We study analytically and numerically the dynamics of the generalized Rosenzweig-Porter model, which is known to possess three distinct phases: ergodic, multifractal and localized phases. Our focus is on the survival probability R(t), the probability of finding the initial state after time t. In particular, if the system is initially prepared in a highly-excited non-stationary state (wave packet) confined in space and containing a fixed fraction of all eigenstates, we show that R(t)can be used as a dynamical indicator to distinguish these three phases. Three main aspects are identified in different phases. The ergodic phase is characterized by the standard power-law decay of R(t) with periodic oscillations in time, surviving in the thermodynamic limit, with frequency equals to the energy bandwidth of the wave packet. In multifractal extended phase the survival probability shows an exponential decay but the decay rate vanishes in the thermodynamic limit in a non-trivial manner determined by the fractal dimension of wave functions. Localized phase is characterized by the saturation value of R(t -> infinity) = k, finite in the thermodynamic limit N -> infinity, which approaches k = R (t -> 0) in this limit.We study analytically and numerically the dynamics of the generalized Rosenzweig-Porter model, which is known to possess three distinct phases: ergodic, multifractal and localized phases. Our focus is on the survival probability R(t), the probability of finding the initial state after time t. In particular, if the system is initially prepared in a highly-excited non-stationary state (wave packet) confined in space and containing a fixed fraction of all eigenstates, we show that R(t)can be used as a dynamical indicator to distinguish these three phases. Three main aspects are identified in different phases. The ergodic phase is characterized by the standard power-law decay of R(t) with periodic oscillations in time, surviving in the thermodynamic limit, with frequency equals to the energy bandwidth of the wave packet. In multifractal extended phase the survival probability shows an exponential decay but the decay rate vanishes in the thermodynamic limit in a non-trivial manner determined by the fractal dimension of wave functions. Localized phase is characterized by the saturation value of R(t -> infinity) = k, finite in the thermodynamic limit N -> infinity, which approaches k = R (t -> 0) in this limit.

Jan 30, 201976 citations

Fading ergodicity and quantum dynamics in random matrix ensembles

Rafał Świętek, Maksymilian Kliczkowski, Miroslav Hopjan +1 more

Recent work has proposed fading ergodicity as a mechanism for many-body ergodicity breaking. Here, we show that two paradigmatic random matrix ensembles -- the Rosenzweig-Porter model and the ultrametric model -- fall within the same universality class of ergodicity breaking when embedded in a many-body Hilbert space of spins-1/2. By calibrating the parameters of both models via their Thouless times, we demonstrate that the matrix elements of local observables display similar statistical properties, allowing us to identify the fractal phase of the Rosenzweig-Porter model with the fading-ergodicity regime. This correspondence is further supported through the analyses of quantum-quench dynamics of local observables, their temporal fluctuations and power spectra, and survival probabilities. Our findings reveal that local observables thermalize within the fading-ergodicity regime on timescales shorter than the Heisenberg time, thus providing a unified framework for understanding ergodicity breaking across these distinct models.Recent work has proposed fading ergodicity as a mechanism for many-body ergodicity breaking. Here, we show that two paradigmatic random matrix ensembles -- the Rosenzweig-Porter model and the ultrametric model -- fall within the same universality class of ergodicity breaking when embedded in a many-body Hilbert space of spins-1/2. By calibrating the parameters of both models via their Thouless times, we demonstrate that the matrix elements of local observables display similar statistical properties, allowing us to identify the fractal phase of the Rosenzweig-Porter model with the fading-ergodicity regime. This correspondence is further supported through the analyses of quantum-quench dynamics of local observables, their temporal fluctuations and power spectra, and survival probabilities. Our findings reveal that local observables thermalize within the fading-ergodicity regime on timescales shorter than the Heisenberg time, thus providing a unified framework for understanding ergodicity breaking across these distinct models.

cond-mat.stat-mechCross-listedarXiv:2603.23616Physical review. B./Physical review. BMar 26, 20261 citation

A random matrix model with localization and ergodic transitions

V E Kravtsov, Ivan M. Khaymovich, E Cuevas +1 more

Motivated by the problem of Many-Body Localization and the recent numerical results for the level and eigenfunction statistics on the random regular graphs, a generalization of the Rosenzweig-Porter random matrix model is suggested that possesses two localization transitions as the parameter $γ$ of the model varies from 0 to $\infty$ . One of them is the Anderson transition from the localized to the extended states that happens at $γ=2$ . The other one at $γ=1$ is the transition from the extended non-ergodic (multifractal) states to the extended ergodic states similar to the eigenstates of the Gaussian Orthogonal Ensemble. We computed the two-level spectral correlation function, the spectrum of multifractality $f(α)$ and the wave function overlap which all show the transitions at $γ=1$ and $γ=2$ .Motivated by the problem of Many-Body Localization and the recent numerical results for the level and eigenfunction statistics on the random regular graphs, a generalization of the Rosenzweig-Porter random matrix model is suggested that possesses two localization transitions as the parameter $γ$ of the model varies from 0 to $\infty$. One of them is the Anderson transition from the localized to the extended states that happens at $γ=2$. The other one at $γ=1$ is the transition from the extended non-ergodic (multifractal) states to the extended ergodic states similar to the eigenstates of the Gaussian Orthogonal Ensemble. We computed the two-level spectral correlation function, the spectrum of multifractality $f(α)$ and the wave function overlap which all show the transitions at $γ=1$ and $γ=2$.

arXiv:1508.01714New Journal of PhysicsDec 16, 2015269 citations

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

Victor Delapalme, Leticia F. Cugliandolo, Alexander K. Hartmann +2 more

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.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.

arXiv:2607.09444Jul 10, 20260 citations

Free probability approach to spectral and operator statistics in Rosenzweig-Porter random matrix ensembles

Viktor Jahnke, Pratik Nandy, Kuntal Pal +2 more

A bstract Utilizing the framework of free probability, we analyze the spectral and operator statistics of the Rosenzweig-Porter random matrix ensembles, which exhibit a rich phase structure encompassing ergodic, fractal, and localized regimes. Leveraging subordination formulae, we develop a perturbative scheme that yields semi-analytic expressions for the density of states up to second order in system size, in good agreement with numerical results. We compute higher-point correlation functions in the ergodic regime using both numerical and suitable analytic approximations. Our analysis of operator statistics for various spin operators across these regimes reveals close agreement with free probability predictions in the ergodic phase, in contrast to persistent deviations observed in the fractal and localized phases, even at late times. Notably, the fractal phase exhibits partial freeness while retaining memory of the initial spectrum, highlighting the importance of non-localized eigenstates and associated with the late-time dynamics of cumulative out-of-time-ordered-correlators (OTOCs). Employing distance measures and statistical tools such as the χ 2 statistic, Kullback-Leibler divergence, and Kolmogorov-Smirnov hypothesis testing, we define a characteristic time scale — the free time — that marks the onset of the validity of free probability predictions for operator spectral statistics in the ergodic phase. Remarkably, our findings demonstrate consistency across these different approaches.A bstract Utilizing the framework of free probability, we analyze the spectral and operator statistics of the Rosenzweig-Porter random matrix ensembles, which exhibit a rich phase structure encompassing ergodic, fractal, and localized regimes. Leveraging subordination formulae, we develop a perturbative scheme that yields semi-analytic expressions for the density of states up to second order in system size, in good agreement with numerical results. We compute higher-point correlation functions in the ergodic regime using both numerical and suitable analytic approximations. Our analysis of operator statistics for various spin operators across these regimes reveals close agreement with free probability predictions in the ergodic phase, in contrast to persistent deviations observed in the fractal and localized phases, even at late times. Notably, the fractal phase exhibits partial freeness while retaining memory of the initial spectrum, highlighting the importance of non-localized eigenstates and associated with the late-time dynamics of cumulative out-of-time-ordered-correlators (OTOCs). Employing distance measures and statistical tools such as the χ 2 statistic, Kullback-Leibler divergence, and Kolmogorov-Smirnov hypothesis testing, we define a characteristic time scale — the free time — that marks the onset of the validity of free probability predictions for operator spectral statistics in the ergodic phase. Remarkably, our findings demonstrate consistency across these different approaches.

arXiv:2506.04520Journal of High Energy PhysicsDec 1, 20257 citations

Dynamical detection of extended nonergodic states in many-body quantum systems

David A. Zarate-Herrada, Isaías Vallejo-Fabila, Léa Ferreira dos Santos +1 more

Fractal dimensions are tools for probing the structure of quantum states and identifying whether they are localized or delocalized in a given basis. These quantities are commonly extracted through finite-size scaling, which limits the analysis to relatively small system sizes. In this work, we demonstrate that the correlation fractal dimension $D_2$ can be directly obtained from the long-time dynamics of interacting many-body quantum systems. Specifically, we show that it coincides with the exponent of the power-law decay of the time-averaged survival probability, defined as the fidelity between an initial state and its time-evolved counterpart. This dynamical approach avoids the need for scaling procedures and enables access to larger systems than those typically reachable via exact diagonalization. We test the method on various random matrix ensembles, including full random matrices, the Rosenzweig-Porter model, and power-law banded random matrices, and extend the analysis to interacting many-body systems described by the one-dimensional Aubry-André model and the disordered spin-1/2 Heisenberg chain. In the case of full random matrices, we also derive an analytical expression for the entire evolution of the time-averaged survival probability.Fractal dimensions are tools for probing the structure of quantum states and identifying whether they are localized or delocalized in a given basis. These quantities are commonly extracted through finite-size scaling, which limits the analysis to relatively small system sizes. In this work, we demonstrate that the correlation fractal dimension $D_2$ can be directly obtained from the long-time dynamics of interacting many-body quantum systems. Specifically, we show that it coincides with the exponent of the power-law decay of the time-averaged survival probability, defined as the fidelity between an initial state and its time-evolved counterpart. This dynamical approach avoids the need for scaling procedures and enables access to larger systems than those typically reachable via exact diagonalization. We test the method on various random matrix ensembles, including full random matrices, the Rosenzweig-Porter model, and power-law banded random matrices, and extend the analysis to interacting many-body systems described by the one-dimensional Aubry-André model and the disordered spin-1/2 Heisenberg chain. In the case of full random matrices, we also derive an analytical expression for the entire evolution of the time-averaged survival probability.

arXiv:2505.23910May 29, 20250 citations

γ

of the model varies from 0 to

\infty

. One of them is the Anderson transition from the localized to the extended states that happens at

γ=2

. The other one at

γ=1

is the transition from the extended non-ergodic (multifractal) states to the extended ergodic states similar to the eigenstates of the Gaussian Orthogonal Ensemble. We computed the two-level spectral correlation function, the spectrum of multifractality

f(α)

and the wave function overlap which all show the transitions at

γ=1

and

γ=2

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.

D_2

can be directly obtained from the long-time dynamics of interacting many-body quantum systems. Specifically, we show that it coincides with the exponent of the power-law decay of the time-averaged survival probability, defined as the fidelity between an initial state and its time-evolved counterpart. This dynamical approach avoids the need for scaling procedures and enables access to larger systems than those typically reachable via exact diagonalization. We test the method on various random matrix ensembles, including full random matrices, the Rosenzweig-Porter model, and power-law banded random matrices, and extend the analysis to interacting many-body systems described by the one-dimensional Aubry-André model and the disordered spin-1/2 Heisenberg chain. In the case of full random matrices, we also derive an analytical expression for the entire evolution of the time-averaged survival probability.