Localization of a quantum particle in a classical one-component plasma.III. Mutual coherence and coherence degradation in Coulomb-disordered media

Yury A. Budkov

arxiv: 2605.18559 · v1 · pith:SXKGVXZOnew · submitted 2026-05-18 · ❄️ cond-mat.stat-mech

Localization of a quantum particle in a classical one-component plasma.III. Mutual coherence and coherence degradation in Coulomb-disordered media

Yury A. Budkov This is my paper

Pith reviewed 2026-05-20 08:28 UTC · model grok-4.3

classification ❄️ cond-mat.stat-mech

keywords mutual coherenceCoulomb disorderelectron microscopyDebye lengthlocalization lengthelectrolyteplasmacoherence degradation

0 comments

The pith

Mutual coherence length in Coulomb-disordered media follows universal scaling with Debye length and localization length.

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

The paper derives the mutual coherence function for an electron beam traveling through static electrolytes or dynamic plasmas and shows that its decay shares the same disorder correlator as the single-particle localization length. The Efimov path-integral approach yields the relation ρ_c ∼ λ_D √(ℓ/L) for both cases, with ℓ scaling as k squared in the static regime and linearly with k in the slow-particle dynamic regime. This supplies an intrinsic mechanism for coherence reduction that is independent of ion thermal velocity. A sympathetic reader would care because the result directly bears on contrast loss in liquid-cell and cryo-electron microscopy of disordered media.

Core claim

What carries the argument

Efimov path-integral formalism, which expresses the mutual coherence function decay through the identical disorder correlator that sets the localization length ℓ.

If this is right

The ion thermal velocity drops out of the final coherence expression.
Exact analytical expressions exist for the phase structure function of a model electrolyte.
Disorder-induced decorrelation contributes measurably to high-spatial-frequency contrast loss in liquid-cell electron microscopy.
The framework extends analytically to the relativistic regime for transmission electron microscopy.

Where Pith is reading between the lines

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

Varying sample thickness in microscopy experiments would directly test the square-root scaling with localization length.
The momentum dependence difference between static and dynamic cases offers a way to distinguish disorder type through energy-dependent contrast measurements.
Similar coherence degradation may appear in other Coulomb-like disordered systems such as soft matter or biological fluids.

Load-bearing premise

The mutual coherence function and its decay can be expressed through the same disorder correlator that governs the single-particle localization length via the path-integral formalism.

What would settle it

Direct measurement of beam coherence length versus sample thickness L in a medium with known Debye length, compared against the predicted square-root dependence on the independently estimated localization length ℓ.

read the original abstract

We derive the mutual coherence function of an electron beam propagating through a static or dynamic Coulomb-disordered medium and show that its decay introduces an intrinsic coherence-reduction mechanism relevant for electron microscopy in Coulomb-disordered media. Using the Efimov path-integral formalism, the coherence length $\rho_c$ is expressed through the same disorder correlator that governs the single-particle localization length $\ell$. For both a static electrolyte and a dynamic plasma we obtain a universal relation $\rho_c \sim \lambda_D \sqrt{\ell/L}$, where $\lambda_D$ is the Debye length and $L$ the sample thickness. In the static case $\ell\propto k^{2}$ (electron momentum), whereas in the dynamic slow-particle regime $\ell\propto k$, leading to qualitatively different energy dependences of the coherence scale. The ion thermal velocity cancels out in the final expression, demonstrating a formal connection between transverse coherence decay and longitudinal localization phenomena. Exact analytical results are given for the phase structure function of a model electrolyte, and numerical estimates indicate that disorder-induced phase decorrelation may contribute appreciably to the attenuation of high-spatial-frequency contrast under experimentally relevant liquid-cell electron microscopy conditions. Possible implications for cryo-EM, disordered liquids, soft condensed matter, and biological media are discussed. In an appendix we extend the theory to the relativistic regime relevant for transmission electron microscopy.

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.

Desk Editor's Note private letter to a colleague

The paper derives a universal scaling ρ_c ∼ λ_D √(ℓ/L) for beam coherence in Coulomb media with different k dependences for static versus dynamic cases, but the direct reuse of the localization correlator for mutual coherence is the step that needs checking.

read the letter

The main thing to know is that this paper extends the localization series by deriving a universal relation for the mutual coherence length of an electron beam in a static electrolyte or dynamic plasma: ρ_c scales as λ_D times the square root of ℓ over sample thickness L. The static case gives ℓ proportional to k squared while the dynamic slow-particle regime gives linear in k, and the ion thermal velocity drops out cleanly. That contrast in energy dependence and the concrete link to microscopy applications are the new pieces here. The author uses the Efimov path-integral approach to tie the coherence decay to the same disorder correlator that sets the localization length, supplies an exact analytical form for the phase structure function in a model electrolyte, and adds numerical estimates showing the effect could matter in liquid-cell conditions. The relativistic appendix is a straightforward extension that fits the scope. Overall the work stays internally consistent and connects the dots to cryo-EM and soft-matter imaging without overclaiming. The soft spot sits in the central mapping. The derivation assumes the disorder correlator transfers straight from the single-particle localization calculation to the two-point mutual coherence function Γ(ρ) without extra path-averaging or cutoff factors that beam propagation over length L might introduce. The stress-test note flags exactly this point, and the visible abstract plus summary do not show intermediate steps or cross-checks against simulations, so it is hard to gauge how robust the identification really is. Minor gaps like missing error estimates or independent verification follow from the same place. This is aimed at researchers who care about coherence loss in electron microscopy of liquids or biological media and at people following quantum transport in disordered Coulomb systems. A reader who wants scaling arguments and practical estimates for high-spatial-frequency contrast would get something usable from it. The paper shows clear thinking and honest use of the prior formalism, so it deserves a serious referee who can examine the path-integral transfer in detail. I would send it to peer review.

Referee Report

2 major / 2 minor

Summary. The manuscript derives the mutual coherence function Γ(ρ) for an electron beam propagating through static electrolytes or dynamic plasmas using the Efimov path-integral formalism. It expresses the coherence length ρ_c through the same disorder correlator that determines the single-particle localization length ℓ, yielding the universal scaling ρ_c ∼ λ_D √(ℓ/L) for both cases. Static and dynamic regimes produce distinct momentum dependences (ℓ ∝ k² vs. ℓ ∝ k), the ion thermal velocity cancels, and analytical expressions are given for the phase structure function; implications for liquid-cell electron microscopy and cryo-EM are discussed, with an appendix extending to the relativistic regime.

Significance. If the central mapping holds, the work establishes a direct link between transverse coherence degradation and longitudinal localization in Coulomb-disordered media, providing a parameter-free relation that could quantify disorder-induced contrast loss in electron microscopy of soft matter and biological samples. The cancellation of thermal velocity and the exact analytical results for the model electrolyte are notable strengths; the approach may also inform studies of disordered liquids and relativistic beam propagation.

major comments (2)

[§3] §3 (Derivation of mutual coherence): The identification of the phase structure function D(ρ) with the identical two-point disorder correlator used for the localization length ℓ requires explicit demonstration that no additional path-averaging or forward-scattering cutoffs arise when extending the Efimov representation from the single-particle return probability to the off-diagonal coherence function Γ(ρ) for a beam of thickness L; the current derivation appears to assume direct transfer without justifying the absence of extra weighting factors.
[Eq. (universal relation)] Eq. (universal relation) and surrounding text: The claim that ρ_c ∼ λ_D √(ℓ/L) is universal across static and dynamic cases rests on the shared correlator; however, the differing k-dependences (ℓ ∝ k² static vs. ℓ ∝ k dynamic) must be shown to propagate consistently into the coherence decay without introducing regime-specific corrections to the square-root scaling.

minor comments (2)

Notation for the disorder correlator should be unified between the localization and coherence sections to avoid potential confusion between single-particle and two-point quantities.
[Appendix] The appendix on the relativistic regime would benefit from a brief comparison table of the non-relativistic and relativistic expressions for ρ_c.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and for the constructive comments. We address each major comment in turn below, providing clarifications on the Efimov formalism and indicating the revisions we will make to strengthen the presentation.

read point-by-point responses

Referee: [§3] §3 (Derivation of mutual coherence): The identification of the phase structure function D(ρ) with the identical two-point disorder correlator used for the localization length ℓ requires explicit demonstration that no additional path-averaging or forward-scattering cutoffs arise when extending the Efimov representation from the single-particle return probability to the off-diagonal coherence function Γ(ρ) for a beam of thickness L; the current derivation appears to assume direct transfer without justifying the absence of extra weighting factors.

Authors: We agree that an explicit demonstration strengthens the argument. In the Efimov path-integral representation the mutual coherence Γ(ρ) is obtained from the disorder-averaged phase factor between a pair of paths separated by transverse distance ρ. Because the phase difference is generated by the identical two-point correlator that enters the single-particle return probability, the structure function D(ρ) carries over directly; the forward-scattering cutoff and the path averaging are already encoded in the definition of the correlator and do not acquire additional weighting factors when the off-diagonal matrix element is considered. The beam thickness L appears simply as the common propagation length in both calculations. To make this equivalence fully transparent we will insert a short subsection in §3 that writes the path integrals for the diagonal and off-diagonal cases side by side and shows that the same disorder average produces D(ρ) in both instances. revision: yes
Referee: [Eq. (universal relation)] Eq. (universal relation) and surrounding text: The claim that ρ_c ∼ λ_D √(ℓ/L) is universal across static and dynamic cases rests on the shared correlator; however, the differing k-dependences (ℓ ∝ k² static vs. ℓ ∝ k dynamic) must be shown to propagate consistently into the coherence decay without introducing regime-specific corrections to the square-root scaling.

Authors: The square-root scaling follows directly from the definition of the coherence length as the transverse separation at which the phase structure function reaches order unity, D(ρ_c) ∼ 1. Because D(ρ) is constructed from the same disorder correlator whose appropriate moment determines 1/ℓ, one obtains ρ_c ∼ λ_D √(ℓ/L) irrespective of the explicit momentum dependence that ℓ itself carries. The static (ℓ ∝ k²) and dynamic (ℓ ∝ k) regimes therefore enter only through the numerical value of ℓ; they do not generate extra factors or alter the functional form of the scaling. We will add a concise derivation immediately after the universal relation that substitutes the two expressions for ℓ into D(ρ) and verifies that the square-root dependence on ℓ/L remains unchanged in both regimes. revision: yes

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The paper derives the mutual coherence function and the universal relation ρ_c ∼ λ_D √(ℓ/L) from the Efimov path-integral formalism applied to the shared disorder correlator, with explicit results for the phase structure function in the electrolyte model and cancellation of ion thermal velocity. This constitutes an independent theoretical step connecting transverse coherence decay to longitudinal localization without reducing the claimed result to a fitted parameter, self-definition, or unverified self-citation chain. The localization length ℓ enters as an input from prior formalism rather than being redefined here, and the derivation remains self-contained against external benchmarks such as the stated analytical expressions and numerical estimates for liquid-cell microscopy.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The derivation rests on the applicability of the Efimov path-integral formalism to the quantum particle in a Coulomb medium and on the existence of a shared disorder correlator for localization and coherence; no new free parameters or invented entities are introduced in the abstract.

axioms (1)

domain assumption Efimov path-integral formalism applies to the propagation of a quantum particle in a Coulomb-disordered medium
Invoked to express the mutual coherence function through the disorder correlator.

pith-pipeline@v0.9.0 · 5779 in / 1314 out tokens · 42045 ms · 2026-05-20T08:28:56.357387+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Using the Efimov path-integral formalism, the coherence length ρc is expressed through the same disorder correlator that governs the single-particle localization length ℓ. … ρc ∼ λD √(ℓ/L)
IndisputableMonolith/Foundation/AlphaCoordinateFixation.lean costAlphaLog_high_calibrated_iff unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Dϕ(ρ) ≃ 4A²L ∫ [K(u)−K(√(u²+ρ²))] du … γ(ρ)=exp(−½ Dϕ(ρ))

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Forward citations

Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

Relativistic Saturation of Coulomb-Limited Electron Coherence
cond-mat.dis-nn 2026-05 unverdicted novelty 7.0

Derives relativistic saturation of the effective coupling for Coulomb-induced phase fluctuations in electron beams, with the coherence-localization relation and algebraic decay of mutual coherence unchanged from the n...
Possible Topological Decoherence Transition in Relativistic Electron Beams Propagating through Coulomb-Disordered Media
cond-mat.dis-nn 2026-05 unverdicted novelty 6.0

Relativistic electron beams in Coulomb-disordered media may undergo a BKT topological transition at critical thickness Lc, switching from algebraic to exponential decoherence via vortex unbinding in an effective 2D ph...

Reference graph

Works this paper leans on

47 extracted references · 47 canonical work pages · cited by 2 Pith papers

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[1] [1]

, title =

de Jonge, Niels and Ross, Frances M. , title =. Nature Nanotechnology , volume =

work page

[2] [2]

Voyles , title =

Shuoyuan Huang and Carter Francis and Po-Cheng Kung and Ajay Annamareddy and Dane Morgan and Paul M. Voyles , title =. Microscopy and Microanalysis , volume =. 2025 , note =

work page 2025

[3] [3]

P. A. Egelstaff , title =

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[4] [4]

Physical review letters , volume=

Static nonlocal dielectric function of liquid water , author=. Physical review letters , volume=. 1996 , publisher=

work page 1996

[5] [5]

Rytov, S. M. and Kravtsov, Yu. A. and Tatarskii, V. I. , title =

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, title =

Bethe, Hans A. , title =. Physical Review , volume =

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Keskin, Sercan , title =

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, title =

Cowley, John M. , title =

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Egerton, R. F. , title =

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and Ross, Frances M

de Jonge, Niels and Houben, Lothar and Dunin-Borkowski, Rafal E. and Ross, Frances M. , title =. Nature Reviews Materials , volume =

work page

[11] [11]

, title =

Maxwell, Louis R. , title =. Journal of the Optical Society of America , volume =

work page

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and K\'alm\'an, E

Lengyel, S. and K\'alm\'an, E. , title =. Nature , volume =

work page

[13] [13]

Yang, Jie and others , title =. Phys. Chem. Chem. Phys. , volume =

work page

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Baryshevsky, V. G. and Feranchuk, I. D. , title =. Physics Reports , volume =

work page

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Ferwerda, H. A. and Hoenders, B. J. and Slump, C. H. , title =. Optica Acta: International Journal of Optics , volume =

work page

[16] [16]

Foundations of Crystallography , volume=

A relativistic n-beam dynamical theory for fast electron diffraction , author=. Foundations of Crystallography , volume=. 1996 , publisher=

work page 1996

[17] [17]

Yang, Jie and others , title =. J. Chem. Phys. , volume =

work page

[18] [18]

Kruh, R. F. , title =. Chemical Reviews , volume =

work page

[19] [19]

Matter , volume =

Mu, Xiaoke and others , title =. Matter , volume =

work page

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and others , title =

Gorelik, Tatiana E. and others , title =. Chemical Reviews , volume =

work page

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Berestetskii, V. B. and Lifshitz, E. M. and Pitaevskii, L. P. , title =. 1982 , isbn =

work page 1982

[22] [22]

Anderson, P. W. , title =. Physical Review , volume =

work page

[23] [23]

, title =

Budkov, Yury A. , title =. submitted to Phys. Rev. E , year =

work page

[24] [24]

Tatarskii, V. I. , title =

work page

[25] [25]

Ishimaru, Akira , title =

work page

[26] [26]

Kornyshev, A. A. and Vorotyntsev, M. A. and Spohr, E. , title =. Electrochim. Acta , volume =

work page

[27] [27]

Kornyshev, A. A. and Sutmann, G. , title =. J. Chem. Phys. , volume =

work page

[28] [28]

Physical Review E , volume=

Nonlocal dielectric response of aqueous electrolytes and decay behavior of ionic correlations , author=. Physical Review E , volume=. 2025 , publisher=

work page 2025

[29] [29]

Akkermans, Eric and Montambaux, Gilles , title =

work page

[30] [30]

Reports on Progress in Physics , volume=

Localization: theory and experiment , author=. Reports on Progress in Physics , volume=

work page

[31] [31]

Marquardt, Florian and von Delft, Jan and Smith, R. A. and Ambegaokar, Vinay , title =. Physical Review B , volume =

work page

[32] [32]

Fried, B. D. and Conte, S. D. , title =

work page

[33] [33]

2018 , publisher=

Hydrodynamic fluctuations, broken symmetry, and correlation functions , author=. 2018 , publisher=

work page 2018

[34] [34]

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work page

[35] [35]

Reviews of modern physics , volume=

Disordered electronic systems , author=. Reviews of modern physics , volume=. 1985 , publisher=

work page 1985

[36] [36]

New Journal of Physics , volume=

Many-body physics with ultracold plasmas: Quenched randomness and localization , author=. New Journal of Physics , volume=. 2019 , publisher=

work page 2019

[37] [37]

physica status solidi (b) , volume=

Quantum Coulomb Glass: Anderson localization in an interacting system , author=. physica status solidi (b) , volume=. 1998 , publisher=

work page 1998

[38] [38]

Physics Reports , volume=

Anomalous localization in low-dimensional systems with correlated disorder , author=. Physics Reports , volume=. 2012 , publisher=

work page 2012

[39] [39]

2012 , publisher=

Physical Kinetics: Volume 10 , author=. 2012 , publisher=

work page 2012

[40] [40]

2010 , publisher=

Path integrals in quantum mechanics , author=. 2010 , publisher=

work page 2010

[41] [41]

1988 , publisher =

Introduction to the Theory of Disordered Systems , author =. 1988 , publisher =

work page 1988

[42] [42]

1980 , publisher=

Statistical Physics, Part 1 , author=. 1980 , publisher=

work page 1980

[43] [43]

1988 , publisher=

Fundamentals of the Theory of Metals , author=. 1988 , publisher=

work page 1988

[44] [44]

2003 , publisher=

Basic concepts for simple and complex liquids , author=. 2003 , publisher=

work page 2003

[45] [45]

Macromolecules , volume=

A statistical theory of weakly charged polyelectrolytes: fluctuations, equation of state and microphase separation , author=. Macromolecules , volume=. 1988 , publisher=

work page 1988

[46] [46]

2024 , publisher=

Statistical Field Theory of Ion-Molecular Fluids , author=. 2024 , publisher=

work page 2024

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Theoretical and Mathematical Physics , volume=

Quantum particle in a random medium , author=. Theoretical and Mathematical Physics , volume=. 2015 , publisher=

work page 2015