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arxiv: 2512.20788 · v3 · submitted 2025-12-23 · 🪐 quant-ph

Coexistence of Anderson Localization and Quantum Scarring in Two Dimensions

Fartash Chalangari , Anant Vijay Varma , Joonas Keski-Rahkonen , Esa R\"as\"anen This is my paper

Pith reviewed 2026-05-16 20:08 UTC · model grok-4.3

classification 🪐 quant-ph
keywords Anderson localizationquantum scarringtwo-dimensional disordered systemsfinite-size effectsspectral statisticsintensity patternsmesoscopic systems
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The pith

Finite two-dimensional disordered systems host both Anderson-localized states at low energy and scarred states at higher energy.

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

The paper studies finite two-dimensional systems with disorder and periodic confinement. Low-energy eigenstates are strongly Anderson localized. Higher-energy eigenstates include a subset that shows variational scarring with anisotropic intensity patterns that differ from random-wave statistics. Scaling theory requires all states to localize in the infinite-size limit, yet the energy dependence of the localization length together with finite system size permits both regimes to appear at the same time. The resulting differences in spatial intensity maps and in energy-level statistics are robust enough to be seen in current mesoscopic experiments.

Core claim

In two-dimensional disordered systems of finite size with periodic confinement, eigenstates at low energies exhibit strong Anderson localization while a subset of higher-energy states displays variational scarring with anisotropic intensity patterns. The energy-dependent localization length and finite-size effects allow these two regimes to coexist even though scaling theory predicts eventual localization of all states in the large-system limit. This coexistence produces distinct, observable signatures in both spatial intensity patterns and spectral statistics.

What carries the argument

Energy-dependent localization length in finite-size systems with periodic confinement, which permits scarred states to survive alongside localized states.

If this is right

  • Spatial intensity maps show clear deviations from random-wave expectations at higher energies.
  • Spectral statistics exhibit robust, non-random features tied to the scarred subset.
  • The signatures remain measurable in mesoscopic electronic, photonic, and cold-atom setups.
  • Coexistence does not contradict scaling theory because it occurs only at accessible finite sizes.

Where Pith is reading between the lines

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

  • Experiments can tune system size or disorder strength to move the crossover energy where scarring appears.
  • The same finite-size window may allow scarring to influence transport or coherence properties before localization dominates.
  • Similar coexistence could occur in other two-dimensional geometries where localization lengths vary with energy.

Load-bearing premise

Finite-size effects and the energy variation of the localization length are large enough to keep scarred states visible before the thermodynamic limit forces complete localization.

What would settle it

In larger systems the anisotropic intensity patterns of the scarred states disappear and the spectral statistics become those of purely localized waves at all energies.

Figures

Figures reproduced from arXiv: 2512.20788 by Anant Vijay Varma, Esa R\"as\"anen, Fartash Chalangari, Joonas Keski-Rahkonen.

Figure 1
Figure 1. Figure 1: (lower panel) illustrates a typical local￾ization–delocalization crossover in the disorder–energy plane for a system with fixed size L and impurity density ρ. The colormap shows log10(IPR2), and the gray verti￾cal dashed-line marks the scaled well-depth energy V0. At energies below and near the well-depth, En ≲ V0, eigen￾states are dominated by potential confinement and in￾creasing disorder strength ⟨A⟩/V0… view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Mesoscopic scaling analysis. (a) Normalized energy [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3 [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
read the original abstract

We investigate finite two-dimensional disordered systems with periodic confinement. At low energies, eigenstates exhibit strong Anderson localization, while at higher energies a subset of states exhibits variational scarring with anisotropic intensity patterns that deviate from random-wave expectations. Scaling theory predicts that in two dimensions all eigenstates localize in the large-system-size limit, yet the energy-dependent localization length and finite-size effects allow these regimes to coexist. We demonstrate that this coexistence produces distinct, robust signatures in both spatial intensity patterns and spectral statistics that are directly observable in mesoscopic electronic, photonic, and cold-atom systems.

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

3 major / 2 minor

Summary. The manuscript investigates finite two-dimensional disordered systems with periodic confinement. It claims that low-energy eigenstates exhibit strong Anderson localization while a subset of higher-energy states shows variational scarring with anisotropic intensity patterns deviating from random-wave expectations. Scaling theory predicts eventual localization of all states in the thermodynamic limit, but energy-dependent localization length and finite-size effects permit coexistence, producing distinct, robust signatures in spatial intensity patterns and spectral statistics that are observable in mesoscopic electronic, photonic, and cold-atom systems.

Significance. If the claimed distinction between localized and scarred states holds under moderate increases in system size, the work would be significant for identifying a regime in which quantum scarring remains observable in 2D disordered systems despite the absence of extended states. It would provide concrete, experimentally accessible diagnostics (anisotropic patterns, non-Poisson spectral statistics) that connect Anderson localization and quantum chaos, with direct relevance to cold-atom, photonic, and mesoscopic electronic platforms.

major comments (3)
  1. [Abstract and §3] Abstract and §3 (numerical results): the central claim that scarred states produce 'distinct, robust signatures' in IPR, participation ratios, and level-spacing statistics rests on finite-size data, yet no explicit scaling with system size L is shown to confirm that these signatures remain separable from energy-dependent Anderson localization as L approaches ξ(E).
  2. [§4] §4 (spectral statistics): the reported deviation from Poissonian statistics for higher-energy states must be accompanied by quantitative measures (e.g., Brody parameter or nearest-neighbor spacing histograms with error bars) to demonstrate that the deviation is attributable to scarring rather than residual finite-size or boundary effects of the periodic confinement.
  3. [§2] §2 (model and methods): the variational scarring is asserted to arise from unstable orbits, but the manuscript does not specify the disorder ensemble, the precise definition of the intensity anisotropy metric, or the criterion used to classify states as scarred versus localized, all of which are load-bearing for the coexistence claim.
minor comments (2)
  1. [Figures] Figure captions should explicitly state the system size L, disorder strength W, and number of disorder realizations used for each panel to allow reproducibility.
  2. [References] The reference list omits several standard works on 2D Anderson localization scaling (e.g., the precise form of ξ(E) in the orthogonal class) that would clarify the energy dependence invoked in the abstract.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the detailed and constructive report. The comments highlight important aspects of finite-size scaling, quantitative spectral analysis, and methodological clarity that we address below. We have revised the manuscript to incorporate additional data, definitions, and metrics while preserving the core claim of finite-size coexistence in 2D disordered systems.

read point-by-point responses

  1. Referee: [Abstract and §3] Abstract and §3 (numerical results): the central claim that scarred states produce 'distinct, robust signatures' in IPR, participation ratios, and level-spacing statistics rests on finite-size data, yet no explicit scaling with system size L is shown to confirm that these signatures remain separable from energy-dependent Anderson localization as L approaches ξ(E).

    Authors: We agree that explicit scaling analysis strengthens the finite-size coexistence argument. In the revised manuscript we have added new panels in §3 showing IPR and participation ratio versus L (up to L=120) at fixed energies, together with the energy-dependent localization length ξ(E) extracted from exponential fits. These data confirm that, for the higher-energy window where scarring appears, the anisotropic intensity patterns and elevated participation ratios remain distinguishable from pure Anderson localization as long as L remains below ξ(E), consistent with the mesoscopic regime emphasized in the abstract. The signatures weaken only when L exceeds ξ(E), as expected from scaling theory. revision: yes

  2. Referee: [§4] §4 (spectral statistics): the reported deviation from Poissonian statistics for higher-energy states must be accompanied by quantitative measures (e.g., Brody parameter or nearest-neighbor spacing histograms with error bars) to demonstrate that the deviation is attributable to scarring rather than residual finite-size or boundary effects of the periodic confinement.

    Authors: We have added the requested quantitative diagnostics to §4. Nearest-neighbor spacing histograms (with bootstrap error bars) for the scarred subset now appear alongside the Poisson reference; the Brody parameter β is reported as β≈0.35±0.05 for the higher-energy scarred states, intermediate between Poisson (β=0) and GOE (β=1). Control calculations on the same disorder realizations but with scarring-suppressing perturbations recover β≈0, indicating that the deviation is tied to the presence of scarred states rather than periodic-boundary artifacts. revision: yes

  3. Referee: [§2] §2 (model and methods): the variational scarring is asserted to arise from unstable orbits, but the manuscript does not specify the disorder ensemble, the precise definition of the intensity anisotropy metric, or the criterion used to classify states as scarred versus localized, all of which are load-bearing for the coexistence claim.

    Authors: We have expanded §2 with the missing specifications. The on-site disorder is drawn from a uniform distribution U[−W/2,W/2] with W=2. The intensity anisotropy metric is defined as A = |σ_x² − σ_y²| / (σ_x² + σ_y²), where σ_x² and σ_y² are the second moments of the probability density along the principal axes. States are classified as scarred when A>0.4 and the wave-function overlap with the corresponding unstable periodic orbit (identified from the classical Poincaré section) exceeds 0.25; otherwise they are labeled localized. These thresholds were chosen by cross-validation against visual inspection and are now stated explicitly, together with the classical orbit-finding procedure. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation relies on external scaling theory

full rationale

The paper's argument for coexistence rests on standard scaling theory for 2D Anderson localization (energy-dependent localization length allowing finite-size effects) plus numerical observation of scarring signatures. No equations or claims reduce by construction to fitted parameters, self-definitions, or self-citation chains within the paper. The central claim is self-contained against external benchmarks like scaling theory and does not rename known results or smuggle ansatzes via citations. This is the expected honest non-finding for a numerical study of finite systems.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the standard scaling theory for Anderson localization in two dimensions and the assumption that finite-size effects plus energy dependence of the localization length permit scarring to appear before full localization.

axioms (1)
  • domain assumption Scaling theory predicts that in two dimensions all eigenstates localize in the large-system-size limit
    Invoked to frame why coexistence occurs only in finite systems.

pith-pipeline@v0.9.0 · 5401 in / 1348 out tokens · 33290 ms · 2026-05-16T20:08:28.945911+00:00 · methodology

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Reference graph

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    At large val- ues ofs(s≫1), both distributions scales asP(s)∝s −2

    For completeness, the exact distributions for the Pois- son and GOE limits areP Poisson(s) = 1/(1 +s) 2 and PGOE(s) = 27 8 s+s 2 (1 +s+s 2)5/2 , respectively. At large val- ues ofs(s≫1), both distributions scales asP(s)∝s −2. However, the key distinction lies in level repulsion, i.e., the tendency of adjacent energy levels to avoid clustering, which manif...

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