🔬 condensed matter

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

This paper investigates the level statistics of generalized Rosenzweig--Porter models in their fractal phase, demonstrating through analytical methods and extensive numerical simulations that the full counting statistics at the Thouless energy scale exhibit a simple, universal form across various model variations.

Original authors: Victor Delapalme, Leticia F. Cugliandolo, Alexander K. Hartmann, Marco Tarzia, Davide Venturelli

Published 2026-07-13

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CC BY 4.0

Original authors: Victor Delapalme, Leticia F. Cugliandolo, Alexander K. Hartmann, Marco Tarzia, Davide Venturelli

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine a giant, chaotic dance floor where thousands of invisible dancers (quantum particles) are trying to find their spots. In some scenarios, they move in perfect, synchronized chaos, scrambling everywhere instantly. In others, they get stuck in a corner, frozen in place. But there's a weird, middle ground where they are neither fully stuck nor fully free—they are "fractal." They spread out, but only in a jagged, self-repeating pattern, like a coastline or a snowflake, occupying just a fraction of the dance floor.

This paper is about mapping the "music" of these dancers. Specifically, the authors are listening to the spacing between the energy levels (the notes) these dancers can hit. They want to know: if you zoom in on a specific range of notes, how do they cluster? Do they repel each other like magnets, or do they ignore each other like strangers in a crowd?

The Universal "Fractal" Rhythm

The researchers started with a famous mathematical model called the Rosenzweig–Porter (RP) model. Think of this model as a recipe for a chaotic dance floor. It mixes two ingredients: a list of random starting positions (the diagonal matrix) and a set of random rules for how dancers can jump to new spots (the off-diagonal matrix).

By tweaking a single knob in the recipe (a parameter called $\gamma$ ), you can change the dance floor from fully chaotic to fully frozen. In the middle zone ( $1 < \gamma < 2$ ), you get that fractal phase.

The big question was: Does the specific type of music playing matter?
In the standard recipe, the "jump rules" are drawn from a very specific, smooth Gaussian distribution (like a perfect bell curve). But what if you changed the music to something spiky, or something with heavy tails, or something that follows different statistical rules? Would the fractal dancers still dance to the same rhythm?

The Main Finding:
The authors say yes. Using advanced math tools (replica method and free probability) and massive computer simulations, they found that the "level statistics" (the rhythm of the notes) in this fractal phase are universal.

No matter if the jump rules are drawn from a smooth Gaussian curve, a "Wigner" matrix (where every jump is independent), or an "orthogonally invariant" matrix (where the jumps have a specific rotational symmetry), the dancers all end up following the exact same scaling law.

They discovered a "master curve" for the level compressibility. Imagine this as a measure of how squishy the crowd is.

At very small scales (tiny windows of energy), the crowd acts like a chaotic fluid (Random Matrix Theory).
At very large scales, the crowd acts like a frozen, independent gas (Poisson statistics).
In the middle, the crowd follows a specific, smooth transition curve that depends only on the size of the window relative to a special energy scale called the Thouless energy ( $E_T$ ).

The paper proves that this transition curve is the same for all the variations of the RP model they tested. It doesn't matter if the "noise" in the system is Gaussian or something else; the fractal dancers all march to the same beat.

What They Ruled Out (and What They Didn't)

The authors were very careful to test the limits of this universality.

They ruled out that the specific distribution of the "jump rules" matters. Whether the jumps are Gaussian, Rademacher (just +1 or -1), sparse, or triangular, the result is the same.
They explicitly tested a different model: The Quantum Random Energy Model (QREM). This is a more realistic model of a disordered quantum system (like a spin glass). It looks very similar to the RP model on paper.
- The Result: The universality breaks down here.
- The QREM dancers do show a transition from chaotic to frozen, but the curve is different. It's broader and slower.
- Why? The authors suggest this is because the QREM dancers are multifractal, not just fractal. While the RP dancers have a single "fractal dimension" (one way of measuring their spread), the QREM dancers have a complex, multi-layered structure where different parts of the wave function scale differently. The simple RP recipe cannot capture this extra complexity.

How Sure Are They?

The Math: For the generalized RP models (Wigner and Orthogonally Invariant), the result is analytically derived. They used rigorous mathematical proofs to show that the scaling function is universal.
The Simulations: They backed this up with exact numerical diagonalization of huge matrices (up to $N=30,000$ ). The data points from the computer simulations lined up perfectly with their mathematical curve.
The Rare Events: They also used a special "large-deviation" algorithm to look at extremely rare events (probabilities as low as $10^{-40}$ ). The data here was a bit noisier due to the limits of computer size, but it still agreed with the theory near the center.
The QREM: For the Quantum Random Energy Model, they simulated the results. They did not prove it mathematically yet; they observed it numerically. The paper suggests that the difference is due to multifractality, but it frames this as a strong hypothesis based on the data, not a final mathematical proof.

The Takeaway

If you have a system that is "fractal" in a specific, simple way (like the generalized Rosenzweig–Porter models), the way its energy levels are spaced is a universal law. It doesn't care about the microscopic details of the noise; it only cares about the scale.

However, if the system is "multifractal" (like the Quantum Random Energy Model), that universal law breaks. The dance floor becomes more complex, and the simple, elegant curve of the RP model no longer fits. The authors conclude that while the RP model is a fantastic toy for understanding chaos, it has limits, and we need new models to capture the full, messy reality of multifractal quantum systems.

Technical Summary: Level Statistics in the Fractal Phase of Generalized Rosenzweig–Porter Models

Problem Statement
The Rosenzweig–Porter (RP) model serves as a minimal framework for studying the crossover between integrability and chaos in quantum many-body systems. Its phase diagram features three distinct regimes: a fully delocalized (ergodic) phase, a fully localized phase, and an intermediate "fractal" phase where eigenstates exhibit non-ergodic, fractal properties. While the level statistics of the standard RP model (where the off-diagonal matrix is Gaussian Orthogonal Ensemble, GOE) are well understood, the universality of these statistics across broader classes of random matrices remains a subject of investigation. Specifically, it was conjectured that the scaling form of level compressibility in the crossover region (around the Thouless energy) is independent of the microscopic details of the off-diagonal matrix elements. Furthermore, it is unclear whether this universality extends to realistic interacting quantum systems, such as the Quantum Random Energy Model (QREM), which exhibits many-body localization but possesses a more complex Hilbert space structure than abstract random matrix ensembles.

Methodology
The authors generalize the RP model by defining the Hamiltonian $H = A + \sqrt{\sigma}M$ , where $A$ is a diagonal matrix with independent and identically distributed (i.i.d.) entries, and $M$ belongs to one of two broad classes:

Orthogonally Invariant Matrices: Matrices statistically invariant under orthogonal rotations (e.g., Wishart-Laguerre ensembles), where entries are generally dependent.
Wigner Matrices: Matrices with independent, identically distributed entries (not necessarily Gaussian).

To compute the full counting statistics (FCS) of the number of eigenvalues $N_{IN}[\alpha, \beta]$ in a given interval, the authors employ the replica method combined with tools from free probability theory.

For orthogonally invariant matrices, they utilize the Harish-Chandra–Itzykson–Zuber (HCIZ) integral and the concept of mutual freeness between $A$ and $M$ .
For Wigner matrices, they perform a direct cumulant expansion of the independent entries.
In both cases, the problem is mapped to a path integral over auxiliary fields, which is solved via a saddle-point approximation in the large- $N$ limit.
The analysis focuses on energy intervals of the order of the Thouless energy ( $E_T$ ), which characterizes the crossover scale between local level repulsion and global Poissonian statistics.

Key Contributions and Results

Universality of Full Counting Statistics:
The authors derive the cumulant generating function (CGF) for the number of eigenvalues in the interval $[-E, E]$ . They demonstrate that in the fractal phase ( $1 < \gamma < 2$ ), the CGF takes a universal scaling form:
$F_{[-E,E]}(s) \approx p_a(0) E_T G\left(s, \frac{E}{E_T}\right)$
where $p_a(0)$ is the density of diagonal entries at zero, $E_T$ is the Thouless energy, and $G$ is a universal scaling function. Crucially, this result holds for both orthogonally invariant and Wigner matrices, differing only by a non-universal rescaling factor related to the second free cumulant (for invariant matrices) or the variance (for Wigner matrices). This confirms the conjecture that the specific choice of the off-diagonal distribution does not affect the scaling function.
Level Compressibility:
From the CGF, the authors derive the level compressibility $\chi(E) = \kappa_2/\kappa_1$ . They show that $\chi(E)$ follows the universal scaling function $\chi_T(E/E_T)$ previously found for the standard GOE RP model:
$\chi_T(y) = \frac{1}{\pi y} \left[ 2y \arctan(y) - \ln(1+y^2) \right]$
This function interpolates between Random Matrix Theory (RMT) statistics at small scales ( $y \ll 1$ ) and Poisson statistics at large scales ( $y \gg 1$ ).
Large-Deviation Analysis:
Using exact diagonalization combined with large-deviation algorithms (biased sampling via Markov Chain Monte Carlo), the authors compute the large-deviation function (LDF) for the counting statistics. They validate the analytical predictions for the rate function near its minimum. They observe that while the data collapses well near the typical value, the behavior at the tails (specifically $k \to 0$ ) exhibits slow finite-size scaling, potentially indicating a logarithmic divergence consistent with theoretical predictions, though numerical resolution limits definitive conclusions for very large systems.
Breakdown of Universality in the QREM:
The authors test the universality hypothesis on the Quantum Random Energy Model (QREM), a simple model of many-body localization. While the QREM admits an effective RP description regarding its phase diagram, the numerical results for level compressibility reveal a quantitative discrepancy.
- The QREM level compressibility collapses onto a scaling function that is distinct from the universal RP curve $\chi_T$ .
- The crossover between RMT and Poisson regimes is significantly broader in the QREM.
- The authors attribute this breakdown to the multifractal nature of QREM eigenstates. Unlike the RP model, where eigenstates are characterized by a single fractal dimension $D_q = D$ , the QREM eigenstates exhibit a spectrum of fractal dimensions ( $D_q$ depends on $q$ ). This multifractality leads to a more complex mini-band structure in the energy spectrum that cannot be captured by the standard RP paradigm.

Significance
The paper establishes that the universal scaling form of level statistics in the fractal phase is robust across different classes of random matrix ensembles (Wigner and orthogonally invariant), provided the eigenstates are characterized by a single fractal dimension. This validates the RP model as a universal effective theory for the integrability-to-chaos crossover in systems with simple fractal eigenstates.

However, the study also delineates the limitations of this universality. By demonstrating that the QREM—a realistic interacting system—deviates from the universal RP scaling due to multifractality, the authors highlight that the standard RP model is insufficient for describing systems with more complex spectral correlations. This suggests that the universality of level compressibility is restricted to ensembles with simple fractal eigenstates, motivating future investigations into models that explicitly incorporate multifractality or correlations between diagonal matrix elements to better capture the phenomenology of realistic many-body localized systems.

The Universal "Fractal" Rhythm

What They Ruled Out (and What They Didn't)

How Sure Are They?

The Takeaway

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