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Many-body localization in a quantum Ising model with the long-range interaction: Accurate determination of the transition point
by Illia Lukin, Andrii Sotnikov, Alexander L. Burin
This is not the latest submitted version.
Submission summary
| Authors (as registered SciPost users): | Alexander Burin |
| Submission information | |
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| Preprint Link: | https://arxiv.org/abs/2505.20159v3 (pdf) |
| Date submitted: | Aug. 26, 2025, 4:25 p.m. |
| Submitted by: | Alexander Burin |
| Submitted to: | SciPost Physics Core |
| Ontological classification | |
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| Academic field: | Physics |
| Specialties: |
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| Approaches: | Theoretical, Computational |
Abstract
Many-body localization (MBL) transition emerges at strong disorder in interacting systems, separating chaotic and reversible dynamics. Although the existence of MBL transition within the macroscopic limit in spin chains with a short-range interaction was proved rigorously, the transition point is not found yet because of the dramatic sensitivity of the transition point to the chain length at computationally accessible lengths, possible due to local fluctuations destroying localization. Here we investigate MBL transition in the quantum Ising model (Ising model in a transverse field) with the long-range interaction suppressing the fluctuations similarly to that for the second-order phase transitions. We estimate the MBL threshold within the logarithmic accuracy using exact results for a somewhat similar localization problem on a Bethe lattice problem and show that our expectations are fully consistent with the estimate of the transition point using exact diagonalization. In spite of unlimited growing of the critical disorder within the thermodynamic limit, this result offers the opportunity to probe the critical behavior of the system near the transition point. Moreover, the model is relevant for the wide variety of physical systems with the long-range dipole-dipole, elastic or indirect exchange interactions.
Author indications on fulfilling journal expectations
- Address an important (set of) problem(s) in the field using appropriate methods with an above-the-norm degree of originality
- Detail one or more new research results significantly advancing current knowledge and understanding of the field.
Current status:
Reports on this Submission
Strengths
- Impressive system sizes achieved in the numerical calculations.
- Clear correspondence between the numerical results and the analytical arguments.
Weaknesses
- Higher-order contributions are not analyzed, even at a qualitative level.
- The study focuses on a single quantity.
Report
These findings are valuable for the MBL community, where well-controlled critical benchmarks are scarce and the very stability of the phase remains under debate. However, the present manuscript is focused on a single quantity, which makes it difficult to assess the robustness of the conclusions. It would strengthen the manuscript to corroborate the transition using complementary quantities and to discuss the consistency across these indicators.
Requested changes
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What is the behavior of other characteristic points of the ergodic–MBL crossover in the studied model? For example, does the Bethe-lattice prediction also describe (i) the crossing point of the $\langle r \rangle$ curves, or (ii) the onset of the deviation of $\langle r \rangle$ from its ergodic value?
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The quantum random energy model (studied recently also in Phys. Rev. B 111, 214206 (2025)) is more directly amenable to Bethe-lattice arguments because energies on neighboring hypercube vertices are uncorrelated. In contrast, the present Ising model exhibits correlations. While the presence of the correlations make the presently studied model more interesting from the many-body perspective, do the authors have any argument why the Bethe lattice argument is applicable despite the correlations?
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For the analytical estimate of the MBL threshold, the authors consider only second-order processes. However, higher-order processes have been argued to play a significant role in MBL physics (see, e.g., arXiv:2005.13558; Phys. Rev. B 104, 184203 (2021); SciPost Phys. 12, 201 (2022); Phys. Rev. Lett. 131, 106301 (2023)). Do the authors have arguments for why such processes do not invalidate the Bethe-lattice threshold in the present model?
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Could the authors comment more extensively on how the variance ($\Delta r^2$) is calculated? Is each $r_i$ taken into account as corresponding to a single disorder sample? Is this variance dependent on the number of energy levels within a single sample?”
Recommendation
Ask for minor revision
Report
The work can be published once the following changes/comments have been addressed.
Requested changes
1- The last and one before last paragraphs on p3 appear to be duplicated 2- The authors use a non-standard form of the SK model where the random couplings are not normalized, so that in the thermodynamic limit, the disordered term of the Hamiltonian is super-extensive (scales like N^2). The authors should at least bring attention to or explain this specific choice. 3- In Eq(4) P' is probably P. If not, it should be defined. 4- For reproducibility, the calculation of the second line in Eq. (4) should be detailed, maybe in the appendix, since Ref. 25 gives a calculation for a uniform distribution. 5- Figure 2 is confusing due to the combined y-axis. I suggest splitting into two panels. This maintains the point of the figure while adding clarity. 6- The authors obtain a power-law decay in system size critical coupling strength, which is consistent with my comment on the dominance of the disordered term in the thermodynamic limit. What would be the scaling of the critical coupling strength for the standard SK normalization of the disordered strength? This point should be discussed in the conclusion/discussion.
Recommendation
Ask for major revision
Below are our responses to the Referee Comments and description of changes. We also have the revised version where the changes are shown in colors. It can be submitted upon request.
We greatly appreciate the remarks and suggestions made by the referee and we did our best to address them all in the revised version as described below.
Referee 1 Comment 1: The last and one before last paragraphs on p3 appear to be duplicated
Author response: Thank you, corrected.
Referee 1 Comment 2: The authors use a non-standard form of the SK model where the random couplings are not normalized, so that in the thermodynamic limit, the disordered term of the Hamiltonian is super-extensive (scales like N^2). The authors should at least bring attention to or explain this specific choice.
Author response: Thank you for pointing this out. Our original choice was motivated by the desire to keep all Hamiltonian parameters independent of the number of spins, which appears to us the most natural convention. The normalization by the square root of the number of spins is required to ensure that the spin-glass transition temperature remains independent of system size. However, this normalization does not make the localization threshold at infinite temperature size-independent in the thermodynamic limit: the critical transverse field vanishes as the number of spins tends to infinity.
Following the referee’s recommendation, we have added a discussion of the normalization choice to Sec. 2. In addition, we introduce an alternative normalization that yields a convergent localization threshold in the thermodynamic limit; this is now presented at the end of Sec. 4 (second paragraph after Eq. (9)).
Referee 1 Comment 3: In Eq(4) P' is probably P. If not, it should be defined.
Author response: Thank you for pointing out this issue. Indeed, it is the derivative of the probability density P. We have clarified its definition in the revised manuscript Appendix, following carefully the original work of Abou-Chacra and coworkers.
Referee 1 Comment 4: For reproducibility, the calculation of the second line in Eq. (4) should be detailed, maybe in the appendix, since Ref. 25 gives a calculation for a uniform distribution.
Author response: The requested details have been added to the Appendix in the revised manuscript, as recommended by the Referee.
Referee 1 Comment 5: Figure 2 is confusing due to the combined y-axis. I suggest splitting into two panels. This maintains the point of the figure while adding clarity.
Author response: Figure 2 is split as recommended by the Referee.
Referee 1 Comment 6: The authors obtain a power-law decay in system size critical coupling strength, which is consistent with my comment on the dominance of the disordered term in the thermodynamic limit. What would be the scaling of the critical coupling strength for the standard SK normalization of the disordered strength? This point should be discussed in the conclusion/discussion.
Author response: The discussion is added to the conclusion as recommended by the Referee.
We sincerely appreciate the insightful and constructive comments provided by Referee 1. We fully agree with all of the suggested revisions, as they will undoubtedly improve the clarity and quality of the manuscript. Each of the proposed changes makes perfect sense, and we are eager to incorporate them into the revised version. We believe these modifications will significantly enhance the readability of the manuscript and ensure that our findings are communicated more effectively to the readers.
We will follow the Referee's guidelines closely and look forward to submitting the updated manuscript.
Author: Alexander Burin on 2025-12-03 [id 6099]
(in reply to Report 2 on 2025-10-19)We greatly appreciate the remarks and suggestions made by the referee and we did our best to address them all in the revised version as described below.
Referee 2 question 1: What is the behavior of other characteristic points of the ergodic–MBL crossover in the studied model? For example, does the Bethe-lattice prediction also describe (i) the crossing point of the r curves, or (ii) the onset of the deviation of r from its ergodic value?
Author response: Following the referee’s suggestion, we have attempted to represent the transition points as the crossing points of the r curves, in line with earlier work by one of us and his graduate student. We find that a reasonable fit is obtained using our previously determined critical transverse fields, under the assumption that the transition width decreases with the number of spins N as N to the power -3/2. This analysis has been incorporated into Sec. 4 of the revised manuscript, and the behavior of the transition width is also discussed in the Conclusion.
Referee 2 question: The quantum random energy model (studied recently also in Phys. Rev. B 111, 214206 (2025) is more directly amenable to Bethe-lattice arguments because energies on neighboring hypercube vertices are uncorrelated. In contrast, the present Ising model exhibits correlations. While the presence of the correlations make the presently studied model more interesting from the many-body perspective, do the authors have any argument why the Bethe lattice argument is applicable despite the correlations?
Author response: This is indeed an important remark. The energies in the system are correlated, so that the energy change due to a single spin flip is much smaller than the typical energy of a Sherrington-Kirkpatrick model state. However, eigenstates with energy E close to zero are formed in both problems by sequences of resonant sites having approximately the same energy (e.g.,
E=0). For the spin-glass problem, the energies of neighboring sites differ by spin-flip energies that approach zero for sites with identical energies. Consequently, the identity of energies along a delocalization path is equivalent to having nearly zero spin-flip energies along this path.
Since spin-flip energies for different spins have negligible correlations at large
N and at infinite temperature, these correlations can be safely neglected, as discussed in earlier work by one of the authors (Ann. Phys., NY 529, 1600292; see also the preprint https://arxiv.org/abs/1610.00811). We refer explicitly to the preprint because it includes a Supporting Information section with detailed and accurate derivations that were omitted in the published paper. The author (AB) has contacted the Annals of Physics editorial office requesting the missing Supporting Information be added to the online version, and they have kindly agreed. We hope that this updated version will be available on the journal website at the time of manuscript consideration.
The discussion of the absence of correlations between spin-flip energies has been added to the revised manuscript (Sec. 2, paragraphs 2 and 3), along with a comparison of our results to those of the quantum random energy model (Sec. 4, paragraph following Eq. (9)). In addition, the Introduction and Conclusion have been revised to emphasize the similarity of the present problem to the Bethe-lattice problem.
Referee 2 question 3: For the analytical estimate of the MBL threshold, the authors consider only second-order processes. However, higher-order processes have been argued to play a significant role in MBL physics (see, e.g., arXiv:2005.13558; Phys. Rev. B 104, 184203 (2021); SciPost Phys. 12, 201 (2022); Phys. Rev. Lett. 131, 106301 (2023)). Do the authors have arguments for why such processes do not invalidate the Bethe-lattice threshold in the present model?
Author response: This is indeed an important question, as it highlights the difference between the present model and systems with short-range interactions, including those cited by the referee. In our model, the breakdown of localization is reasonably well determined—within logarithmic accuracy—by the condition of one resonance per eigenstate of the static Hamiltonian
H0 (Eq. (1)). This is analogous to the definition of the Anderson localization threshold in three dimensions or on the Bethe lattice, and it is fundamentally different from the MBL transition in paradigmatic short-range spin chains.
In short-range spin chains, the single-resonance criterion yields a localization threshold for the dynamic interaction that vanishes in the thermodynamic limit as 1/N, which is not the case. Considering second-order processes in short-range models produces vanishing two-spin hopping amplitudes (Eq. (5)) for the vast majority of spin pairs, since most of them are non-interacting. This dramatically alters the single-spin-flip estimate of the localization threshold, in stark contrast to our model, where all spins interact with each other. A note on this distinction has been added to the Introduction and at the end of Sec. 3.
As discussed in earlier work (Ann. Phys., NY 529, 1600292; see also https://arxiv.org/abs/1610.00811), the inclusion of higher-order (n>2) spin transitions in the present model effectively rescales the argument of the logarithm in the localization threshold by the square root of the number of spin flips. Together with the reduced number of non-self-intersecting paths compared to the Bethe lattice, this is expected to modify the localization threshold only by a numerical factor relative to the Bethe-lattice result. This expectation is confirmed by the numerical results presented in the current work.
A discussion of higher-order processes has been added to Sec. 3 (just before the last paragraph).
Referee 2 question 4: Could the authors comment more extensively on how the variance Δr^2 is calculated? Is each ri taken into account as corresponding to a single disorder sample? Is this variance dependent on the number of energy levels within a single sample?”
Author response: We thank the Referee for the questions. These points were not explained clearly in the previous version. Note that, first, we average the gap ratio over a certain energy window to obtain its value at a fixed disorder realization. Next, we compute the variance of the obtained gap ratios over disorder realizations. The chosen size of the energy window can slightly alter the resulting value of the variance, because small window sizes typically introduce additional source of fluctuation. We added the corresponding text with clarification to the paper.