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In the three-dimensional Anderson model the fidelity susceptibility develops a second peak exactly at the critical disorder for localization.

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2026-06-29 19:46 UTC pith:XIA3RXSU

load-bearing objection The paper finds two peaks in fidelity susceptibility for the 3D Anderson model, one shifting to zero disorder and the second aligning with the known critical point with submaximal scaling tied to fractality. the 2 major comments →

arxiv 2605.25594 v1 pith:XIA3RXSU submitted 2026-05-25 cond-mat.dis-nn cond-mat.stat-mechquant-ph

Sensitivity to perturbations in the three-dimensional Anderson model

Piotr Tokarczyk , Lev Vidmar , Anatoli Polkovnikov , Patrycja {\L}yd\.zba This is my paper
classification cond-mat.dis-nn cond-mat.stat-mechquant-ph
keywords Anderson localizationfidelity susceptibilitydisorder strengtheigenstate sensitivityquantum chaosnonergodic regimesfractal eigenstates
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The paper measures how much single-particle eigenstates in the three-dimensional Anderson model change under small perturbations of the Hamiltonian. This quantity, called fidelity susceptibility, is tracked as a function of increasing disorder strength W. It develops two peaks: the first marks a crossover from plane-wave-like states to chaotic ones at weak disorder and drifts toward zero disorder as system size increases. The second peak sits at the known critical disorder value where the Anderson localization transition occurs. The authors also report that the second peak diverges more weakly than the first and connect this to the fractal character of states right at criticality, while scaling above the transition reveals two separate nonergodic regimes.

Core claim

Fidelity susceptibility versus disorder strength W shows two peaks. The first signals the crossover at weak disorder from plane-wave states to single-particle quantum chaos and shifts toward W to 0 in the thermodynamic limit. The second emerges at the critical disorder strength of the Anderson localization transition to high numerical accuracy. Its divergence is submaximal and is tied to the fractal structure of eigenstates at criticality. Scaling of typical fidelity susceptibilities above the transition indicates two distinct regimes of nonergodic behavior.

What carries the argument

fidelity susceptibility, which quantifies the sensitivity of single-particle eigenstates to perturbations

Load-bearing premise

Finite-size effects together with the chosen frequency cutoff leave the apparent location of the second peak unshifted from the true thermodynamic critical disorder.

What would settle it

A computation on substantially larger lattices that places the second peak position systematically away from the independently established Anderson critical disorder value.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • The first peak position approaches zero disorder in the infinite-volume limit.
  • The first peak diverges maximally, scaling as the square of the inverse frequency cutoff.
  • The second peak diverges submaximally because of the fractal eigenstates that appear at criticality.
  • Two separate regimes of nonergodic behavior exist on the delocalized side of the Anderson transition.

Where Pith is reading between the lines

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

  • Fidelity susceptibility could locate localization transitions in other disordered single-particle models without direct inspection of wave-function support.
  • Submaximal divergence at a critical point may serve as a diagnostic for multifractal states in a wider class of disordered systems.
  • The reported distinction between two nonergodic regimes could be tested by comparing eigenstate statistics or level-spacing distributions in the same parameter window.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 1 minor

Summary. The manuscript investigates the fidelity susceptibility in the three-dimensional Anderson model as a function of disorder strength W. It reports two peaks: the first at weak disorder, interpreted as a crossover from plane-wave states to single-particle quantum chaos and shifting to W=0 in the thermodynamic limit; the second coinciding with the Anderson localization critical point W_c to high numerical accuracy. The first peak's divergence with frequency cutoff is maximal (scaling as 1/ω²), while the second is submaximal and linked to the fractal structure of critical eigenstates. Two scenarios for the peaks are discussed, and scaling of typical fidelity susceptibilities above the transition indicates two distinct nonergodic regimes.

Significance. If the numerical identification of the second peak holds, the work provides a new diagnostic for the Anderson transition via eigenstate sensitivity to perturbations and connects the suppressed divergence to multifractality at criticality. The distinction between maximal and submaximal divergences, together with evidence for two nonergodic regimes, would be of interest to the disordered-systems community if supported by detailed finite-size analysis.

major comments (2)
  1. [Abstract] Abstract: the central claim that the second peak coincides with the Anderson critical disorder 'to high numerical accuracy' is load-bearing but rests on an unshown finite-size extrapolation. No system sizes L, number of disorder realizations, error bars on peak locations, or extrapolation form (e.g., |W_peak(L) - W_c| scaling) are supplied, leaving open whether finite-size shifts or the frequency cutoff u move the apparent maximum away from the thermodynamic W_c.
  2. [Numerical results (assumed §4)] The protocol used to locate the second peak (direct maximum versus fit, choice of frequency cutoff) and the quantitative comparison to independently known W_c values must be specified; without this the identification cannot be assessed independently of the data used to define the peak.
minor comments (1)
  1. [Abstract] Abstract: the phrase 'high numerical accuracy' is used without a quantitative measure (e.g., relative error or number of decades in L); a brief parenthetical statement would improve clarity.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments. We address the two major points below and will revise the manuscript accordingly to include the requested details on finite-size analysis and numerical protocols.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the central claim that the second peak coincides with the Anderson critical disorder 'to high numerical accuracy' is load-bearing but rests on an unshown finite-size extrapolation. No system sizes L, number of disorder realizations, error bars on peak locations, or extrapolation form (e.g., |W_peak(L) - W_c| scaling) are supplied, leaving open whether finite-size shifts or the frequency cutoff u move the apparent maximum away from the thermodynamic W_c.

    Authors: We agree that the supporting finite-size data and extrapolation were not presented with sufficient detail. In the revised manuscript we will add the system sizes L employed, the number of disorder realizations per size, error bars on the extracted peak locations, and the explicit extrapolation form (including any scaling of |W_peak(L) - W_c|) that demonstrates convergence to the accepted thermodynamic W_c. This will allow the reader to assess the robustness of the identification independently of the frequency cutoff. revision: yes

  2. Referee: [Numerical results (assumed §4)] The protocol used to locate the second peak (direct maximum versus fit, choice of frequency cutoff) and the quantitative comparison to independently known W_c values must be specified; without this the identification cannot be assessed independently of the data used to define the peak.

    Authors: We concur that the precise numerical protocol must be stated explicitly. The revised manuscript will describe whether the peak position is obtained from a direct maximum search or from a fit, the concrete value(s) of the frequency cutoff u employed, and a direct quantitative comparison of the extrapolated peak location against independently established literature values of W_c. These additions will make the identification reproducible and verifiable. revision: yes

Circularity Check

0 steps flagged

No circularity: numerical peaks identified against external Anderson critical point

full rationale

The paper reports direct numerical computation of fidelity susceptibility in the 3D Anderson model, finding two peaks as a function of disorder W. The second peak is stated to coincide with the known Anderson localization critical disorder to high numerical accuracy. No equations, fitted parameters, or self-citations are shown that reduce this identification to a quantity defined by the same data or by prior work of the authors; the result rests on independent numerical evaluation against the externally established W_c. The derivation chain is therefore self-contained and does not exhibit any of the enumerated circularity patterns.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on standard single-particle quantum mechanics and numerical diagonalization of the Anderson Hamiltonian; no free parameters, ad-hoc axioms, or new entities are introduced in the abstract.

axioms (2)
  • domain assumption Fidelity susceptibility is a well-defined linear-response quantity that quantifies eigenstate sensitivity to perturbations
    Standard definition invoked when the paper states it 'quantifies the sensitivity of single-particle eigenstates to perturbations'
  • domain assumption The Anderson localization transition occurs at a known critical disorder strength W_c in 3D
    The paper positions the second peak 'at the critical disorder strength associated with the Anderson localization transition'

pith-pipeline@v0.9.1-grok · 5718 in / 1506 out tokens · 31772 ms · 2026-06-29T19:46:40.063205+00:00 · methodology

0 comments
read the original abstract

We investigate the fidelity susceptibility, which quantifies the sensitivity of single-particle eigenstates to perturbations, in the three-dimensional Anderson model. As a function of disorder strength $W$, it exhibits two distinct peaks. The first peak signals a crossover at weak disorder strength from plane-wave states to single-particle quantum chaos, and its position shifts toward $W\to 0$ in the thermodynamic limit. The second peak emerges, to high numerical accuracy, at the critical disorder strength associated with the Anderson localization transition. We further show that the divergence of the first peak is maximal, scaling as the square of the inverse frequency cutoff, whereas the divergence of the second peak is submaximal. We relate the latter suppression to the fractal structure of single-particle eigenstates at criticality. We discuss two distinct scenarios that give rise to the peaks in the fidelity susceptibilities. Moreover, studying the scaling of typical fidelity susceptibilities above the Anderson transition, we find evidence of two distinct regimes of nonergodic behavior.

Figures

Figures reproduced from arXiv: 2605.25594 by Anatoli Polkovnikov, Lev Vidmar, Patrycja {\L}yd\.zba, Piotr Tokarczyk.

Figure 1
Figure 1. Figure 1: FIG. 1. (a) [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. The weak-disorder crossover. (a) The finite-size scal [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. The spectral functions for the 3D Anderson model [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. The Anderson localization transition. Results in (a,b) [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. The weak-disorder crossover: (a) the peak position, [PITH_FULL_IMAGE:figures/full_fig_p008_6.png] view at source ↗
Figure 5
Figure 5. Figure 5: (b) marks W∗ 1 ≈ 3.8 √µ, obtained from the second￾order polynomial fits to the numerical results. Note that the variation of χ˜ r typ with µ can be accessed experimen￾tally by probing the nonequilibrium dynamics of a sys￾tem at time t = µ −1 , which is far shorter than the longest quantum time scale τH = ω −1 typ. The signatures of a two-parameter scaling of fidelity susceptibility near the Anderson locali… view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. The weak-disorder crossover: (a) [PITH_FULL_IMAGE:figures/full_fig_p009_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: (a), χ˜ r av and χ˜ r typ are plotted against W using grey and red color scales, respectively. We consider one sys￾tem size, V = 383 , along with different frequency cut￾offs, 10−4 < µ < 1, which are independent of disorder strength, W. Fixing the frequency cutoff, µ, and vary￾ing the system size, V , yields qualitatively similar results (not shown). It is clear from [PITH_FULL_IMAGE:figures/full_fig_p009… view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9 [PITH_FULL_IMAGE:figures/full_fig_p011_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10. (a) [PITH_FULL_IMAGE:figures/full_fig_p011_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: FIG. 11. The spectral functions from the phenomenologi [PITH_FULL_IMAGE:figures/full_fig_p013_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: FIG. 12. A cross-section of a cubic lattice with [PITH_FULL_IMAGE:figures/full_fig_p014_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: FIG. 13. The average ratio [PITH_FULL_IMAGE:figures/full_fig_p015_13.png] view at source ↗
Figure 15
Figure 15. Figure 15: FIG. 15. Spectral function [PITH_FULL_IMAGE:figures/full_fig_p016_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: FIG. 16. Distribution of fidelity susceptibilities, [PITH_FULL_IMAGE:figures/full_fig_p017_16.png] view at source ↗
Figure 17
Figure 17. Figure 17: FIG. 17 [PITH_FULL_IMAGE:figures/full_fig_p018_17.png] view at source ↗

discussion (0)

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Reference graph

Works this paper leans on

96 extracted references · 4 canonical work pages · 1 internal anchor

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    fidelity suscep- tibilities

    In this section, we focus ex- clusively on the rescaled quantities and therefore refer to ˜χtyp and˜χr typ/av from Eq. (19) simply as “fidelity suscep- tibilities”. In contrast, we invoke the unscaled quantities to study the localized phase in Sec. IV. We parametrize ˜χr typ/av ∝µ −a, wherea= 1−¯ain Eq. (14). Weplottheunregularizedtypicalfidelitysusceptib...

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    All results were established from second-order polynomial fits to numerical results from Fig

    (b)˜χr typ evaluated atW ∗ 1 and plotted againstµ−1. All results were established from second-order polynomial fits to numerical results from Fig. 1(b). Lines in (a) and (b) are least-squares fits ofcxandbx a toV≥18 3, respectively. We obtainc≈41anda≈0.98. from the discussion of the weak-disorder crossover. As expected, the peak of˜χ r typ for ˆTs, which ...

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    (b)˜χ typ evaluated at W ∗ 2 and plotted versusω −1 typ. (d)˜χr typ evaluated atW ∗ 2 and plotted versusµ −1. Lines in panels (a,c) and (d) are least- squares fits ofdx+candbx a, respectively. These fits are performed forV≥18 3 [with the exception ofˆnin panels (a,b), whereV≥30 3 is considered]. For ˆTs, ˆT ,ˆn, we obtain c= 16.42,16.45,16.48in (a),c= 16....

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    Conversely, in the lo- calized regime, the results indicate that the ratio remains different from unity even for large frequency cutoffs, i.e., µ≫ω typ. This observation suggests that the lack of self averaging of the spectral functions (and so the fi- delity susceptibilities) is not only related to the lowest frequency scales associated withωtyp. The dif...

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    Perturbation theory To quantify the effects of resonances on the typical fi- delity susceptibility, we compare numerical results with perturbation theory, which is expected to be valid at 11 20 33 55 90 W 0 5 10log[χr typ] (a) ˆTs, µ = 0 L = 20 L = 24 L = 28 L = 32 L = 36 20 33 55 90 W −2 −1 0 1 log[χr typ] (b) ˆTs, µ = 10 −1 −5 −4 −3 −2 −1 log10 µ 2 4 6 ...

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    Finite-size scaling ofW ∗ 3 Finally, we investigate whether the described crossover atW ∗ 3 shifts towardW ∗ 2, saturates at a value greater thanW ∗ 2, or diverges toward infinite disorder strength in the thermodynamic limit. For this reason, we determine W ∗ 3 from the crossing points of˜χr typ (again rescaled) for neighboringµ, which are equally spaced ...

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