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Emergent quantum chaos from correlations on a random graph

T0 review · 2 major / 4 minor · reviewed 2026-07-14 · grok-4.5

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Pith's one-line read Sparse long-range random bonds alone on a 1D ring produce quantum-chaotic spectra and a localization transition without on-site disorder.

desk verdict Sparse Bernoulli long-range bonds alone produce a clean GOE–Poisson crossover on a 1D ring; the numerics are solid, the Gaussian theory is honest, and the precise σ_c plus “new class” claim still need FSS. read the letter →

arxiv 2607.11662 v1 pith:SA6OFG7C submitted 2026-07-13 cond-mat.dis-nn cond-mat.stat-mech

Mrinal Sarkar , Valerio Pagni , Tilman Enss , Nicolò Defenu This is my paper
classification cond-mat.dis-nncond-mat.stat-mech
keywords quantumchaosAndersonlocalizationlong-rangehoppingBernoullidisorderGOElevelstatisticsrandomgraphsnonlinearsigmamodel
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 shows that a one-dimensional ring whose only randomness is the presence or absence of long-range bonds, each carrying the same unit hopping, is enough to generate both ergodic quantum-chaotic spectra and Anderson-like localization. Bond probabilities fall as a power of distance set by a single parameter σ. At small σ the spectrum follows GOE level statistics and eigenstates are extended; at larger σ the statistics become Poisson and the states localize. The transition sits near σc ≈ 0.8–0.85, well above the threshold suggested by the average hopping and below the prediction of a Gaussian truncation of the bond disorder. The authors argue that the sparse, non-Gaussian character of the bonds is essential and places the model outside the usual power-law random-matrix and short-range Anderson classes. The result matters because it demonstrates that purely geometric disorder can drive quantum chaos and localization without any on-site potential or many-body interactions.

What carries the argument

The 1DLR3 graph: a ring in which every pair of sites is joined independently with probability p_ij = d_ij^(-(1+σ)) and every present bond carries unit hopping. The resulting sparse Bernoulli adjacency matrix is the sole source of disorder; its mean, variance, and higher cumulants all decay with the same power, and a supersymmetric nonlinear sigma model truncated at the second cumulant is used to locate the Gaussian prediction σc = 1.

What would settle it

A finite-size scaling collapse of the gap-ratio or fractal-dimension data that places the thermodynamic critical point at or above σ = 1, or an explicit non-Gaussian field theory whose higher-order vertices restore the threshold exactly to σ = 1.

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Extended reading notes

Core claim

Sparse Bernoulli long-range bonds of unit strength on a one-dimensional ring, with no on-site disorder, generate GOE spectral statistics and ergodic eigenstates for small σ and Poisson statistics with localized eigenstates for large σ. The transition occurs in the window 0.80 ≲ σc ≲ 0.85, far above the summability threshold σ = 0 of the mean hopping and below the Gaussian-field-theory value σ = 1, indicating that higher cumulants of the bond distribution are infrared-relevant and define a distinct universality class.

Load-bearing premise

That the numerical shift of the transition below the Gaussian prediction of σ = 1 is caused by infrared-relevant higher cumulants of the Bernoulli bonds, rather than by finite-size effects or incomplete scaling.

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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

2 major / 4 minor

Summary. The manuscript studies a one-dimensional ring (1DLR3) whose only randomness is geometric: each pair of sites is connected independently with probability p_ij = d_ij^{-(1+\sigma)} and every occupied bond carries unit hopping; on-site disorder is absent. Exact diagonalization (shift-invert Lanczos, L up to 2^{20}, 100 mid-spectrum states, 100 realizations) shows GOE level statistics (P(s), ⟨r⟩, std(s)) and fractal dimensions D_q o 1 at small σ, and Poisson statistics with D_q o 0 at large σ. The crossover of ⟨r⟩(σ) is reported in the window 0.80 ≲ σ_c ≲ 0.85. A Gaussian truncation of the Bernoulli disorder is used to construct a nonlocal nonlinear sigma model whose infrared stiffness K(q) ∼ |q|^σ places the metal-insulator threshold at σ = 1; the numerical shift below this value is attributed to infrared-relevant higher cumulants, suggesting a universality class distinct from both the power-law random banded matrix ensemble and the ordinary Anderson transition.

Significance. If the reported transition and its location survive finite-size scaling, the work establishes that purely geometric sparse long-range bonds are sufficient to generate both quantum-chaotic spectral correlations and a localization transition in a non-interacting Hamiltonian. This is a clean and conceptually interesting addition to the long-range localization literature: the disorder is inseparable from the kinetic network, the model is free of on-site potentials, and the Gaussian NLsM derivation (including the End Matter) is carefully executed and correctly predicts σ = 1 under truncation. The large-scale ED diagnostics (GOE/Poisson benchmarks, fractal dimensions) are standard and reproducible. The claim of a distinct universality class remains a suggestion rather than a demonstrated result, but the numerical observation that sparse Bernoulli geometry alone produces a clear ergodic-to-localized crossover already constitutes a solid contribution.

major comments (2)
  1. The central quantitative claim that the transition lies in 0.80 ≲ σ_c ≲ 0.85 (and therefore below the Gaussian NLsM threshold σ = 1) rests on the visual crossover of ⟨r⟩(σ) in Fig. 4 for L = 2^{15}–2^{20}. No finite-size scaling collapse, crossing-point analysis, or extrapolation of drift is presented. Because the Gaussian theory cleanly yields criticality at σ = 1 (Eqs. 9–12 and End Matter), the downward shift is load-bearing for the higher-cumulant and distinct-universality interpretations. Without a scaling analysis it remains possible that the apparent window is a finite-size artifact that drifts toward 1 (or is biased by band-center selection and sparse-graph sampling). A proper FSS study, or at least a quantitative estimate of residual drift, is required before the location and the associated universality claim can be regarded as established.
  2. The interpretation that higher Bernoulli cumulants are infrared-relevant and responsible for the shift from σ = 1 to ≈ 0.8 is stated as a suggestion (effective-theory section and Conclusions) without a non-Gaussian calculation or even a controlled estimate of the leading cumulant correction. While the observation that all cumulants share the same long-distance tail d^{-(1+σ)} is correct, the manuscript does not demonstrate that these vertices actually renormalize the stiffness kernel. The distinct-universality-class claim therefore remains an untested conjecture; either a non-Gaussian field-theory argument or an explicit statement that the claim is only heuristic should be supplied.
minor comments (4)
  1. The abstract and introduction quote the transition window as 0.80 ≲ σ_c ≲ 0.85 while the body text sometimes writes the equivalent α_c range; consistent use of one parametrization would improve readability.
  2. Fig. 1 caption and the surrounding text refer to both “Bernoulli” and “Gaussian truncation” matrices; a brief remark that the Gaussian matrices are dense (and therefore not sparse graphs) would avoid possible confusion for readers scanning the figure alone.
  3. The distance definition d_ij = sin(π|i-j|/L)/sin(π/L) is standard for rings but is introduced only by citation; a one-line reminder that it reduces to |i-j| at short distance would help non-specialists.
  4. In the End Matter the identity C(0)Π(0) = 1 is used to rewrite the stiffness; an explicit cross-reference to the corresponding equation in the main text would make the appendix self-contained.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: ED diagnostics and Gaussian NLσM threshold are independent; mismatch is not forced by construction or self-citation.

full rationale

The central claims rest on two independent pillars that do not reduce to each other. Exact diagonalization of the Bernoulli 1DLR3 Hamiltonian (shift-invert Lanczos, 100 mid-spectrum states, 100 realizations, L up to 2^20) yields standard external benchmarks—⟨r⟩ o0.5307 (GOE) or 0.3863 (Poisson), Wigner/Poisson P(s), and D_q o1 or 0—without any fitted free parameter that is later re-used as a prediction. The Gaussian truncation of the same model (retaining only mean hopping p_ij and variance C_ij∼d^{-(1+σ)}) is derived from scratch in the End Matter via Hubbard–Stratonovich + saddle-point + soft-mode expansion, producing the stiffness K(q)∼|q|^σ and the criticality condition K(q)∼|q| that places the threshold at σ_c^{(2)}=1. The paper simply records that the numerical crossover window 0.80≲σ_c≲0.85 lies below this analytic value and therefore suggests (without claiming a derivation) that higher Bernoulli cumulants are infrared-relevant. No quantity is defined in terms of the result it is said to predict, no parameter is fitted to a subset of the data and then re-presented as a first-principles forecast, and the few self-citations ([30,31]) supply only the chordal distance on the ring, not the location of the transition or the universality-class claim. The derivation chain is therefore self-contained against external RMT and Anderson-localization benchmarks; the acknowledged mismatch is an honest discrepancy, not a circular reduction.

Assumptions & free parameters 0 free parameters · 5 assumptions · 1 invented entities

The load-bearing content is a defined random-graph Hamiltonian plus standard spectral diagnostics and a truncated supersymmetric NLσM. No free parameters are fitted to force σ_c; the transition window is read from ED. Background axioms are standard RMT localization diagnostics and the Gaussian cumulant truncation. The main invented object is the 1DLR3 ensemble itself.

assumptions (5)
  • domain assumption Adjacent gap ratio ⟨r⟩ and nearest-neighbor spacing P(s) converge to GOE (Wigner surmise) for ergodic single-particle spectra and to Poisson for localized spectra in the thermodynamic limit.
    Used throughout Numerical results and Fig. 2–4 as the primary phase diagnostic; standard in Anderson/RMT literature but not re-derived here.
  • domain assumption Fractal dimensions D_q=τ(q)/(q−1) from IPR moments satisfy D_q→1 for ergodic states and D_q→0 for localized states for all q>1.
    Invoked in eigenstate statistics section and Fig. 3 to corroborate GOE/Poisson phases.
  • ad hoc to paper Retaining only mean hopping and second cumulant of Bernoulli bonds yields a valid Gaussian NLσM whose IR stiffness K(q)∼|q|^σ places the metal-insulator threshold at σ=1.
    Effective theory section and End Matter; the truncation is an explicit approximation whose failure is then used to argue higher cumulants matter.
  • domain assumption Criticality of the nonlocal NLσM occurs when the stiffness scales as K(q)∼|q|, following the PRBM/Anderson NLσM literature.
    Cited via Evers-Mirlin and Mirlin et al.; used to convert |q|^σ into σ_c=1 under Gaussian truncation.
  • domain assumption Ring chordal distance d_ij=sin(π|i−j|/L)/sin(π/L) correctly implements translation-invariant long-range geometry on a finite ring.
    Model section; standard in long-range network papers cited by the authors.
invented entities (1)
  • 1DLR3 (one-dimensional long-range random ring) ensemble
    purpose: Minimal Hamiltonian whose only quenched randomness is independent Bernoulli long-range bonds with p_ij=d_ij^{-(1+σ)} and unit hopping, used to isolate geometric disorder as the driver of chaos and localization.
    Defined in the Introduction and Model sections; not previously standard in the cited PRBM/Anderson literature. Independent evidence is limited to the paper's own ED and the suggestion of photonic-graph experiments.

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Pith. "Pith review of Emergent quantum chaos from correlations on a random graph." pith.science (2026). https://pith.science/paper/SA6OFG7C

@misc{pith2026260711662,
  author       = {Pith},
  title        = {Pith review of: Emergent quantum chaos from correlations on a random graph},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/SA6OFG7C}},
  note         = {Machine review of arXiv:2607.11662}
}
abstract

This work demonstrates that sparse long-range random bonds on a one-dimensional lattice alone can generate quantum-chaotic spectral correlations and also drive a localization transition in a noninteracting single-particle Hamiltonian. The model is a one-dimensional ring in which each pair of sites is connected independently with a probability $p_{ij}= d_{ij}^{-(1+\sigma)}$. Each bond carries identical unit hopping and on-site disorder is absent. Despite the absence of on-site disorder and interaction, the model displays quantum chaotic spectra with Gaussian orthogonal ensemble (GOE) level statistics at small $\sigma$ and localized eigenstates with Poisson statistics at larger $\sigma$. The transition occurs in the range $ 0.80 \lesssim \sigma_c \lesssim 0.85$, far above the summability threshold of the mean hopping profile ($\sigma=0$). A Gaussian field theory retaining only the mean and variance of the Bernoulli bonds instead predicts a threshold at $\sigma=1$, suggesting that higher cumulants are infrared-relevant. Our findings hint towards a universality class that is distinct from both the power-law random banded matrix model and the standard Anderson transition.

Figures

Figures reproduced from arXiv: 2607.11662 by the authors.

Figure 1
Figure 1. FIG. 1. Hopping matrices for a typical disorder realization [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Eigenvalue statistics for [PITH_FULL_IMAGE:figures/full_fig_p002_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Scaling of the fractal dimensions [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗

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Cited by 1 Pith paper

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  1. Quantum Chaos and Diffusive Transport from Geometric Randomness

    cond-mat.stat-mech 2026-07 conditional novelty 7.0 of 10

    Geometric randomness on random locally tree-like layered graphs produces quantum chaos and diffusion in the extensive-layer regime, and mixed localization with ballistic transport in the quasi-1D limit.

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