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In the fractal phase of generalized Rosenzweig–Porter models, level counting statistics around the Thouless energy collapse to one universal scaling function shared by all the variants studied.

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2026-07-13 02:57 UTC pith:MOD7PG3Y

load-bearing objection Solid extension of the RP full-counting-statistics calculation to free and Wigner classes, with clean numerics and a useful QREM contrast.

arxiv 2607.09444 v1 pith:MOD7PG3Y submitted 2026-07-10 cond-mat.dis-nn cond-mat.stat-mech

Level statistics in the fractal phase of generalized Rosenzweig--Porter models

Victor Delapalme , Leticia F. Cugliandolo , Alexander K. Hartmann , Marco Tarzia , Davide Venturelli This is my paper
classification cond-mat.dis-nn cond-mat.stat-mech
keywords Rosenzweig-Porter modelfractal eigenstateslevel compressibilityThouless energyfull counting statisticsfree probabilityreplica methodquantum random energy model
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The Rosenzweig–Porter ensemble is a simple random-matrix toy for the crossover from quantum chaos to localization. Between the fully extended and fully localized regimes sits a fractal phase whose eigenstates occupy only a vanishing fraction of Hilbert space. This paper generalizes the off-diagonal piece of the model to two broad classes (rotationally invariant matrices and Wigner matrices) and computes the full counting statistics of eigenvalues inside an energy window of width set by the Thouless energy. Using free probability and the replica method it shows that the cumulant generating function collapses onto a single, closed-form scaling function that is insensitive to the microscopic distributions of both matrices. Exact diagonalization of large samples and large-deviation sampling down to probabilities of order 10^{-40} confirm the analytic form. When the same observable is measured in the quantum random energy model, a scaling collapse still appears but the shape is different; the authors attribute the discrepancy to multifractal rather than simply fractal eigenstates. The result therefore both sharpens the random-matrix description of the intermediate phase and marks its limitations for more realistic many-body systems.

Core claim

In the fractal phase 1<γ<2 the cumulant generating function of the number of eigenvalues lying in an interval of width of order the Thouless energy ET takes the universal scaling form F(s)=pa(0)ET G(s,E/ET)+O(σ^{2}), where the function G is independent of the distributions of the diagonal matrix A and of the off-diagonal matrix M (provided M is either orthogonally invariant or a finite-variance Wigner matrix).

What carries the argument

The replica path-integral representation of the cumulant generating function, reduced by free-probability (HCIZ) techniques or by direct averaging over independent entries, to a saddle-point action that at leading order in the Thouless scale depends only on the second free (or ordinary) cumulant of M.

Load-bearing premise

The calculation keeps only the second cumulant of the off-diagonal matrix and discards higher-order terms; this is controlled only when the energy window is of order the Thouless energy and the matrix entries have finite variance.

What would settle it

Compute the level compressibility of a generalized RP matrix whose off-diagonal entries have infinite variance (or of a model with strongly correlated diagonal entries) at the Thouless scale and check whether it still collapses onto the same analytic curve χT(y).

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Editorial analysis

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Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

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Summary. The paper studies the full counting statistics of eigenvalues in generalized Rosenzweig–Porter ensembles H = A + √σ M, where A is diagonal with i.i.d. entries and M is either orthogonally invariant (mutually free with A) or a finite-variance Wigner matrix. In the fractal phase 1 < γ < 2, a replica path-integral calculation combined with free-probability tools shows that the cumulant generating function of the number of eigenvalues in an interval of width E ∼ E_T takes the universal scaling form F[−E,E](s) = p_a(0) E_T G(s, E/E_T) + O(σ²), with G given explicitly by Eq. (78). The associated level compressibility χ_T(y) (Eq. 85) is likewise universal. The analytic predictions are checked by exact diagonalization (N up to 3·10^4) across several ensembles and by large-deviation sampling of the rate function down to probabilities ∼10^{-40}. A numerical contrast with the quantum random energy model shows a different, broader crossover, attributed to multifractality of the QREM eigenstates.

Significance. The work supplies a controlled analytic derivation of a previously conjectured universal scaling function for the level compressibility and full counting statistics of RP-type models. The derivation unifies the GOE, Wishart and finite-variance Wigner cases under a single saddle-point action that retains only the second free (or ordinary) cumulant of M, and recovers earlier results as special cases. High-quality numerical support—including large-deviation sampling to 10^{-40}—and a clear negative result for the QREM strengthen the claim that the universality class is tied to simple fractal (rather than multifractal) eigenstates. The result is a useful benchmark for random-matrix models of the integrability-to-chaos crossover.

minor comments (4)
  1. In §5.1 the finite-size drift of Φ(0) is left open (logarithmic growth versus possible saturation). A short remark clarifying that the analytic prediction Φ(k)∼2/k as k→0 is expected only for N→∞, and that the observed slow drift is consistent with the independent-level benchmark of Appendix B, would help the reader.
  2. Eq. (48) defines E_T with the second cumulant of M; it would be useful to state explicitly in the caption of Fig. 3 how this prefactor is fixed for each ensemble (especially the sparse-Gaussian and tent cases) so that the collapse is fully reproducible.
  3. The discussion of Lévy matrices in §5.4 notes that they empirically obey the same χ_T even though the finite-variance assumption is violated. A one-sentence pointer to the density–density correlator of Ref. [59] would make the connection sharper.
  4. Typographical: the arXiv date stamp “13 July 2026” and a few missing spaces around inline math (e.g. “1<γ<2”) should be cleaned before final production.

Circularity Check

0 steps flagged

No significant circularity: the universal CGF scaling is obtained from a controlled saddle-point evaluation of the replica action, recovering prior special cases rather than assuming them.

full rationale

The central claim is the universal form of the cumulant generating function F[−E,E](s)=pa(0)ET G(s,E/ET)+O(σ²) with G given by Eq. (78) for both orthogonally invariant and finite-variance Wigner matrices in the fractal phase. Sections 3.1–3.2 construct the replica path-integral actions for the two classes and show they coincide at O(σ) once only the second free (or ordinary) cumulant of M is retained; the subsequent saddle-point evaluation in §4.1 then produces a scaling function independent of pa and of all higher cumulants of M. The O(σ²) truncation is self-consistently justified for E∼ET≪1. Prior results of overlapping authors (the GOE case of Ref. [1] and the Wishart case of Ref. [66]) are recovered as special cases of the same calculation rather than used as load-bearing premises. The Thouless energy is the standard definition already present in the RP literature; it is not fitted to the data being predicted. Numerical validation (exact diagonalization at N=3·10⁴ and large-deviation sampling to 10^{-40}) is independent of the analytic derivation. The QREM contrast is presented as a negative result outside the claimed class. The only mild self-reference is the recovery of earlier special cases, which does not force the general result. Hence the derivation is self-contained against its own inputs.

Axiom & Free-Parameter Ledger

3 free parameters · 4 axioms · 0 invented entities

The central claim rests on standard free-probability and replica technology plus the modeling assumption that M has finite second cumulant and is either free or entry-wise independent of A. No new particles or forces are postulated; the only free parameters are the conventional RP control parameters γ and ν that define the phase diagram.

free parameters (3)
  • γ (RP exponent)
    Controls the variance of off-diagonal elements and selects the fractal phase 1<γ<2; chosen by hand to sit inside that phase (typically γ=1.5).
  • ν (overall scale of M)
    Sets the absolute magnitude of the Thouless energy; fixed to O(1) values (usually ν=1) without fitting.
  • effective fractal dimension D (QREM only)
    Extracted by requiring best collapse of χ(ω) curves for different system sizes; used only for the QREM comparison, not for the RP universality claim.
axioms (4)
  • standard math Low-rank HCIZ integral approximation for orthogonally invariant matrices (Eq. 23)
    Standard free-probability result used to average over M when rank(T)≪N.
  • domain assumption Replica-symmetric saddle point for the auxiliary matrix Q
    Assumed by continuity with the GOE calculation of Ref. [1]; not proved to be the unique extremum.
  • domain assumption Analytic continuation n±→±is/π of the replica partition function
    Standard but non-rigorous step of the replica method; justified a posteriori by numerical agreement.
  • domain assumption Finite second cumulant of the entries of M (κ₂^{(ζ)} or κ₂^{(free)})
    Required for the expansion of the action to O(σ); excludes Lévy matrices (which nevertheless empirically obey the same scaling).

pith-pipeline@v1.1.0-grok45 · 42030 in / 2542 out tokens · 41091 ms · 2026-07-13T02:57:06.457456+00:00 · methodology

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read the original abstract

The Rosenzweig--Porter (RP) random matrix ensemble has emerged as a minimal model for the integrability-to-chaos crossover in quantum many-body systems. Its phase diagram features a region with fractal eigenstates, exhibiting intermediate spectral and localization properties between the fully localized and fully delocalized regimes. In this work, we explore several generalizations of the RP model and determine their level statistics at the scale of the Thouless energy $E_T$, which characterizes the crossover. Using tools from free probability theory and the replica method, we compute the full counting statistics in the limit of large system size, and show that it takes a simple, universal scaling form around $E_T$, shared across all variations of the model. We validate our analytical predictions using exact numerical diagonalization of large samples, and large-deviation algorithms that resolve the full counting statistics down to probabilities as low as $10^{-40}$. We also contrast our predictions with measurements on the quantum random energy model, which is the simplest model displaying many-body localization.

Figures

Figures reproduced from arXiv: 2607.09444 by Alexander K. Hartmann, Davide Venturelli, Leticia F. Cugliandolo, Marco Tarzia, Victor Delapalme.

Figure 1
Figure 1. Figure 1: The phase diagram of the GRP model (see Section 2). the local chaotic or non-chaotic behavior of the system. This issue is conventionally addressed through the unfolding procedure, which normalizes the spectrum to enforce a uniform mean level spacing, or more conveniently by considering the ratios of consecutive spacings [63], defined as rn = min(δn, δn+1)/ max(δn, δn+1), which are by construction independ… view at source ↗
Figure 2
Figure 2. Figure 2: (a) The distribution of the full counting statistics IN [−E, E] for W = √ 3, ν = 1, γ = 1.5, E = 3ET , and matrix sizes N = 20, 50, 100, 200, 500, 1000 and 2000. (b) The corresponding shifted rate function Φ(k) − Φmin as a function of the counting statistics k = IN /N pa(0)ET , rescaled according to Eq. (50). The solid line shows the theoretical prediction, while the dashed line the rate function obtained … view at source ↗
Figure 3
Figure 3. Figure 3: Level compressibility χ(E) for (a) the Symmetrized Wishart–RP model, and (b) for Wigner–RP matrices. Symbols denote data from exact numerical diagonalization of random sample matrices of size N = 30000. For energies of the scales of the Thouless energy ET (48), all curves collapse on the universal prediction χ(E) = χT (E/ET ) given in Eq. (85) and drawn with a dash-dotted line. In (b), the first three sets… view at source ↗
Figure 4
Figure 4. Figure 4: Schematic out-of-equilibrium phase diagram of the QREM (see also Refs. [76–81]). The transition line separating the fully ergodic phase from the intermediate partially delocalized, non-ergodic phase (blue), and the transition line separating the intermediate phase from the localized phase (red), are obtained from the numerical solution of Eqs. (104), (105), and (106) (see Ref. [79] for details). The dashed… view at source ↗
Figure 5
Figure 5. Figure 5: Level compressibility χ(ω) for the QREM, Eq. (96), at Γ = 0.4 for (a) ε = 0.18, (b) ε = 0.24, and (c) ε = 0.28, spanning the intermediate non-ergodic delocalized phase of the model (see [PITH_FULL_IMAGE:figures/full_fig_p032_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Fractal dimensions D1 (blue) and D2 (red) of the QREM eigenstates for Γ = 0.4 and |ε| = 0.28, computed numerically from Eq. (110). The horizontal black dotted line indicates the value of the fractal exponent D that yields the best collapse of the level compressibility data for different system sizes (see [PITH_FULL_IMAGE:figures/full_fig_p033_6.png] view at source ↗

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