pith. machine review for the scientific record. sign in

arxiv: 2605.05445 · v1 · submitted 2026-05-06 · ❄️ cond-mat.dis-nn · cond-mat.quant-gas· cond-mat.stat-mech

Recognition: unknown

Resonance Proliferation Across Localization Transitions

Carlo Vanoni , David M. Long , Anushya Chandran
Authors on Pith no claims yet

Pith reviewed 2026-05-08 15:40 UTC · model grok-4.3

classification ❄️ cond-mat.dis-nn cond-mat.quant-gascond-mat.stat-mech
keywords many-body localizationresonance proliferationstatistical Jacobi approximationflow equationlocalization transitionsAnderson modelthermalizationparticipation ratio
0
0 comments X

The pith

A flow equation for the resonance density exponent θ(w) derived in the statistical Jacobi approximation distinguishes localized phases as stable fixed points from those undergoing resonance proliferation and thermalization.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper develops a statistical method to predict how resonances form at different frequency scales in many-body localization models. Working inside the statistical Jacobi approximation, it derives a flow equation for the power-law exponent θ(w) that tracks the density of resonances at frequency scale w. Localized phases correspond to a line of fixed points with θ(w) positive and stable, whereas an instability in the flow marks the proliferation of resonances and the onset of thermalization. The predicted θ(w) agrees with numerical data from Anderson models on random regular graphs, the Lévy-Rosenzweig-Porter ensemble, and real-space MBL systems at intermediate disorder. A sympathetic reader would care because the approach supplies a concrete mechanism for the slow finite-size drifts toward delocalization that appear in MBL numerics.

Core claim

Working within the statistical Jacobi approximation (SJA), we derive a flow equation for a power-law exponent θ(w) characterizing the density of resonances at frequency scale w. A localized phase is described by a line of fixed points with θ(w)>0, while an instability of the flow signals resonance proliferation and the onset of thermalization. The predicted θ(w) matches numerics on the Anderson model on random regular graphs and the Lévy-Rosenzweig-Porter random matrix ensemble, both of which host resonance-driven delocalization transitions. We further connect the flow to eigenstate properties such as the participation ratio and to dynamical observables such as the return probability. The θ

What carries the argument

The statistical Jacobi approximation (SJA) and the derived flow equation for the power-law exponent θ(w) that characterizes resonance density at frequency scale w.

If this is right

  • The flow equation connects resonance statistics to eigenstate properties such as the participation ratio.
  • It also links to dynamical observables such as the return probability.
  • The predicted θ(w) accounts for finite-size drifts observed in real-space many-body localization models at intermediate disorder.
  • Resonance-driven delocalization transitions occur in the Anderson model on random regular graphs and the Lévy-Rosenzweig-Porter ensemble.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • If the flow description is accurate, larger-system MBL simulations should exhibit the flow instability at progressively lower disorder strengths.
  • The same flow analysis could be applied to other disordered quantum systems where resonance formation controls localization transitions.
  • Quantitative comparison of measured θ(w) across different lattice geometries would test how universal the fixed-point structure is.

Load-bearing premise

The statistical Jacobi approximation must correctly capture how resonances form and how their frequency scales interact statistically.

What would settle it

A numerical extraction of the resonance density exponent θ(w) versus frequency scale in a many-body localized system at intermediate disorder that deviates from the predicted flow or fixed-point structure would falsify the central claim.

Figures

Figures reproduced from arXiv: 2605.05445 by Anushya Chandran, Carlo Vanoni, David M. Long.

Figure 1
Figure 1. Figure 1: FIG. 1. (a) Schematic representation of two of the models we study using the Jacobi algorithm, namely the Anderson model view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Numerical solution of the heuristic flow equations, view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Sketch of the integrated number of resonances view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Jacobi flow of view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. The exponent view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. The exponent view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. Plot of the solution of the functional flow equations, view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9. (a) Number of resonances view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10. (a) The analytical prediction for the return probabil view at source ↗
Figure 11
Figure 11. Figure 11: FIG. 11 view at source ↗
Figure 12
Figure 12. Figure 12: FIG. 12. Flow of view at source ↗
read the original abstract

Models of many-body localization (MBL) exhibit slow numerical drifts towards delocalization with increasing system size, for which no satisfactory theory exists. Numerics indicates that these drifts are driven by the proliferation of many-body resonances at intermediate disorder strengths. We develop a statistical method to predict the distribution of resonance oscillation frequencies which captures how the formation of resonances at larger frequency scales subsequently affects the formation of resonances at lower frequencies. Working within the statistical Jacobi approximation (SJA), we derive a flow equation for a power-law exponent $\theta(w)$ characterizing the density of resonances at frequency scale $w$. A localized phase is described by a line of fixed points with $\theta(w)>0$, while an instability of the flow signals resonance proliferation and the onset of thermalization. The predicted $\theta(w)$ matches numerics on the Anderson model on random regular graphs and the L\'evy-Rosenzweig-Porter random matrix ensemble, both of which host resonance-driven delocalization transitions. We further connect the flow to eigenstate properties such as the participation ratio and to dynamical observables such as the return probability. The predicted $\theta(w)$ also matches what is numerically measured in real-space models of MBL at intermediate disorder strengths, representing a significant step towards explaining the finite-size drifts observed in MBL.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript develops a statistical method within the statistical Jacobi approximation (SJA) to derive a flow equation for a power-law exponent θ(w) that characterizes the density of resonances at frequency scale w. A localized phase is described by a line of fixed points with θ(w)>0, while flow instability signals resonance proliferation and the onset of thermalization. The predicted θ(w) is reported to match numerics for the Anderson model on random regular graphs and the Lévy-Rosenzweig-Porter ensemble; the same quantity is also compared to numerical measurements in real-space MBL models at intermediate disorder to address finite-size drifts toward delocalization. Connections are drawn to eigenstate properties such as participation ratio and dynamical observables such as return probability.

Significance. If the central derivation and its numerical matches hold, the work supplies a parameter-free flow description of resonance proliferation that could explain observed finite-size drifts in MBL without ad-hoc fitting. The explicit links to participation ratios and return probabilities strengthen the connection between the flow and measurable quantities. The quantitative agreement reported for the Anderson model on random regular graphs and the Lévy-Rosenzweig-Porter ensemble provides concrete support for the SJA framework in those settings.

major comments (2)
  1. [Derivation of the flow equation (main text section introducing the SJA flow)] The flow equation for θ(w) is obtained entirely inside the SJA by assuming statistical independence of resonance formation across frequency scales together with a power-law ansatz. Because this construction is load-bearing for the fixed-point analysis, the instability criterion, and all subsequent claims, the manuscript must supply a self-contained derivation (including the precise statistical averaging steps and any truncation) rather than leaving the steps implicit.
  2. [Application to real-space MBL models] The abstract and the section applying the flow to real-space MBL models assert that the SJA-derived θ(w) matches numerically measured resonance densities at intermediate disorder and thereby explains finite-size drifts. This extension is load-bearing for the paper’s central claim about MBL, yet the SJA assumes statistical independence and absence of spatial clustering; the manuscript should demonstrate that local correlations, level repulsion, or resonance clustering in interacting real-space Hamiltonians do not invalidate the power-law ansatz or the flow.
minor comments (2)
  1. [Notation and figures] The definition of the frequency scale w and the precise meaning of the exponent θ(w) should be restated explicitly when first introduced and again in the figure captions to prevent ambiguity when comparing across models.
  2. [Numerical comparisons] Quantitative measures of agreement (e.g., mean-square deviation or χ² per degree of freedom) between the predicted θ(w) curves and the numerical data points should be reported in the figure captions or a table, rather than relying solely on visual inspection.

Simulated Author's Rebuttal

2 responses · 1 unresolved

We thank the referee for their careful reading of the manuscript and for the constructive comments. We appreciate the positive evaluation of the work's significance. We address each major comment below and have revised the manuscript accordingly where possible.

read point-by-point responses

  1. Referee: [Derivation of the flow equation (main text section introducing the SJA flow)] The flow equation for θ(w) is obtained entirely inside the SJA by assuming statistical independence of resonance formation across frequency scales together with a power-law ansatz. Because this construction is load-bearing for the fixed-point analysis, the instability criterion, and all subsequent claims, the manuscript must supply a self-contained derivation (including the precise statistical averaging steps and any truncation) rather than leaving the steps implicit.

    Authors: We agree that the derivation of the flow equation must be fully self-contained. In the revised manuscript we have substantially expanded the section introducing the SJA flow. The new text now contains an explicit, step-by-step derivation that spells out the statistical averaging procedure under the independence assumption, the adoption of the power-law ansatz for the resonance density, and the truncations that are performed. Supporting algebraic details have been added to a new appendix so that every step can be followed without reference to external material. revision: yes

  2. Referee: [Application to real-space MBL models] The abstract and the section applying the flow to real-space MBL models assert that the SJA-derived θ(w) matches numerically measured resonance densities at intermediate disorder and thereby explains finite-size drifts. This extension is load-bearing for the paper’s central claim about MBL, yet the SJA assumes statistical independence and absence of spatial clustering; the manuscript should demonstrate that local correlations, level repulsion, or resonance clustering in interacting real-space Hamiltonians do not invalidate the power-law ansatz or the flow.

    Authors: We acknowledge that the SJA is formulated under the assumption of statistical independence and without explicit spatial structure. In the revised manuscript we have inserted a dedicated discussion subsection that examines the possible influence of local correlations, level repulsion, and resonance clustering on the power-law ansatz. We point out that the quantitative agreement between the SJA-predicted θ(w) and the numerically extracted resonance densities at intermediate disorder already provides empirical support that these effects do not destroy the flow structure in the regimes of interest. A complete analytic proof that spatial correlations can never invalidate the ansatz would require an extended theory that incorporates real-space information; such an extension lies outside the scope of the present work. The added discussion therefore clarifies the approximation's domain of applicability while preserving the central claim based on the observed numerical match. revision: partial

standing simulated objections not resolved
  • A rigorous demonstration that spatial correlations, level repulsion, and resonance clustering never invalidate the power-law ansatz or the flow equation in every regime of real-space MBL would require additional theoretical machinery or extensive new simulations that are not contained in the current manuscript.

Circularity Check

0 steps flagged

Derivation of θ(w) flow equation is self-contained within SJA; no reduction to inputs by construction

full rationale

The central derivation obtains a flow equation for the resonance density exponent θ(w) by modeling successive frequency scales as statistically independent under the stated SJA assumptions and a power-law form for the density. Fixed-point analysis and instability criteria then follow directly from the resulting differential equation without reference to the target numerical data. Agreement with exact resonance counting on the Anderson model on random regular graphs, the Lévy-Rosenzweig-Porter ensemble, and real-space MBL is presented strictly as external validation. No load-bearing step equates a fitted parameter to a prediction, imports a uniqueness theorem via self-citation, or renames an input as an output; the SJA framework supplies an independent statistical model whose assumptions are explicitly declared and tested against separate benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the statistical Jacobi approximation (SJA) as the domain framework for deriving the flow equation, with no free parameters or invented entities explicitly introduced in the abstract.

axioms (1)
  • domain assumption The statistical Jacobi approximation (SJA) is valid for predicting the distribution of resonance oscillation frequencies and deriving the flow for θ(w).
    All derivations and predictions are performed within the SJA as stated.

pith-pipeline@v0.9.0 · 5536 in / 1391 out tokens · 113925 ms · 2026-05-08T15:40:00.507834+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

82 extracted references · 5 canonical work pages · 1 internal anchor

  1. [1]

    The Hamil- tonianH LRP of the model is given by HLRP =L (µ,γ) +D,(30) whereD ij =ϵ iδij,i= 1,

    L´ evy-Rosenzweig-Porter As a first model, we consider a modification of the Rosenzweig-Porter (RP) random matrix ensemble [53], with off-diagonal matrix elements extracted from a long- tailed L´ evy (power-law) distribution [54–60]. The Hamil- tonianH LRP of the model is given by HLRP =L (µ,γ) +D,(30) whereD ij =ϵ iδij,i= 1, . . . , N, is a diagonal ma- ...

  2. [2]

    That is, the adja- cency matrix of an RRG

    Random Regular Graph As a second model, we consider the Anderson model on a random regular graph (RRG), with Hamiltonian HRRG =A+D.(32) 9 The diagonal matrixDhas the same definition as for the Levy RP model, whileAis the adjacency matrix of a graph uniformly sampled from the set of all graphs withNvertices and fixed degree. That is, the adja- cency matrix...

  3. [3]

    random numbersh i are dis- tributed uniformly between−WandW

    XXZ spin chain As a third model, we consider the random field spin- 1/2 XXZ chain at the Heisenberg point, often called the standard model of MBL, defined by the Hamiltonian HXXZ =J LX i=1 Si ·S i+1 + LX i=1 hiSz i ,(33) where we introduced the spin-1/2 operatorsS i = ( ˆSx i , ˆSy i , ˆSz i ) and the i.i.d. random numbersh i are dis- tributed uniformly b...

  4. [4]

    the ramp

    Ramp The monotonic increase of the exponentθ(w) asw decreases from its starting value is dubbed “the ramp” (right panel of Fig. 3). This feature is visible only in the large-wand large-disorder regime, and appears in both the RRG and XXZ models. The ramp is not captured by the heuristic flow equa- tion because the off-diagonal matrix elements of the Hamil...

  5. [5]

    In particular, starting from largerw, all plots show a flow towards more negativeθwith decreasingw

    Flow equation regime Once the power law distribution of Hamiltonian matrix elements has been initialized, there is an intermediate range ofwwhich is well described by the heuristic flow equations. In particular, starting from largerw, all plots show a flow towards more negativeθwith decreasingw. Models which are more strongly disordered (largerWor smaller...

  6. [6]

    4 departs from the predictions of the flow equation

    Bounce Oncewbecomes very small,O(1/ √ N), Fig. 4 departs from the predictions of the flow equation. The asymp- totic divergence (θ <0) or saturation (θ >0) ofθ(w) is interrupted by a bounce back toθ= 1. The bounce oc- curs in correspondence with the “knee” innres [see Fig. 3]. The bounce is a signature of the models exiting the sparse regime and crossing ...

  7. [7]

    comoving

    Scaling with matrix size Lastly, we comment on howθ(w) depends on the ma- trix size, via the Hilbert space dimensionN. The flow equations do not directly predict theNde- pendence forθ(w). The SJA analysis indicates that the flow equation Eq. (28) should be independent ofNin the sparse regime. 1 However, the sparse regime scaling breaks down whenθ(w) begin...

  8. [8]

    P. W. Anderson, Absence of diffusion in certain random lattices, Phys. Rev.109, 1492 (1958)

    1958

  9. [9]

    I. V. Gornyi, A. D. Mirlin, and D. G. Polyakov, Interact- ing electrons in disordered wires: Anderson localization and low-ttransport, Phys. Rev. Lett.95, 206603 (2005)

    2005

  10. [10]

    Basko, I

    D. Basko, I. Aleiner, and B. Altshuler, Metal–insulator transition in a weakly interacting many-electron system with localized single-particle states, Ann. Phys.321, 1126 (2006)

    2006

  11. [11]

    Oganesyan and D

    V. Oganesyan and D. A. Huse, Localization of interacting fermions at high temperature, Phys. Rev. B75, 155111 (2007)

    2007

  12. [12]

    Pal and D

    A. Pal and D. A. Huse, Many-body localization phase transition, Phys. Rev. B82, 174411 (2010)

    2010

  13. [13]

    J. Z. Imbrie, On many-body localization for quantum spin chains, Journal of Statistical Physics163, 998 (2016)

    2016

  14. [14]

    Schreiber, S

    M. Schreiber, S. S. Hodgman, P. Bordia, H. P. L¨ uschen, M. H. Fischer, R. Vosk, E. Altman, U. Schneider, and I. Bloch, Observation of many-body localization of inter- acting fermions in a quasirandom optical lattice, Science 349, 842 (2015)

    2015

  15. [15]

    Rubio-Abadal, J.-y

    A. Rubio-Abadal, J.-y. Choi, J. Zeiher, S. Hollerith, J. Rui, I. Bloch, and C. Gross, Many-body delocaliza- tion in the presence of a quantum bath, Phys. Rev. X9, 041014 (2019)

    2019

  16. [16]

    Nandkishore and D

    R. Nandkishore and D. A. Huse, Many-body localization and thermalization in quantum statistical mechanics, An- nual Review of Condensed Matter Physics6, 15 (2015)

    2015

  17. [17]

    J. Z. Imbrie, V. Ros, and A. Scardicchio, Local integrals of motion in many-body localized systems, Ann. Phys. 529, 1600278 (2017)

    2017

  18. [18]

    D. A. Abanin, E. Altman, I. Bloch, and M. Serbyn, Col- loquium: Many-body localization, thermalization, and entanglement, Rev. Mod. Phys.91, 021001 (2019)

    2019

  19. [19]

    Sierant, M

    P. Sierant, M. Lewenstein, A. Scardicchio, L. Vidmar, and J. Zakrzewski, Many-body localization in the age of classical computing, Rep. Prog. Phys.88, 026502 (2025), arXiv:2403.07111 [cond-mat.dis-nn]

    work page arXiv 2025

  20. [20]

    D. J. Luitz, N. Laflorencie, and F. Alet, Many-body local- ization edge in the random-field Heisenberg chain, Phys. Rev. B91, 081103 (2015)

    2015

  21. [21]

    ˇSuntajs, J

    J. ˇSuntajs, J. Bonˇ ca, T. c. v. Prosen, and L. Vidmar, Quantum chaos challenges many-body localization, Phys. Rev. E102, 062144 (2020)

    2020

  22. [22]

    Sierant, D

    P. Sierant, D. Delande, and J. Zakrzewski, Thouless time analysis of anderson and many-body localization transi- tions, Phys. Rev. Lett.124, 186601 (2020)

    2020

  23. [23]

    S. R. Taylor and A. Scardicchio, Subdiffusion in a one- dimensional anderson insulator with random dephasing: Finite-size scaling, griffiths effects, and possible impli- cations for many-body localization, Phys. Rev. B103, 184202 (2021)

    2021

  24. [25]

    Sels, Bath-induced delocalization in interacting disor- dered spin chains, Phys

    D. Sels, Bath-induced delocalization in interacting disor- dered spin chains, Phys. Rev. B106, L020202 (2022)

    2022

  25. [26]

    Sierant and J

    P. Sierant and J. Zakrzewski, Challenges to observation of many-body localization, Phys. Rev. B105, 224203 (2022)

    2022

  26. [27]

    Abanin, J

    D. Abanin, J. Bardarson, G. De Tomasi, S. Gopalakr- ishnan, V. Khemani, S. Parameswaran, F. Pollmann, A. Potter, M. Serbyn, and R. Vasseur, Distinguishing lo- 23 calization from chaos: Challenges in finite-size systems, Annals of Physics427, 168415 (2021)

    2021

  27. [28]

    R. K. Panda, A. Scardicchio, M. Schulz, S. R. Taylor, and M. ˇZnidariˇ c, Can we study the many-body localisation transition?, Europhysics Letters128, 67003 (2020)

    2020

  28. [29]

    D. J. Luitz, N. Laflorencie, and F. Alet, Extended slow dynamical regime close to the many-body localization transition, Phys. Rev. B93, 060201 (2016)

    2016

  29. [30]

    P. J. D. Crowley and A. Chandran, A constructive the- ory of the numerically accessible many-body localized to thermal crossover, SciPost Phys.12, 201 (2022)

    2022

  30. [31]

    Morningstar, L

    A. Morningstar, L. Colmenarez, V. Khemani, D. J. Luitz, and D. A. Huse, Avalanches and many-body res- onances in many-body localized systems, Phys. Rev. B 105, 174205 (2022)

    2022

  31. [32]

    D. M. Long, P. J. D. Crowley, V. Khemani, and A. Chan- dran, Phenomenology of the prethermal many-body lo- calized regime, Phys. Rev. Lett.131, 106301 (2023)

    2023

  32. [33]

    Sels and A

    D. Sels and A. Polkovnikov, Dynamical obstruction to localization in a disordered spin chain, Phys. Rev. E104, 054105 (2021)

    2021

  33. [34]

    De Roeck and F

    W. De Roeck and F. m. c. Huveneers, Stability and insta- bility towards delocalization in many-body localization systems, Phys. Rev. B95, 155129 (2017)

    2017

  34. [35]

    D. J. Luitz, F. m. c. Huveneers, and W. De Roeck, How a small quantum bath can thermalize long localized chains, Phys. Rev. Lett.119, 150602 (2017)

    2017

  35. [36]

    P. J. D. Crowley and A. Chandran, Avalanche induced coexisting localized and thermal regions in disordered chains, Phys. Rev. Research2, 033262 (2020)

    2020

  36. [37]

    Gopalakrishnan, M

    S. Gopalakrishnan, M. M¨ uller, V. Khemani, M. Knap, E. Demler, and D. A. Huse, Low-frequency conductivity in many-body localized systems, Phys. Rev. B92, 104202 (2015)

    2015

  37. [38]

    Khemani, S

    V. Khemani, S. P. Lim, D. N. Sheng, and D. A. Huse, Critical properties of the many-body localization transi- tion, Phys. Rev. X7, 021013 (2017)

    2017

  38. [39]

    Villalonga and B

    B. Villalonga and B. K. Clark, Eigenstates hybridize on all length scales at the many-body localization transition (2020), arXiv:2005.13558 [cond-mat.dis-nn]

    work page arXiv 2020

  39. [40]

    S. J. Garratt, S. Roy, and J. T. Chalker, Local resonances and parametric level dynamics in the many-body local- ized phase, Phys. Rev. B104, 184203 (2021)

    2021

  40. [41]

    H. Ha, A. Morningstar, and D. A. Huse, Many-body res- onances in the avalanche instability of many-body local- ization, Phys. Rev. Lett.130, 250405 (2023)

    2023

  41. [42]

    J. K. Jiang, F. M. Surace, and O. I. Motrunich, Charge transport capacity as a probe of resonances in models of many-body localization (2026), arXiv:2604.18710 [cond- mat.dis-nn]

    work page internal anchor Pith review Pith/arXiv arXiv 2026

  42. [43]

    Tikhonov and A

    K. Tikhonov and A. Mirlin, From Anderson localization on random regular graphs to many-body localization, Ann. Phys.435, 168525 (2021)

    2021

  43. [44]

    A. D. Luca and A. Scardicchio, Ergodicity breaking in a model showing many-body localization, Europhys. Lett. 101, 37003 (2013)

    2013

  44. [45]

    D. M. Long, D. Hahn, M. Bukov, and A. Chandran, Be- yond Fermi’s golden rule with the statistical Jacobi ap- proximation, SciPost Phys.15, 251 (2023)

    2023

  45. [46]

    D. Hahn, D. M. Long, M. Bukov, and A. Chandran, Pre- dicting dynamics from flows of the eigenstate thermaliza- tion hypothesis (2025), arXiv:2504.01073 [quant-ph]

    work page arXiv 2025

  46. [47]

    V. L. Berezinskiˇi, Destruction of long-range order in one- dimensional and two-dimensional systems possessing a continuous symmetry group. ii. quantum systems, Sov. J. Exp. Theor. Phys.34, 610 (1972)

    1972

  47. [48]

    J. M. Kosterlitz and D. J. Thouless, Ordering, metasta- bility and phase transitions in two-dimensional systems, J. Phys. C: Solid State Physics6, 1181 (1973)

    1973

  48. [49]

    C. Jacobi, ¨Uber ein leichtes verfahren die in der theo- rie der s¨ acularst¨ orungen vorkommenden gleichungen nu- merisch aufzul¨ osen*)., Journal f¨ ur die reine und ange- wandte Mathematik1846, 51 (1846)

  49. [50]

    J. M. Deutsch, Quantum statistical mechanics in a closed system, Phys. Rev. A43, 2046 (1991)

    2046

  50. [51]

    Srednicki, Chaos and quantum thermalization, Phys

    M. Srednicki, Chaos and quantum thermalization, Phys. Rev. E50, 888 (1994)

    1994

  51. [52]

    J. M. Deutsch, Eigenstate thermalization hypothesis, Re- ports on Progress in Physics81, 082001 (2018)

    2018

  52. [53]

    D. A. Huse, R. Nandkishore, and V. Oganesyan, Phe- nomenology of fully many-body-localized systems, Phys. Rev. B90, 174202 (2014)

    2014

  53. [54]

    Vanoni, B

    C. Vanoni, B. L. Altshuler, V. E. Kravtsov, and A. Scardicchio, Renormalization group analysis of the anderson model on random regular graphs, Proceedings of the National Academy of Sciences121, e2401955121 (2024)

    2024

  54. [55]

    Goremykina, R

    A. Goremykina, R. Vasseur, and M. Serbyn, Analytically solvable renormalization group for the many-body local- ization transition, Phys. Rev. Lett.122, 040601 (2019)

    2019

  55. [56]

    P. T. Dumitrescu, A. Goremykina, S. A. Parameswaran, M. Serbyn, and R. Vasseur, Kosterlitz-Thouless scaling at many-body localization phase transitions, Phys. Rev. B99, 094205 (2019)

    2019

  56. [57]

    Morningstar and D

    A. Morningstar and D. A. Huse, Renormalization-group study of the many-body localization transition in one di- mension, Phys. Rev. B99, 224205 (2019)

    2019

  57. [58]

    Niedda, G

    J. Niedda, G. B. Testasecca, G. Magnifico, F. Balducci, C. Vanoni, and A. Scardicchio, Renormalization group analysis of the many-body localization transition in the random-field XXZ chain, Phys. Rev. B112, 144201 (2025)

    2025

  58. [59]

    Morningstar, D

    A. Morningstar, D. A. Huse, and J. Z. Imbrie, Many- body localization near the critical point, Phys. Rev. B 102, 125134 (2020)

    2020

  59. [60]

    V. E. Kravtsov, I. M. Khaymovich, E. Cuevas, and M. Amini, A random matrix model with localization and ergodic transitions, New Journal of Physics17, 122002 (2015)

    2015

  60. [61]

    C. Monthus, Multifractality of eigenstates in the delo- calized non-ergodic phase of some random matrix mod- els: Wigner–Weisskopf approach, Journal of Physics A: Mathematical and Theoretical50, 295101 (2017)

    2017

  61. [62]

    V. E. Kravtsov, I. M. Khaymovich, B. L. Altshuler, and L. B. Ioffe, Localization transition on the ran- dom regular graph as an unstable tricritical point in a log-normal Rosenzweig-Porter random matrix ensemble (2020), arXiv:2002.02979 [cond-mat.dis-nn]

    work page arXiv 2020

  62. [63]

    I. M. Khaymovich, V. E. Kravtsov, B. L. Altshuler, and L. B. Ioffe, Fragile extended phases in the log-normal rosenzweig-porter model, Phys. Rev. Res.2, 043346 (2020)

    2020

  63. [64]

    Biroli and M

    G. Biroli and M. Tarzia, L´ evy-Rosenzweig-Porter random matrix ensemble, Phys. Rev. B103, 104205 (2021)

    2021

  64. [65]

    multifractal

    I. M. Khaymovich and V. E. Kravtsov, Dynamical phases in a “multifractal” Rosenzweig-Porter model, SciPost Phys.11, 045 (2021). 24

    2021

  65. [66]

    Kutlin and I

    A. Kutlin and I. M. Khaymovich, Anatomy of the eigen- states distribution: A quest for a genuine multifractality, SciPost Phys.16, 008 (2024)

    2024

  66. [67]

    Kutlin and C

    A. Kutlin and C. Vanoni, Investigating finite-size ef- fects in random matrices by counting resonances, SciPost Phys.18, 090 (2025)

    2025

  67. [68]

    Sierant, M

    P. Sierant, M. Lewenstein, and A. Scardicchio, Univer- sality in Anderson localization on random graphs with varying connectivity, SciPost Phys.15, 045 (2023)

    2023

  68. [69]

    K. S. Tikhonov, A. D. Mirlin, and M. A. Skvortsov, Anderson localization and ergodicity on random regular graphs, Phys. Rev. B94, 220203 (2016)

    2016

  69. [70]

    Garc´ ıa-Mata, J

    I. Garc´ ıa-Mata, J. Martin, O. Giraud, B. Georgeot, R. Dubertrand, and G. Lemari´ e, Critical properties of the anderson transition on random graphs: Two-parameter scaling theory, Kosterlitz-Thouless type flow, and many- body localization, Phys. Rev. B106, 214202 (2022)

    2022

  70. [71]

    Vanoni and V

    C. Vanoni and V. Vitale, Analysis of localization transi- tions using nonparametric unsupervised learning, Phys. Rev. B110, 024204 (2024)

    2024

  71. [72]

    Gopalakrishnan and D

    S. Gopalakrishnan and D. A. Huse, Instability of many- body localized systems as a phase transition in a non- standard thermodynamic limit, Phys. Rev. B99, 134305 (2019)

    2019

  72. [73]

    S. Bera, G. De Tomasi, I. M. Khaymovich, and A. Scardicchio, Return probability for the anderson model on the random regular graph, Phys. Rev. B98, 134205 (2018)

    2018

  73. [74]

    S. Iyer, V. Oganesyan, G. Refael, and D. A. Huse, Many- body localization in a quasiperiodic system, Phys. Rev. B87, 134202 (2013)

    2013

  74. [75]

    Khemani, D

    V. Khemani, D. N. Sheng, and D. A. Huse, Two univer- sality classes for the many-body localization transition, Phys. Rev. Lett.119, 075702 (2017)

    2017

  75. [76]

    Agrawal, S

    U. Agrawal, S. Gopalakrishnan, and R. Vasseur, Uni- versality and quantum criticality in quasiperiodic spin chains, Nature Communications11, 2225 (2020)

    2020

  76. [77]

    Agrawal, R

    U. Agrawal, R. Vasseur, and S. Gopalakrishnan, Quasiperiodic many-body localization transition in di- mensiond≥1, Phys. Rev. B106, 094206 (2022)

    2022

  77. [78]

    Y.-T. Tu, D. Vu, and S. Das Sarma, Avalanche stabil- ity transition in interacting quasiperiodic systems, Phys. Rev. B107, 014203 (2023)

    2023

  78. [79]

    Y.-T. Tu, D. M. Long, and S. Das Sarma, Interacting quasiperiodic spin chains in the prethermal regime, Phys. Rev. B109, 214309 (2024)

    2024

  79. [80]

    Biroli, G

    G. Biroli, G. Semerjian, and M. Tarzia, Anderson model on bethe lattices: Density of states, localization prop- erties and isolated eigenvalue, Progress of Theoretical Physics Supplement184, 187 (2010)

    2010

  80. [81]

    GitHub repository athttps://github.com/ CarloVanoni/JacobiFlow.git

Showing first 80 references.