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arxiv: 2606.28262 · v1 · pith:7OWMYSW7new · submitted 2026-06-26 · ❄️ cond-mat.dis-nn · cond-mat.soft· cond-mat.stat-mech

Classical versus quantum Anderson localization in disordered systems

Stefano Mossa , Giancarlo Ruocco , Walter Schirmacher This is my paper

Pith reviewed 2026-06-29 01:36 UTC · model grok-4.3

classification ❄️ cond-mat.dis-nn cond-mat.softcond-mat.stat-mech
keywords Anderson localizationclassical waveselectronic Anderson modelacoustic sum rulemass disorderforce-constant disorderlocalization phase diagramthree-dimensional systems
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The pith

Classical-wave systems form a distinct constrained disorder class where the acoustic sum rule blocks direct mapping to standard electronic Anderson models.

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

The paper establishes that the commonly used mapping between classical-wave localization and the electronic Anderson model with diagonal disorder is not mathematically justified. The correct modulus-type formulation shows classical-wave systems as a distinct constrained disorder class. In this class, the acoustic sum rule correlates diagonal and off-diagonal matrix elements, preventing direct correspondence to standard electronic models. A unified eigenvalue framework determines localization phase diagrams for classical waves and electronic models, revealing that localized states in classical waves occur only near band edges.

Core claim

Classical-wave systems with mass and force-constant disorder constitute a distinct constrained disorder class in which the acoustic sum rule correlates diagonal and off-diagonal matrix elements and prevents any direct correspondence with the standard electronic disorder models. This is shown by comparing to electronic tight-binding models with diagonal and off-diagonal disorder using complementary spectral, eigenvector, and level-statistics diagnostics within a unified eigenvalue framework.

What carries the argument

The modulus-type formulation of classical waves incorporating the acoustic sum rule that correlates diagonal and off-diagonal elements.

If this is right

  • Classical-wave systems exhibit localized states only near a band edge, similar to electronic off-diagonal disorder.
  • For mass disorder, an extended localized regime appears near the upper band edge, differing from conventional potential-type results.
  • Localization topologies in classical waves differ fundamentally from both types of electronic disorder systems.
  • The approach provides new insight into conditions for Anderson localization in three-dimensional photonic and acoustic media.

Where Pith is reading between the lines

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

  • This suggests that theories for localization in photonic and acoustic media should account for the constrained disorder class separately.
  • Numerical simulations of 3D acoustic systems could directly test the predicted differences in phase diagrams.
  • The distinction may extend to other classical wave systems obeying similar sum rules.
  • Experimental observations in disordered media could confirm the band-edge localization preference.

Load-bearing premise

The premise that the modulus-type formulation rather than the conventional potential-type approach is the mathematically correct representation for classical waves with mass and force-constant disorder.

What would settle it

Numerical computation of the localization length or participation ratio in a three-dimensional system with mass disorder to check if it matches the modulus-type phase diagram or previous potential-type predictions.

Figures

Figures reproduced from arXiv: 2606.28262 by Giancarlo Ruocco, Stefano Mossa, Walter Schirmacher.

Figure 1
Figure 1. Figure 1: FIG. 1. Arithmetic (solid lines) and typical (dashed lines) densities of states for the four disorder classes introduced in Table I: [PITH_FULL_IMAGE:figures/full_fig_p009_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Localization phase diagrams for classical-wave and electronic disordered systems: a) classical waves with force-constant [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Force-constant distribution [PITH_FULL_IMAGE:figures/full_fig_p012_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Phase diagrams for classical mass disorder (MD) (panel a) ) and electronic diagonal disorder (DD) (panel b), including [PITH_FULL_IMAGE:figures/full_fig_p013_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Mass-disorder phase diagram obtained with two different eigenvector conventions. In panel (a) the observables are [PITH_FULL_IMAGE:figures/full_fig_p017_5.png] view at source ↗
read the original abstract

We investigate Anderson localization in three-dimensional disordered systems by comparing scalar classical waves with mass and force-constant disorder to electronic tight-binding models with diagonal and off-diagonal disorder. We show that the commonly employed mapping between classical-wave localization and the electronic Anderson model with diagonal disorder is not mathematically justified. Instead, the correct modulus-type formulation reveals that classical-wave systems constitute a distinct constrained disorder class, in which the acoustic sum rule correlates diagonal and off-diagonal matrix elements and prevents any direct correspondence with the standard electronic disorder models. Within a unified eigenvalue framework, we determine localization phase diagrams for all four disorder classes using complementary spectral, eigenvector, and level-statistics diagnostics. We find that classical-wave systems share a key qualitative feature with electronic off-diagonal disorder: localized states occur only near a band edge, while extended states persist in the central part of the spectrum even at strong disorder. At the same time, the acoustic sum rule produces localization topologies that differ fundamentally from both diagonal- and off-diagonal-disorder electronic systems. In particular, for mass disorder we obtain a phase diagram that differs qualitatively from previous results based on the conventional potential-type approach and reveals an extended localized regime near the upper band edge. Our results establish a unified perspective on localization in quantum and classical wave systems and provide new insight into the conditions under which Anderson localization may occur in three-dimensional photonic and acoustic media.

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 / 1 minor

Summary. The paper claims that the commonly employed mapping between classical-wave localization and the electronic Anderson model with diagonal disorder is not mathematically justified. The correct modulus-type formulation instead reveals classical-wave systems as a distinct constrained disorder class in which the acoustic sum rule correlates diagonal and off-diagonal matrix elements. Within a unified eigenvalue framework, the authors determine localization phase diagrams for classical waves (mass and force-constant disorder) and electronic models (diagonal and off-diagonal disorder) using spectral, eigenvector, and level-statistics diagnostics. They report that classical waves exhibit localized states only near band edges (sharing this feature with off-diagonal electronic disorder) but with localization topologies that differ from both electronic classes, including a qualitatively new phase diagram for mass disorder that differs from prior potential-type results.

Significance. If the central distinction and revised phase diagrams hold, the work supplies a unified perspective on Anderson localization across quantum and classical wave systems and new insight into conditions for localization in three-dimensional photonic and acoustic media. The use of complementary diagnostics across four disorder classes is a methodological strength.

major comments (2)
  1. [model definition / acoustic sum rule argument] The load-bearing premise is that the modulus-type formulation (rather than the conventional potential-type approach) is the mathematically correct representation for scalar classical waves with mass/force-constant disorder. The manuscript must supply an explicit derivation from the equations of motion (likely in the model-construction section) showing how the acoustic sum rule enforces the claimed correlations between diagonal and off-diagonal elements and why this blocks any direct correspondence to standard electronic Anderson models; without this step the distinction, the revised phase diagrams, and the contrast with both diagonal and off-diagonal electronic disorder cannot be assessed.
  2. [phase-diagram results] The claim of a qualitatively different phase diagram for mass disorder (extended localized regime near the upper band edge) is presented as differing from previous potential-type results. Direct side-by-side quantitative comparison (e.g., critical disorder strengths or mobility-edge locations) with the earlier literature must be provided, together with error bars or convergence checks on the spectral/eigenvector/level-statistics diagnostics used to locate the boundaries.
minor comments (1)
  1. [notation / model section] Clarify the precise definition of the modulus-type versus potential-type matrix elements at first introduction to ensure readers can follow the subsequent sum-rule argument without ambiguity.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the thorough review and constructive feedback. We address the two major comments below and will revise the manuscript to incorporate the requested additions while preserving the core claims.

read point-by-point responses

  1. Referee: [model definition / acoustic sum rule argument] The load-bearing premise is that the modulus-type formulation (rather than the conventional potential-type approach) is the mathematically correct representation for scalar classical waves with mass/force-constant disorder. The manuscript must supply an explicit derivation from the equations of motion (likely in the model-construction section) showing how the acoustic sum rule enforces the claimed correlations between diagonal and off-diagonal elements and why this blocks any direct correspondence to standard electronic Anderson models; without this step the distinction, the revised phase diagrams, and the contrast with both diagonal and off-diagonal electronic disorder cannot be assessed.

    Authors: We agree that an explicit derivation from the equations of motion is required to substantiate the distinction. In the revised manuscript we will insert a new subsection (in the model-construction section) that starts from the scalar wave equation of motion, introduces the modulus-type discretization, and derives the acoustic sum rule constraint on the matrix elements. This will explicitly demonstrate the enforced correlations between diagonal and off-diagonal terms and the resulting absence of a direct mapping to either diagonal or off-diagonal electronic Anderson models. revision: yes

  2. Referee: [phase-diagram results] The claim of a qualitatively different phase diagram for mass disorder (extended localized regime near the upper band edge) is presented as differing from previous potential-type results. Direct side-by-side quantitative comparison (e.g., critical disorder strengths or mobility-edge locations) with the earlier literature must be provided, together with error bars or convergence checks on the spectral/eigenvector/level-statistics diagnostics used to locate the boundaries.

    Authors: We will add a new figure and accompanying table that directly compares our mass-disorder mobility edges and critical disorder strengths with the corresponding quantities reported in the prior potential-type literature. The revised text will also report statistical error bars obtained from multiple disorder realizations and include convergence tests with respect to system size and number of disorder samples for all three diagnostics (spectral, eigenvector, and level statistics). revision: yes

Circularity Check

0 steps flagged

No circularity: derivation rests on independent matrix construction from equations of motion

full rationale

The paper's central claim—that the modulus-type formulation is required by the acoustic sum rule and blocks mapping to standard electronic Anderson models—is presented as following from the wave equations of motion rather than from any self-definition, fitted parameter, or self-citation chain. The abstract explicitly states that the conventional mapping 'is not mathematically justified' and that the modulus-type version 'reveals' the constrained class; this is framed as a direct consequence of the sum rule correlating diagonal and off-diagonal elements. No quoted step reduces a prediction to its own input by construction, and the phase-diagram results are obtained from a unified eigenvalue framework applied to the four disorder classes. The derivation chain therefore remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

No free parameters, axioms, or invented entities are mentioned or required in the abstract; the work relies on standard numerical diagnostics for localization without introducing new fitted quantities or postulated entities.

pith-pipeline@v0.9.1-grok · 5777 in / 1159 out tokens · 38069 ms · 2026-06-29T01:36:52.013354+00:00 · methodology

discussion (0)

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Reference graph

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