Circular Rosenzweig-Porter random matrix ensemble

This paper proposes and numerically validates a unitary (circular) analogue of the Rosenzweig-Porter random matrix ensemble, defined via Dyson Brownian motion, to model the level statistics and eigenstate fractality of many-body localization in periodically driven (Floquet) systems.

The Gist

Imagine you have a giant, chaotic dance floor filled with thousands of dancers. In the world of quantum physics, these dancers represent the possible states of a complex system (like a group of interacting atoms).

This paper is about understanding how these dancers move and interact when the music changes in two different ways:

1. Static Systems: The music is a steady, unchanging hum (like a normal, still room).
2. Floquet (Periodically Driven) Systems: The music is a rhythmic, repeating beat that keeps changing the rules every few seconds (like a strobe light or a pulsing laser).

For a long time, physicists had a great "rulebook" (called the Rosenzweig-Porter ensemble) for the first scenario (the static room). This rulebook helps them predict whether the dancers will mix freely (chaos) or stay stuck in their own little corners (localization).

However, nobody had a good rulebook for the second scenario (the pulsing, rhythmic room). Since the rules of quantum mechanics change when things are driven by a rhythm, the old math didn't quite fit.

The New Idea: A Circular Dance Floor

The authors of this paper asked: "Can we build a version of that old rulebook that works for the rhythmic, pulsing systems?"

They created a new model they call the Circular Rosenzweig-Porter ensemble.

Here is how they built it, using a simple analogy:

• The Old Way (Brownian Motion): Imagine the dancers moving randomly on a flat, straight line. If you nudge them randomly over time, they spread out in a predictable way. This is how the old model worked.
• The New Way (Circular Motion): For the rhythmic systems, the authors realized the dancers don't move on a straight line; they move on a circle. Think of the dancers running around a circular track. Their positions are measured by angles (like a clock face) rather than straight distances.

They defined their new model as the result of a "random walk" that happens specifically on this circle. They didn't just guess the math; they simulated this process on a computer to see what happens.

What They Found

The authors ran massive computer simulations (involving up to 1,000 dancers) to see if their new "circular" model behaved like the old "straight-line" model. They checked two main things:

1. The Spacing of the Dancers (Energy Levels)
They looked at the gaps between the dancers.

• In the "Chaotic" zone: The dancers are spread out evenly, and the gaps between them follow a specific, complex pattern (like a crowded party where everyone is jostling).
• In the "Localized" zone: The dancers cluster together or stay far apart in a very predictable, simple way (like people standing in a line).
• The Result: Their new circular model showed the exact same switch from "chaotic" to "clustered" as the old model did. The "tipping point" where the behavior changes happened at the same spot.

2. The Shape of the Dancers (Eigenstates)
They looked at how "spread out" a single dancer's influence is.

• Spread out: A dancer's energy is shared among many others.
• Fractal: A dancer is in a weird middle ground—spread out, but not fully. It's like a cloud that has a fuzzy, self-similar shape.
• Localized: A dancer is stuck in one spot.
• The Result: The circular model reproduced these exact same shapes. Whether the dancers were fully mixed, partially mixed (fractal), or stuck, the new model matched the old one perfectly.

The Bottom Line

The paper claims that they have successfully built a unitary (circular) version of the famous Rosenzweig-Porter model.

By treating the system as a circle rather than a straight line, they created a tool that accurately describes the behavior of periodically driven (pulsing) quantum systems. Just like the old model was a "phenomenological" (descriptive) tool for static systems, this new circular model serves as a descriptive tool for systems that are being rhythmically shaken or driven.

They proved this by showing that the statistical "fingerprints" of their new model (how the levels are spaced and how the states are shaped) are indistinguishable from the fingerprints of the original, well-understood model. This gives physicists a new, reliable way to study complex, rhythmic quantum systems without having to solve incredibly difficult equations from scratch.

Technical Summary

Technical Summary: Circular Rosenzweig-Porter Random Matrix Ensemble

Problem Statement
The Rosenzweig-Porter (RP) random matrix ensemble is a well-established phenomenological model used to describe the level statistics and eigenstate fractality across the many-body localization (MBL) transition in static, disordered interacting quantum systems. In static systems, dynamics are governed by Hermitian Hamiltonians. However, periodically driven (Floquet) systems are described by unitary Floquet operators, where eigenvalues lie on the unit circle in the complex plane. While MBL has been extended to Floquet systems, a direct unitary (circular) analogue of the RP ensemble capable of capturing the same phenomenology for periodically driven systems had not been constructed. This paper addresses the question of whether such a unitary analogue can be defined and if it shares the key statistical properties of the original RP ensemble.

Methodology
The authors propose a construction of the Circular Rosenzweig-Porter (CRP) ensemble based on the observation that the standard RP ensemble can be viewed as the outcome of a finite-time Dyson Brownian motion (DBM) process.

1. Construction: The authors define the CRP ensemble as the result of a stochastic evolution of a time-dependent, symmetric unitary matrix $S(t)$ of dimension $N$. The evolution follows a Dyson Brownian motion process adapted for unitary matrices:
   $$dS(t) = i\sqrt{dt} U^T(t) X U(t)$$
   where $X$ is a Gaussian Orthogonal Ensemble (GOE) matrix sampled at each infinitesimal time step, and $U(t)$ is derived from the decomposition $S(t) = U^T(t)U(t)$.
2. Initial Conditions: The process starts with a diagonal matrix $S(0) = \text{diag}(e^{i\theta_1(0)}, \dots, e^{i\theta_N(0)})$, where the initial phases $\theta_n(0)$ are sampled independently from a uniform distribution over $[-\pi, \pi)$. This ensures a uniform density of states, consistent with generic Floquet operators.
3. Ensemble Definition: The CRP ensemble is defined as the state of the matrix $S(t)$ at a specific time $t = \epsilon^2 / N^\gamma$, where $\epsilon$ is a perturbation strength and $\gamma$ controls the relative strength of the diagonal and off-diagonal terms.
4. Numerical Implementation: The authors numerically evaluate the stochastic process using an algorithm that updates the matrix via $S(t+dt) = S(t)e^{i\sqrt{dt}X}$. This approach preserves unitarity for finite time steps and avoids explicit matrix decompositions. Simulations were performed for matrix dimensions $N \leq 1000$, averaging over at least $10^6$ eigenvalues or eigenstates.

Key Contributions

• Definition of CRP: The paper introduces the Circular Rosenzweig-Porter ensemble as a unitary analogue of the standard RP ensemble, defined explicitly through a circular Dyson Brownian motion process.
• Theoretical Mapping: The authors provide a heuristic argument that the spectral statistics of the CRP ensemble at the middle of the spectrum should match those of the RP ensemble, provided the density of states is scaled appropriately (specifically, mapping the RP parameter $\epsilon$ to $\epsilon/\sqrt{2\pi}$ in the CRP context).
• Numerical Verification: The study provides comprehensive numerical evidence characterizing the statistical properties of the CRP ensemble's eigenvalues and eigenstates.

Results
The numerical investigation confirms that the CRP ensemble exhibits statistical behaviors analogous to the RP ensemble across different regimes of the parameter $\gamma$:

1. Level Spacing Statistics: The average ratio of consecutive level spacings, $r$, serves as a diagnostic for quantum chaos.
   • For $\gamma < 2$, the system exhibits Wigner-Dyson statistics ($r \approx 0.530$), characteristic of chaotic systems.
   • For $\gamma \geq 2$, the system transitions to Poissonian statistics ($r \approx 0.386$), characteristic of integrable systems.
   • A phase transition occurs at the critical point $\gamma_c = 2$ with a critical exponent $\nu = 1$. The data shows a scaling collapse of $r$ as a function of $(\gamma - \gamma_c) \ln(N)^{1/\nu}$, consistent with the RP ensemble.
2. Eigenstate Fractality: The inverse participation ratio (IPR) of the eigenstates was analyzed in the basis where $S(0)$ is diagonal.
   • For $\gamma \leq 1$, eigenstates are extended ($IPR_2 \sim N^{-1}$).
   • For $\gamma \geq 2$, eigenstates are localized ($IPR_2 \sim O(1)$).
   • In the intermediate regime $1 < \gamma < 2$, eigenstates are fractal, scaling as $IPR_2 \sim N^{-(2-\gamma)}$.
3. Eigenstate Shape: The distribution of eigenstate amplitudes $|\langle m | \psi_n \rangle|^2$ was found to follow Breit-Wigner statistics. The data aligns with the theoretical prediction:
   $$|\langle m | \psi_n \rangle|^2 \sim \frac{1}{(\theta_n - \theta_m^{(0)})^2 + \Gamma^2}$$
   where $\Gamma$ is the spreading width. While finite-size effects and continuum approximations lead to slight quantitative mismatches, the qualitative agreement and the distinct structural differences between the $\gamma = 0.5$ and $\gamma = 1.5$ regimes are clearly observed.

Significance and Claims
The authors claim that the Circular Rosenzweig-Porter ensemble serves as a valid phenomenological model for the level statistics and eigenstate fractality observed across the many-body localization transition in periodically driven (Floquet) systems. By establishing that the CRP ensemble shares key statistical properties with the RP ensemble, the work provides a tractable theoretical tool for studying non-ergodic phases in driven quantum systems.

The paper modestly frames this work as a "first step" toward constructing more generalizations, such as models incorporating multifractality or other Floquet models with multifractal eigenstates. The authors suggest that future generalizations could be achieved by considering stochastic processes with correlated increments, similar to recent proposals for the standard RP ensemble. Additionally, they note the potential utility of the CRP ensemble as a non-maximally random building block in studies of random quantum circuits.