issue numbers.
Please correct them.
Overall I recommend the manuscript for the publication in SciPost Physics.
Requested changes
Please correct the following minor reference issues:
a. [23, 27, 75, 97, 110] page numbers are missing
b. [30, 57, 68, 76, 78, 79, 85, 86, 93, 101, 108, 117] (doi) link is missing
c. Links to some arXiv are missing and the format is different: compare [81] to [25, 32, 99]
d. The format of references is heterogeneous in terms of the presence/absence of issue numbers.
Anonymous on 2023-09-22 [id 4001]
Provide a DOI for all references whenever possible (e.g., Refs. [27,30,55,57,75,78,83,85,88,98,106]).