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REVIEW 4 major objections 6 minor 110 references

Dynamical detection of extended nonergodic states in many-body quantum systems

T0 review · 4 major / 6 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read The paper argues that the exponent of the long-time power-law decay of the time-averaged survival probability equals the correlation fractal dimension $D_2$, so fractal dimensions of many-body quantum states can be read off directly from…

desk verdict Useful dynamical probe of D2 in random-matrix settings with a solid GOE benchmark, but the many-body extension rests on single-size hand-picked fits and needs stability checks before it can be trusted. read the letter →

arxiv 2505.23910 v1 pith:STPWDDRS submitted 2025-05-29 cond-mat.dis-nn

David A. Zarate-Herrada , Isaías Vallejo-Fabila , Lea F. Santos , E. Jonathan Torres-Herrera This is my paper
classification cond-mat.dis-nn PACS 05.45.Mt71.30.+h
keywords fractaldimensionsurvivalprobabilitytime-averagedmultifractalityextendednonergodicstatesmany-bodylocalizationAubry-AndrémodelRosenzweig-Porter
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

Fractal dimensions of quantum states are usually extracted by scaling the inverse participation ratio across many system sizes, a costly procedure that struggles near critical points. The paper argues that the exponent $\nu$ of the long-time power-law decay of the time-averaged survival probability — the fidelity between an initial state and its time-evolved version, averaged over time — equals the correlation fractal dimension $D_2$ of the initial state in the energy eigenbasis. This equality is demonstrated analytically for Gaussian orthogonal random matrices and numerically for Rosenzweig-Porter matrices, power-law banded random matrices, the interacting Aubry-André model, and the disordered Heisenberg chain, where $\nu$ matches the box-counting value $D^{\mathrm{box}}_2$ across ergodic, extended nonergodic, and localized regimes. Because a single time trace replaces finite-size scaling, the method opens access to larger systems and gives an unambiguous basis in which the fractal dimension is defined. A practical consequence is that extended nonergodic phases can be identified dynamically, without state tomography.

What carries the argument

The central object is the time-averaged survival probability $\langle SP(t)\rangle = \frac{1}{t}\int_0^t \langle SP(\tau)\rangle\, d\tau$, whose power-law tail exponent $\nu$ is claimed to equal $D_2$. Its role is to act as a filter: time averaging removes the spectral contributions that contaminate the raw survival probability — the Bessel-function oscillations from the edges of the energy distribution and the correlation hole from level repulsion — leaving only the correlations among the components $C^{(0)}_n$ of the initial state in the Hamiltonian eigenbasis. The independent definition of $D_2$ used for comparison is the box-counting method, $P(l)\propto l^{D_2}$, with $P(l)$ built from the squared projections of the initial state inside energy boxes of linear size $l$. In the Gaussian orthogonal ensemble case the machinery is fully explicit: integrating the analytical survival probability (Eq. 10) yields Eq. (12), whose large-time limit is $\langle SP(t\to\infty)\rangle \approx \frac{8}{9\pi}\frac{N}{\Gamma t} SP - \frac{1}{6}\frac{N}{\Gamma t} SP + SP$, giving $\nu = 1$ with no correlation hole.

What would settle it

Compute $\nu$ from $\langle SP(t)\rangle$ and $D^{\mathrm{box}}_2$ from the same initial states in the interacting Aubry-André model at $L=18$ or $20$ with a time-evolution algorithm: a systematic discrepancy, or the disappearance of a clean power-law window in the intermediate regime $0.7<h<1.7$, would falsify the claimed equality.

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Extended reading notes

Core claim

The paper's central claim is that the long-time power-law exponent $\nu$ of the time-averaged survival probability, $\langle SP(t)\rangle \propto t^{-\nu}$, coincides with the correlation fractal dimension $D_2$ obtained by box counting, $P(l)\propto l^{D_2}$, where $P(l)$ sums the squared amplitudes of the initial state inside energy boxes of size $l$. The claim is tested in five settings: full random matrices from the Gaussian orthogonal ensemble, the Rosenzweig-Porter ensemble, power-law banded random matrices, the interacting Aubry-André model, and the disordered spin-$1/2$ Heisenberg chain. In every case, $\nu$ agrees with $D^{\mathrm{box}}_2$ across the ergodic, extended nonergodic, and localized phases, even where the raw survival-probability exponent $\gamma$ or the IPR-based fractal dimension $D^{\mathrm{IPR}_0}_2$ deviate. For the Gaussian orthogonal ensemble the entire evolution of $\langle SP(t)\rangle$ is derived analytically (Eq. 12): Bessel oscillations and the correlation hole are averaged away, and the asymptotic decay is exactly $t^{-1}$, so $\nu = D_2 = 1$. The conclusion is that the time-averaged survival probability filters out spectral correlations and leaves a decay controlled solely by correlations among eigenstate components, making it a direct dynamical observable for $D_2$.

Load-bearing premise

The method assumes that in the time window chosen for the fit, the time-averaged survival probability has already lost all spectral correlations (level repulsion, spectral edges, and the correlation hole), so its decay exponent is governed only by correlations among the eigenstate components; the paper proves this filtering only for the GOE and checks it numerically at $L=16$ for the interacting models.

Editorial extensions

If this is right

  • If $\nu = D_2$ holds, the fractal dimension of an initial state can be extracted from a single long-time dynamical trace, replacing the need to diagonalize many system sizes and perform finite-size scaling.
  • Because the method uses time evolution rather than full exact diagonalization, it is compatible with large-scale evolution algorithms and can reach larger system sizes than IPR-based scaling studies.
  • The basis is fixed to the energy eigenbasis, so the extracted $D_2$ is not affected by the choice of real-space or momentum-space basis that complicates eigenstate-based IPR analyses.
  • In extended nonergodic phases, $\nu$ and $D^{\mathrm{box}}_2$ agree even when the IPR-based $D_2$ does not, so the dynamical exponent is a sharper indicator of nonergodicity.
  • Since the survival probability is a fidelity, the approach may be usable in experiments without state tomography, as a direct probe of multifractal and extended nonergodic phases.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • A testable extension is to push the equality $\nu = D_2$ to larger system sizes in the interacting Aubry-André and Heisenberg models using time-evolution algorithms, checking whether a clean power-law window persists near the many-body localization crossover; the paper tests only $L=16$ for the interacting models.
  • If the equality holds across the transition region, the time-averaged survival probability could become a dynamical order parameter for many-body localization, locating the transition by where $\nu$ (and hence $D_2$) extrapolates to zero.
  • The faster decay of $\nu$ and $D^{\mathrm{box}}_2$ relative to $D^{\mathrm{IPR}_0}_2$ in intermediate regimes hints that box-counting and dynamics are more sensitive to weak multifractality than IPR scaling; a possible reason is that energy-box sums weight clusters of nearby components, an interpretation the paper does not develop.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 6 minor

Summary. The manuscript proposes that the long-time power-law exponent ν of the time-averaged survival probability, Eq. (5), equals the box-counting fractal dimension Dbox2 of Eq. (8), and that this equality allows D2 to be extracted from dynamics alone, without finite-size scaling. The authors test this claim on GOE random matrices, the Rosenzweig-Porter (RP) model, power-law banded random matrices (PBRM), the interacting Aubry-André model, and the disordered Heisenberg spin chain. For GOE they provide an explicit analytical derivation of the full time-averaged survival probability in Appendix A and verify it numerically. For the other models they compare γ, ν, DIPR0_2, and Dbox2 as functions of the relevant control parameter and report excellent agreement between ν and Dbox2 in all cases, while γ and DIPR0_2 deviate in extended nonergodic and chaotic regimes. The conclusion is that the time-averaged survival probability is a robust and scalable dynamical probe of multifractality in many-body systems.

Significance. If the central equality ν = Dbox2 holds for interacting many-body systems, the paper offers a genuinely useful tool: it would replace multi-size scaling analyses of eigenstates with a single dynamical simulation, and it connects to tensor-network time-evolution methods that reach larger sizes than exact diagonalization. The GOE derivation is a concrete strength: the analytical expression in Appendix A is explicit, the asymptotic ν = 1 is derived, and the numerical curves agree with it. The RP and PBRM results are also valuable benchmarks because they scan ergodic, extended nonergodic, and localized regimes. However, the many-body part of the claim, which is the advertised main advance, rests on a single system size and on fitted exponents without uncertainty estimates; this is the load-bearing gap that prevents the paper from being accepted in its present form.

major comments (4)
  1. [Sec. IV A/B, Figs. 6–8] The central many-body claim, ν ≈ Dbox2, is demonstrated only at L = 16 (N = 12,870), a single system size. Since both ν from Eq. (5) and Dbox2 from Eq. (8) are extracted from the same finite system with hand-picked fitting windows, the agreement could be a common finite-size renormalization rather than the thermodynamic equality asserted in the abstract and conclusions. Please provide results for at least two or three additional sizes (e.g., L = 12, 14, 16) with error bars, showing that ν and Dbox2 converge to the same value, or demonstrate the dynamical method on larger systems using the tensor-network/MPO time evolution advertised in the Introduction. Without this, the scalable extraction claim is not supported by the data.
  2. [Sec. II A and Appendix A] The filtering property that time averaging removes spectral correlations and leaves only correlations among eigenstate components is derived analytically only for GOE matrices (Eq. (12) and Appendix A). For the interacting Aubry-André and Heisenberg models (Secs. IV A/B) no analogous argument is given, and at L = 16 the discrete spectrum and finite correlation-hole time scale can contaminate the fitted power-law window. The manuscript should state a concrete criterion for choosing the fitting windows for γ, ν, and the P(l) scaling region, and should quantify sensitivity to window endpoints for the many-body data. As written, ν from Eq. (5) may reflect the pre-saturation dynamics of a finite system rather than the asymptotic fractal dimension D2.
  3. [Figs. 2, 8, 9] No error bars or statistical uncertainties are reported for the fitted exponents γ, ν, DIPR0_2, or Dbox2, and the fitting windows are described only as 'the same for both exponents' with no procedure for their selection. Because every conclusion in the paper is a comparison of fitted exponents, the absence of uncertainty quantification makes it impossible to judge whether the reported agreement between ν and Dbox2 is significant or within expected numerical scatter. Please report bootstrap or realization-to-realization errors and explicitly list the fitting intervals used in each figure.
  4. [Introduction and Conclusions] The paper's stated practical advantage is that the dynamical approach 'avoids the need for scaling procedures and enables access to larger systems than those typically reachable via exact diagonalization.' This is not demonstrated for the many-body models: all many-body data in Secs. IV A/B come from exact diagonalization at L = 16, the same scale as standard finite-size scaling studies. A concrete demonstration on a larger system, even for one parameter point, using the tensor-network methods referenced in the Introduction would be needed to substantiate this claim.
minor comments (6)
  1. [Eq. (4)] The same angle brackets ⟨·⟩ are used for the disorder average and for the time average in Eq. (4), which is confusing; consider a different notation such as an overline for the time average.
  2. [Sec. IV B] In the sentence 'The figures also reiterates that' the verb should agree with the plural subject; it should read 'The figures also reiterate that.'
  3. [Appendix B] The text says 'The bottom left and right Fig. 8(d) show the analysis of ⟨P(l)⟩,' but the P(l) panels are labeled as Fig. 8(e); please correct the cross-reference.
  4. [Fig. 3 caption] The color assignments for the α values in the caption of Fig. 3 are ambiguous; labeling the curves directly or using distinct markers would improve readability.
  5. [Refs. [106] and [107]] References [106] and [107] appear to be the same article by Xu et al. with slightly different titles; please merge or remove the duplicate.
  6. [Sec. II A] The phrase '10 4 samples' appears twice and should be written as '10^4 samples'.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the exponent nu and the box-counting dimension Dbox_2 are extracted from independent temporal and static quantities, and their agreement is benchmarked against external results.

full rationale

The central claim of the paper is that the power-law exponent nu of the time-averaged survival probability, defined in Eqs. (4)-(5), coincides with the box-counting fractal dimension Dbox_2 extracted from P(l) in Eqs. (7)-(8). These are not the same quantity by construction: nu is a temporal decay exponent obtained by fitting the time average of the fidelity, while Dbox_2 is a static exponent obtained by binning the initial state's energy components over box sizes l. The two quantities live on different axes (time versus energy scale) and are fitted independently. For GOE matrices, the paper provides an explicit analytical derivation in Appendix A, showing nu = 1 and separately noting that Dbox_2 = 1 for random vectors; this is a genuine derivation, not a renaming. For the Rosenzweig-Porter and power-law banded random matrix ensembles, and for the interacting Aubry-Andre and Heisenberg models, the agreement between nu and Dbox_2 is established numerically from separate fits, and the paper explicitly cites the external result [4] for the general relation between Dbox_2 and the decay of the time-averaged survival probability. The self-citations to the authors' earlier expressions for the GOE survival probability [21, 67, 68] serve as inputs to the analytical GOE calculation, but they are not the sole support for the many-body claim, which is checked against independently computed static Dbox_2 values. No fitted parameter is relabeled as a prediction, and no uniqueness theorem or ansatz is smuggled in through a self-citation chain. The main limitation, namely that the interacting-model verification is performed only at L = 16 with manually selected fitting windows, is a finite-size and fitting-robustness concern rather than a circularity. Thus no specific circular step can be exhibited, and the appropriate finding is no significant circularity.

Assumptions & free parameters 1 free parameters · 4 assumptions · 0 invented entities

The central derivation introduces no free parameters beyond the manually chosen fitting windows. The main assumptions are the spectral-filtering property of the time-averaged survival probability, the existence of a valid box-counting scaling regime, and the prior phase diagrams for the interacting models. No new particles, forces, or dimensions are postulated.

free parameters (1)
  • Power-law fitting windows for gamma, nu, and box-counting scaling ranges = Not specified; chosen by hand
    The exponents that form the central comparison are extracted by fitting over intervals selected by the authors. The paper says the time intervals are the same for gamma and nu, but it does not state the criterion, so the extracted values and the agreement could depend on this choice.
assumptions (4)
  • domain assumption The time-averaged survival probability filters out spectral details and reflects only correlations among the eigenstate components.
    Invoked in Sec. I and used to justify nu = D2; proven analytically only for GOE in Appendix A, assumed for RP, PBRM, Aubry-André, and Heisenberg models. Location: Sec. I, Sec. II A, Eqs. (4)-(5).
  • domain assumption The box-counting scaling P(l) proportional to l^D2 holds over the accessible box-size window.
    The extraction of Dbox2 in Sec. II B assumes a linear region in the log-log plot; the paper selects this region by eye, without error bars. Location: Eq. (8), Figs. 3(b), 8(e), 9(e).
  • domain assumption The phase boundaries of the interacting Aubry-André model (0.7 < h < 1.7 for extended nonergodic) and the Heisenberg model's delocalization/localization behavior are as given by cited prior work.
    The interpretation of the results depends on these phase diagrams; the Heisenberg MBL transition itself is debated. Location: Sec. IV, refs. [85,106-111].
  • standard math Standard quantum mechanical time evolution with a closed Hamiltonian and initial states near the center of the spectrum.
    The derivations assume unitary evolution, Eq. (1), and the choice of initial states with energy near the band center. This is standard and unproblematic.

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Cite this review

Pith. "Pith review of Dynamical detection of extended nonergodic states in many-body quantum systems." pith.science (2026). https://pith.science/paper/STPWDDRS

@misc{pith2026250523910,
  author       = {Pith},
  title        = {Pith review of: Dynamical detection of extended nonergodic states in many-body quantum systems},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/STPWDDRS}},
  note         = {Machine review of arXiv:2505.23910}
}
abstract

Fractal dimensions are tools for probing the structure of quantum states and identifying whether they are localized or delocalized in a given basis. These quantities are commonly extracted through finite-size scaling, which limits the analysis to relatively small system sizes. In this work, we demonstrate that the correlation fractal dimension $D_2$ can be directly obtained from the long-time dynamics of interacting many-body quantum systems. Specifically, we show that it coincides with the exponent of the power-law decay of the time-averaged survival probability, defined as the fidelity between an initial state and its time-evolved counterpart. This dynamical approach avoids the need for scaling procedures and enables access to larger systems than those typically reachable via exact diagonalization. We test the method on various random matrix ensembles, including full random matrices, the Rosenzweig-Porter model, and power-law banded random matrices, and extend the analysis to interacting many-body systems described by the one-dimensional Aubry-Andr\'e model and the disordered spin-1/2 Heisenberg chain. In the case of full random matrices, we also derive an analytical expression for the entire evolution of the time-averaged survival probability.

Figures

Figures reproduced from arXiv: 2505.23910 by the authors.

Figure 1
Figure 1. FIG. 1. Analytical expression for the survival probability [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Survival probability [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. (a) Scaling analysis of [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figures from the paper (5 more)
Figure 5
Figure 5. Figure 5: FIG. 5. Study of fractality for the power-law banded random [PITH_FULL_IMAGE:figures/full_fig_p006_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6. Power-law exponents [PITH_FULL_IMAGE:figures/full_fig_p007_6.png]
Figure 7
Figure 7. Figure 7: shows γ, ν, D IPR0 2 , and Dbox 2 as a function of the disorder strength h for the Heisenberg model. Once again we confirm that ν ≈ Dbox 2 for interacting many￾body quantum systems. The figures also reiterates that, similarly to the Aubry-Andr´e model, γ → 2 deep in th…
Figure 8
Figure 8. Figure 8: FIG. 8. Interacting Aubry-Andr´e model (left panels) and disordered Heisenberg model (right panels) for [PITH_FULL_IMAGE:figures/full_fig_p008_8.png]
Figure 9
Figure 9. Figure 9: FIG. 9. Power-law banded random matrix model at the criti [PITH_FULL_IMAGE:figures/full_fig_p009_9.png]

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