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REVIEW 5 major objections 8 minor 1 cited by

Planckian Diffusion: The Ghost of Anderson Localization

T0 review · 5 major / 8 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read The paper claims that a moving random medium turns Anderson localization into a universal Planckian diffusion, with $D = \alpha\hbar/m$ and $\alpha$ about 0.5 to 2.

desk verdict Plausible and potentially important claim that moving impurities convert Anderson localization into universal D ~ hbar/m diffusion, but the numerical support is under-documented and the paper overreaches in its universality language. read the letter →

arxiv 2411.18768 v1 pith:3ILWJH7A submitted 2024-11-27 quant-ph cond-mat.dis-nn

Yubo Zhang , Anton M. Graf , Alhun Aydin , Joonas Keski-Rahkonen , Eric J. Heller This is my paper · ORCID
classification quant-phcond-mat.dis-nn
keywords AndersonlocalizationPlanckiandiffusiondynamicdisordermovingimpuritiesquantumtransportlinearresistivitystrangemetalswavepacketsimulation
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

The paper sets out to show that Anderson localization, the freezing of quantum waves in a static disordered medium, is replaced by a universal Planckian diffusion, $D = \alpha\hbar/m$ with $\alpha$ of order one, as soon as the disorder starts to move. It argues that this constant is independent of how fast the impurities move over a wide range, how strongly they scatter, their mass, and the temperature, and that it does not even require thermal equilibrium. If true, this would mean the much-studied Planckian scattering time $\tau \approx \hbar/(k_B T)$ is a corollary of a deeper, more general transport constant. The evidence comes from numerical wavepacket propagation in two dimensions, a simple chamber model, and a re-analysis of an existing experiment.

What carries the argument

The load-bearing mechanism is the moving random potential: impurity motion scrambles the phase relations among multiply scattered waves before Anderson localization can lock them in, so the wavefunction diffuses instead of localizing. The key identity in the chamber model is the quantum dwell-time relation: a two-dimensional box of area $A$ has density of states $\rho = 2\pi m A/h^2$, and a single-channel leak broadens each level by one spacing, giving dwell time $\tau = m A/h$; with a random-walk step $\sqrt{A}$ this yields $D = d^2/4\tau = 2\pi\hbar/4m \approx \hbar/m$, independent of $A$. In the simulations, the diffusion coefficient is extracted from the time-derivative of the mean-square displacement of a Gaussian wavepacket propagated numerically.

What would settle it

A direct check would be to repeat the moving-impurity simulation for many independent disorder realizations and much longer times: if the ensemble-averaged slope of the mean-square displacement does not approach a stable value in the range $0.5$ to $2$ times $\hbar/m$, or if the slope drifts with system size and runtime, the claimed universal plateau is a finite-window artifact.

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Extended reading notes

Core claim

The paper's central claim is that the destruction of Anderson localization by a time-dependent random medium gives rise to a universal quantum diffusion constant, $D = \alpha\hbar/m$, with $\alpha$ of order unity, rather than a material-specific transport coefficient. This Planckian diffusion is claimed to be insensitive to impurity speed, carrier-impurity coupling strength, impurity mass, and temperature, and to hold even when no thermal equilibrium exists. The claim is supported by a two-dimensional wavepacket simulation with thousands of moving impurities, by a phenomenological morphing-chambers model based on the scaling theory of localization, and by re-interpreting an experiment on electrons on solid hydrogen. In thermal systems the same constant implies the conventional Planckian scattering rate $\tau = \alpha\hbar/(k_B T)$, but the paper argues that the diffusion constant itself is the more fundamental statement.

Load-bearing premise

The load-bearing premise is that the slope of the mean-square displacement over the finite simulation window equals the true long-time diffusion constant, even though no ensemble averaging, error bars, or convergence checks are reported.

Editorial extensions

If this is right

  • In a thermal system, the mobility-diffusion relation turns $D = \alpha\hbar/m$ into the Planckian scattering time $\tau = \alpha\hbar/(k_B T)$, so resistivity is linear in temperature down to at least 1 K.
  • The universal diffusion does not require thermal equilibrium; it holds for driven, nonthermal impurity motion as well, making it a broader statement than the Planckian speed limit.
  • Only about 10% of impurities need to move to break Anderson localization and produce diffusion at the Planckian rate.
  • The diffusion coefficient remains near $\hbar/m$ across impurity speeds from roughly sound speed upward, potential heights, impurity masses from 10 to 10000 electron masses, and temperatures from 1 K to 500 K.
  • The chamber model gives $D = 2\pi\hbar/4m \approx \hbar/m$ independent of chamber area, showing a geometric mechanism for the same constant.

Reading between the lines

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

  • If the universality holds, the Planckian scattering rate $\tau \approx \hbar/(k_B T)$ would be a symptom of the same mechanism rather than an independent bound, and could appear in non-thermal driven systems at zero temperature.
  • A direct experimental test could use ultracold atoms in a disorder potential whose pattern is made to drift or fluctuate; the measured diffusion constant should fall near $\hbar/m$ and be insensitive to the drift speed over a wide window.
  • A natural numerical follow-up is to ensemble-average over many disorder realizations and extend the time window; this would map the boundaries of the plateau and test whether it is a true long-time limit or a finite-window crossover.
  • The chamber model suggests searching for Planckian diffusion in other wave systems, such as acoustic or photonic media with moving scatterers, where the predicted diffusion constant should again be near $\hbar/m_\text{eff}$.
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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

5 major / 8 minor

Summary. The paper claims that when Anderson localization is destroyed by the motion of a disordered medium, the ensuing diffusion is universal: D = αℏ/m with α of order unity, independent of impurity speed, coupling strength, impurity mass, temperature, and even of the existence of thermal equilibrium. The argument combines (i) time-dependent Schrödinger simulations of a Gaussian wavepacket among thousands of moving Bessel-shaped impurities in two dimensions, with D read off the slopes of the mean square displacement (Figures 3–5); (ii) a phenomenological chamber model; (iii) an earlier experiment on electrons on solid hydrogen; and (iv) the authors' prior strange-metal simulations. A Drude/Einstein conversion turns the temperature-independent D into a Planckian scattering rate τ = αℏ/k_BT and linear-in-T resistivity. The paper distinguishes a semi-adiabatic regime (D ≪ ℏ/m), a broad 'ghost' Planckian regime (0.5–2ℏ/m), and a high-velocity classical regime (D ≫ 2ℏ/m).

Significance. If the claim holds, it would be a striking unification: a universal quantum diffusion constant ℏ/m emerging from broken Anderson localization, independent of coupling and temperature and not requiring thermal equilibrium, with direct consequences for the Planckian dissipation phenomenology. The design of the central computation is honest—D is measured from MSD slopes with parameters varied rather than fitted—and the paper makes a concrete, falsifiable experimental proposal (deliberately disturbing the localized states of ref. [26] and measuring D). It also flags its own limitations, including the absence of data in the adiabatic regime. However, as presented, the numerical evidence does not yet distinguish an asymptotic universal regime from a finite-window crossover, the 'Planckian window' is defined to coincide with the claimed range of α, and one of the experimental supporting figures is an uncorrected placeholder. The significance is potentially high but conditional on the statistical and convergence analysis requested below.

major comments (5)
  1. [Section II, Section III B, Figs. 3–5] The central numerical claim—that D plateaus at αℏ/m over a wide parameter range—is built on slopes of the mean square displacement taken over finite simulation windows, but the manuscript provides no ensemble averaging, no statistical error bars, no system-size scan, and no demonstration that these slopes have converged to the t→∞ limit defining D in Section II. This is not a routine omission: the text itself notes that the wavepacket is kept away from the boundary 'to make the most use of the window size,' so the reported slopes are window-averaged quantities, and a zero-mean Gaussian wavepacket spreads ballistically at early times, so its local MSD slope can cross ℏ/m during a crossover even when the asymptotic diffusivity is different. Since the claimed independence of D from impurity speed, potential height, impurity mass, and temperature is inferred from these slopes, the universality statement currently rests on unquantified finite-window data; I ask for ensemble-averaged MSDs with errors, a check that the local slope is stationary in a linear regime (e.g., a local log-log exponent consistent with 1 over a substantial time interval), and a convergence test in system size and window duration.
  2. [Abstract and Section III B] The acceptance window for the claim is the claim itself: the abstract asserts α ∈ [0.5, 2], and Section III B defines the 'Planckian regime' as D between 0.5ℏ/m and 2ℏ/m. With no error bars on the individual points in Figures 3–5, every plotted datum inside the pink box automatically counts as agreement, and the phrase 'diffusion coefficient quickly rises to the Planckian regime' carries no quantitative content beyond 'the points are inside the box.' The case for universality would be materially strengthened by reporting fitted values of α (with uncertainties) as a function of the control parameters and showing that they cluster around a single value rather than merely lying within a factor-of-two band.
  3. [Section IV] The chamber model derives D ≈ ℏ/m by assuming that a single-channel leak broadens each level by one level spacing (τ = ℏρ = mA/h) and then combining this with d = √A and D = d²/(4τ). Because the assumption fixes τ to be of order ℏρ, the final result D = h/(4m) is independent of A by construction; the model illustrates how a Planckian D could arise but provides no independent check on the simulations, since the one-level-spacing broadening is asserted rather than derived or tested. Section VI counts this model among the 'four cases' supporting universality, so I ask that it be explicitly relabeled as a consistency check and that the level-broadening assumption be justified (or at least tested) rather than posited.
  4. [Section III B and Section VI] The manuscript reports no data in the adiabatic/semi-adiabatic regime—'No exact data points are plotted in the adiabatic regime in Figure 3, because for such very slow movement we cannot accurately gauge diffusion'—yet Section VI states that the diffusion coefficient 'rises very rapidly but smoothly from zero to the Planckian rate' and uses the adiabatic limit to engage with the conjectured Planckian bound on transport (refs. [24,25]). The assertion of a smooth (or any particular) transition, and the claimed bearing on the Planckian bound, are unsupported by plotted measurements; either the low-velocity data should be provided or these statements should be restricted to what the data show.
  5. [Section V A, Figure 8] Figure 8, which presents the solid-hydrogen experiment as a confirming case, cannot currently be evaluated: its caption contains the literal placeholder text 'Lorem ipsum00', and no data plot with axes is identifiable. Since Section VI explicitly counts this experiment among the 'four cases' that 'all indicate' a universal outcome, the evidential base for the experimental claim is incomplete; the figure must be restored, and the caption should clarify whether the quoted slope D = 0.3ℏ/me is the original experiment's interpretation or the present authors' re-analysis of the inverse residual mobility.
minor comments (8)
  1. [Section VIII] The heading 'AKNOWLEDGEMENTS' should read 'ACKNOWLEDGEMENTS'.
  2. [Throughout] Typos should be corrected: 'Panckian' (first word of Section VI), 'intricaces' (Section III B), 'givn' and 'stange metals' (Figure 9 caption), 'supercedes' (Abstract), 'furthers tests' (Section VI), and 'we have find' (Section VII).
  3. [References] References [15] and [16] are the same article (Aydin et al., PNAS 121:2404853121, 2024) and should be merged or one removed.
  4. [Section VII] The sentence beginning 'Referring to recent work We have shown...' is grammatically incomplete and should be rewritten.
  5. [Section II] Please report the numerical simulation parameters (box size, grid spacing, time step, number of impurities, number of disorder realizations) so that the MSD measurements are reproducible.
  6. [Section V A] Add one sentence explaining how the diffusion coefficient is extracted from the inverse residual mobility in the Corbino geometry; as written, the connection between the measured mobility and D = 0.3ℏ/me is asserted without derivation.
  7. [Abstract] The phrase 'Planckian diffusion supercedes the Planckian speed limit' should be softened to reflect that the relation τ = αℏ/k_BT follows from D together with the Einstein relation in thermal systems; as written the implication structure is unclear.
  8. [Section VI] Clarify the direction of the inequality when connecting the adiabatic regime to the Planckian bound in refs. [24,25]: as stated, going 'below the Planckian limit' in the adiabatic regime is a statement about a nonuniversal crossover, and the relevance to the bound on transport needs a sentence of elaboration.

Circularity Check

2 steps flagged · score 4.0 of 10

Central numerics are non-circular, but the Section IV chamber model builds D≈ℏ/m in by assuming a one-level-spacing lifetime, and Section V.B leans on the authors' own prior results, yielding partial circularity.

  1. self definitional [Section IV, 'Phenomenological model' (τ = ℏρ; D = d²/4τ)]
    "A single channel “leak” allowing escape from a chaotic box gives a decay lifetime that broadens the levels by one level spacing, or τ = ℏρ, or τ = mA/h. So, if escape from one chamber to the next leads to a 2D random walk, with stepsize d = √A, and time between steps τ = mA/h, the 2D random walk diffusion constant is D = d²/4τ = 2πℏ/4m ∼ ℏ/m, independent of the confinement area A."

    The assumed escape lifetime τ = ℏρ is exactly a one-level-spacing/Planckian time expressed in the chamber geometry. Substituting ρ = 2πmA/h² and d² = A gives D = A / (4 mA/h) = h/(4m) = (π/2)ℏ/m. Thus the advertised output D ∼ ℏ/m is algebraically the same statement as the assumed τ, up to a constant; the area cancels. The section opens by saying the model 'obeys Planckian diffusion D≈ℏ/m', so the result is built in rather than derived from independent inputs. It illustrates a possible mechanism but cannot independently confirm universality.

  2. self citation load bearing [Section V.B, 'Lattice wave on quantum wave' (Figure 9 and ref. [16])]
    "In Figure 9 we cite earlier results reinforcing the conclusions drawn here, that linear resistivity at the correct Planckian slope prevails in the three strange metals investigated [16]."

    One of the four claimed confirming cases is drawn from reference [16], a prior paper by the same group (Aydin, Keski-Rahkonen, Heller). The manuscript does not re-derive or externally benchmark those strange-metal curves; it explicitly says it 'cites earlier results reinforcing the conclusions drawn here.' Thus that leg of the universality claim rests on the authors' own prior conclusions rather than on a new independent test. It is secondary rather than sole support, because the central numerical simulation is independent, but it is still load-bearing in the paper's 'three additional confirming cases' structure.

full rationale

The core simulation is not circular: D is extracted from slopes of the mean-square displacement while impurity velocity, fraction of moving impurities, potential parameters, temperature, and impurity mass are varied, and the reported values are classified against 0.5–2ℏ/m rather than fitted to that window. The Einstein-relation conversion D → τ = k_BT/(mD) is algebraic and does not by itself inject the Planckian result. The main circularity is in Section IV, where the chamber model assumes τ = ℏρ (one-level-spacing broadening) and then obtains D ∼ ℏ/m as a rearrangement of that assumption. Section V.B additionally cites the authors' own PNAS 2024 paper as a confirming case, adding a secondary self-citation load. Concerns about finite-window MSD slopes, lack of ensemble averaging, and missing convergence checks are important but belong to correctness/reliability rather than to circularity. Overall the central derivation has independent content, so the paper is only partially circular, not fully reduced to its inputs.

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

The central simulation rests on standard quantum evolution plus domain assumptions such as single-particle dynamics, classical impurity motion, and the semiclassical Einstein relation. The main ad hoc step is the chamber model's one-level-spacing broadening assumption, which encodes the Planckian result rather than deriving it. Alpha is an observed order-one number, not a parameter-free prediction. No new physical entities are introduced.

free parameters (1)
  • alpha in D = alpha hbar / m = about 1 (band 0.5 to 2)
    Not predicted with a first-principles value; the observed diffusion coefficients cluster in the hand-defined Planckian regime 0.5 to 2 hbar / m. The boundaries are chosen post hoc to bracket the data.
assumptions (4)
  • domain assumption A single noninteracting Schrodinger particle in a time-dependent classical potential captures the essential transport physics.
    The entire simulation in Section II models one electron wavepacket; no carrier-carrier interactions and no impurity back-action.
  • ad hoc to paper A single-mode leak between chaotic chambers broadens each level by one level spacing, giving tau = hbar rho = m A / h.
    This is the key assumption in Section IV; it is exactly the critical Thouless condition that produces D of order hbar / m independent of A.
  • domain assumption The Einstein relation 1 / tau = k_B T / (m D) holds for the simulated 2D carrier system.
    Used in Section III.C to convert the temperature-independent diffusion constant into a linear-in-T resistivity and to identify the Planckian scattering rate.
  • domain assumption The computed MSD slope reaches the true long-time diffusion limit within the simulation window.
    Section II asserts the long-time limit forgets initial conditions, but no convergence test or finite-size analysis is shown.

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Cite this review

Pith. "Pith review of Planckian Diffusion: The Ghost of Anderson Localization." pith.science (2026). https://pith.science/paper/3ILWJH7A

@misc{pith2026241118768,
  author       = {Pith},
  title        = {Pith review of: Planckian Diffusion: The Ghost of Anderson Localization},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/3ILWJH7A}},
  note         = {Machine review of arXiv:2411.18768}
}
abstract

We find that Anderson localization ceases to exist when a random medium begins to move, but another type of fundamental quantum effect, Planckian diffusion $D = \alpha\hbar/m$, rises to replace it, with $\alpha $ of order of unity. Planckian diffusion supercedes the Planckian speed limit $\tau= \alpha \hbar/k_B T,$ as it not only implies this relation in thermal systems but also applies more generally without requiring thermal equilibrium. Here we model a dynamic disordered system with thousands of itinerant impurities, having random initial positions and velocities. By incrementally increasing their speed from zero, we observe a transition from Anderson localization to Planckian diffusion, with $\alpha$ falling within the range of $0.5$ to $2$. Furthermore, we relate the breakdown of Anderson localization to three additional, distinctly different confirming cases that also exhibit Planckian diffusion $D\sim \hbar/m$, including one experiment on solid hydrogen. Our finding suggests that Planckian diffusion in dynamic disordered systems is as universal as Anderson localization in static disordered systems, which may shed light on quantum transport studies.

Figures

Figures reproduced from arXiv: 2411.18768 by the authors.

Figure 1
Figure 1. FIG. 1. Static impurities versus dynamical impurities. Ran [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Activation of disorder dynamics at 10 ps. Prior to 10 [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 4
Figure 4. FIG. 4. Dependence of diffusion on proportion of moving im [PITH_FULL_IMAGE:figures/full_fig_p004_4.png] view at source ↗
Figures from the paper (4 more)
Figure 6
Figure 6. Figure 6: FIG. 6. Linear Resistivity down to 1 K. Calculation based [PITH_FULL_IMAGE:figures/full_fig_p004_6.png]
Figure 8
Figure 8. Figure 8: FIG. 8. A re-drawn plot of the inverse residual mobility as a [PITH_FULL_IMAGE:figures/full_fig_p005_8.png]
Figure 7
Figure 7. Figure 7: FIG. 7. A model presented as a realization of a Thouless [PITH_FULL_IMAGE:figures/full_fig_p005_7.png]
Figure 9
Figure 9. Figure 9: FIG. 9. For parameters appropriate to three stange metals, [PITH_FULL_IMAGE:figures/full_fig_p006_9.png]

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Forward citations

Cited by 1 Pith paper

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