---
title: Rosenzweig-Porter Random Matrix Ensemble
url: https://www.emergentmind.com/topics/rosenzweig-porter-random-matrix-ensemble
type: topic
---

# Rosenzweig-Porter Random Matrix Ensemble

The Rosenzweig–Porter (RP) Random Matrix Ensemble is a one-parameter family of large random matrices that realizes distinct ergodic, fractal (non-ergodic extended), and localized phases of eigenstates as a function of the scaling parameter $\gamma$. The Lévy Rosenzweig–Porter ensemble, a fat-tailed extension, provides a framework for analytically tractable non-ergodic extended phases in systems where off-diagonal matrix elements are $\mu$-stable Lévy random variables. The model evidences sharp ergodic–fractal and fractal–localized transitions, described by nontrivial spectral and multifractal properties, and exhibits a rich $\mu$–$\gamma$ phase diagram. The analytical study of its spectral density relies on functional Hubbard–Stratonovich transformations combined with Efetov’s supersymmetry formalism, yielding a self-consistent order-parameter-driven description of the ergodic–fractal transition [2408.15072]. 

## 1. Construction and Definition of the Lévy Rosenzweig–Porter Ensemble

The Lévy RP ensemble is defined by the Hamiltonian $H = H^{(1)} + H^{(2)}$, where
- $H^{(1)}$ is diagonal, with elements $H^{(1)}_{kk}$ drawn independently from a Gaussian distribution $P_1(H^{(1)}_{kk}) = (2\pi W^2)^{-1/2}\exp[-(H^{(1)}_{kk})^2/2W^2]$,
- $H^{(2)}$ has full off-diagonal structure, $H^{(2)}_{ij}$ ($i<j$) independently drawn from a symmetric $\mu$-stable Lévy law $L_\mu(x)$ with characteristic function $\tilde{L}_\mu(k) = \exp[-|k|^\mu]$, $0 < \mu \leq 2$.

Off-diagonal elements are rescaled as
\[
P_2^{(\mu,\gamma)}(H^{(2)}_{ij}) = N^{\gamma/\mu}L_\mu(N^{\gamma/\mu} H^{(2)}_{ij}),
\]
so that their typical scale is $N^{-\gamma/\mu}$, with $\gamma > 0$ tuning the hopping-to-diagonal-disorder ratio. For $\mu=2$, $L_2(x)$ reduces to a Gaussian and the model reduces to the classical (Gaussian) RP ensemble; for $\mu<2$, the power-law tails render the second moment of off-diagonal elements divergent.

## 2. Analytical Framework: Supersymmetry and Functional HS Transformation

The non-Gaussian, power-law nature of the off-diagonal entries invalidates the standard (quadratic) Hubbard–Stratonovich transformation typically used for Gaussian RP models. Instead, the analytic approach proceeds as follows [2408.15072]:
- The partition function in the Efetov supervector representation is
  \[
  Z(E, J) = \int \prod_i [d\phi_i] \exp\left[\frac{1}{2}\sum_{ij} \phi_i^\dagger[(E + J K)\delta_{ij} - H_{ij}]\phi_j\right],
  \]
  where $\phi_i = (S_{i1}, S_{i2}, \chi_i, \chi_i^*)^T$ combines bosonic and Grassmann components.
- Averaging over $H^{(2)}$, the cumulant generates a non-analytic $|\phi_i^\dagger \phi_j|^\mu$ inter-site action term:
  \[
  C(\phi_i^\dagger \phi_j) \equiv N\int dx\, P_2(x) [e^{-i (\phi_i^\dagger\phi_j)x} - 1] \approx -N^{1-\gamma}|\phi_i^\dagger\phi_j|^\mu.
  \]
- The required functional HS transformation linearizes the non-analytic action:
  \[
  \exp\left[\frac{1}{2N}\sum_{ij} C(\phi_i^\dagger\phi_j)\right]
  = \int Dg\, \exp\left[-\frac{N}{2}\iint d\psi\,d\psi' g(\psi)C^{-1}(\psi, \psi')g(\psi') + \sum_i g(\phi_i)\right],
  \]
  and the large-$N$ saddle point yields a functional self-consistency equation for $g(\phi)$.

## 3. Density of States and Scaling as an Order Parameter

The mean spectral density $\rho(E)$ is controlled by a nontrivial transcendental equation for $f_\mu(E)$,
\[
f_\mu(E) = \frac{N^{1-\gamma}}{B(1/2,\mu/2)}\int_0^\infty dy\, y^{\mu/2-1}
\exp\left(\frac{iE}{2}y - f_\mu(E)y^{\mu/2} + \ln\int d\epsilon\, P_1^{(W)}(\epsilon) e^{-\frac{i}{2}\epsilon y}\right),
\]
with $B$ the Euler beta function. Then,
\[
\rho(E) = \operatorname{Re} \left\{ \frac{1}{2\pi} \int_0^\infty dy\, \exp\left( \frac{iE}{2}y - f_\mu(E)y^{\mu/2} + \ln\int d\epsilon\, P_1^{(W)}(\epsilon) e^{-\frac{i}{2}\epsilon y}\right) \right\}.
\]
For Gaussian $P_1$, the logarithmic term simplifies: $\ln\int d\epsilon\, P_1^{(W)}(\epsilon) e^{-\frac{i}{2}\epsilon y} = -W^2y^2/8$.

Rescaling to $\tilde{\rho}(\epsilon) = \rho(E) N^{(1-\gamma)/\mu}$ with $\epsilon = E N^{(1-\gamma)/\mu}$ shows that all $N$ and $W$ dependence enters through the single variable $\xi = W^{-1}N^{(1-\gamma)/\mu}$. In this way, the spectral density acts as a "spectral order parameter" distinguishing the ergodic and fractal regimes:
- $\tilde{\rho}(0)\to 0$ for $\gamma<1$ (ergodic phase),
- $\tilde{\rho}(0)\to\text{const}>0$ for $\gamma>1$ (fractal phase).

The critical point $\gamma=1$ marks a nontrivial, $N$-independent line; finite-size scaling for the order parameter yields a scaling exponent $\nu=1$.

## 4. Phase Diagram and Multifractality

The Lévy-RP model displays a phase diagram governed jointly by $\mu$ and $\gamma$, with three principal regimes [2012.12841]:
- For $\gamma<1$, eigenstates are fully delocalized (ergodic): level statistics are Wigner–Dyson and spectral density displays band spreading $\sim N^{(1-\gamma)/\mu}$;
- For $1<\gamma<\mu$, eigenstates are non-ergodic extended (fractal or multifractal): they occupy $N^{D_1}$ sites, with
  \[
  D_1 = \frac{\mu-\gamma}{\mu-1}, \qquad D_\infty = 2 - \frac{2\gamma}{\mu},
  \]
  reflecting a nontrivial multifractal spectrum $D_q$;
- For $\gamma>\mu$, eigenstates are Anderson localized.

At $\mu<1$, a mobility edge appears, and at $\mu=1$, $\gamma=1$ is a tricritical point. For $\mu<2$ (Lévy tails), genuine multifractality emerges; for $\mu=2$, fractal dimensions coincide and the classical Gaussian RP scaling $D_1 = D_\infty = 2-\gamma$ is recovered.

## 5. Spectral, Dynamical, and Correlation Properties

Physical signatures of the three regimes are manifest in both spectral correlations and dynamics [2410.14437]:
- **Local density of states (LDOS) correlations** $\beta(\omega)$ exhibit universal scaling:
  \[
  \beta(\omega \ll \Gamma_0) \sim W/\Gamma_0,\qquad \beta(\omega \gg \Gamma_0) \sim (W/\Gamma_0) (\omega/\Gamma_0)^{-\mu},
  \]
  with $\Gamma_0$ the characteristic width of a local state.
- **Return probability** $R(t)$ at long times displays a stretched-exponential decay governed by the Lévy index,
  \[
  \ln R(t \gg \Gamma_0^{-1}) \sim -(\Gamma_0 t)^{\mu/2}.
  \]
  For $\mu>2$ (Gaussian RP), this becomes a simple exponential, $R(t)\sim e^{-\Gamma_0 t}$; for $1<\mu<2$, the stretched exponential reflects non-Fermi's Golden Rule (FGR) dynamics.
- Level compressibility and long-range spectral correlations at the Thouless scale are universal across Gaussian, Lévy, and Wishart RP variants.

## 6. Physical Interpretation and Applications

The Lévy RP ensemble encapsulates a minimal yet analytically tractable model for the breakdown of ergodicity via fractal or multifractal mini-bands, where non-ergodic extended states retain overlap on a sub-extensive number of sites, yielding Wigner–Dyson statistics on the mini-band scale but anomalous global features [2012.12841]. This ensemble provides a robust synthetic benchmark for non-ergodic extended behavior in many-body disordered systems, including analogs of "bad metallicity," sub-diffusive transport, and multifractal eigenstates observed in Hilbert space for interacting quantum systems.

Numerical studies confirm the predictions for $\rho(E)$ and the scaling of LDOS correlations up to system sizes $N\sim 5000$ [2408.15072]. The established connection between the universal spectral order parameter and transitions in eigenfunction statistics (ergodicity breaking at $\gamma=1$, localization at $\gamma=\mu$) provides analytic tools for probing nontrivial intermediate regimes in both single-particle and many-body localization contexts. 

## 7. Summary of Key Scaling Relations and Table of Phase Diagram

Below, the (μ, γ) phase diagram is summarized:

| Regime                 | Condition                        | Eigenfunction Scaling                | Level Statistics                 |
|------------------------|----------------------------------|--------------------------------------|----------------------------------|
| Ergodic                | $\gamma < 1$                     | $D_q = 1$                            | Wigner–Dyson                     |
| Fractal/Multifractal   | $1 < \gamma < \mu$               | $D_1 = (\mu-\gamma)/(\mu-1)$         | WD on mini-band, Poisson outside |
| Localized              | $\gamma > \mu$                   | $D_q = 0$                            | Poisson                          |

The critical lines at $\gamma=1$ (ergodic–fractal transition) and $\gamma=\mu$ (fractal–localized transition) are validated analytically and numerically, with a universal finite-size scaling exponent $\nu=1$ at the ergodic–fractal line. The order parameter $O = \tilde\rho(0)$ transitions smoothly across the phase boundary [2408.15072].

The Lévy Rosenzweig–Porter ensemble thus stands as a paradigmatic, analytically tractable model for exploring ergodicity breaking, multifractality, and localization transitions in strongly disordered quantum systems.

Source: https://www.emergentmind.com/topics/rosenzweig-porter-random-matrix-ensemble