Motivated by a series of recent works, an interest in multifractal phases has risen as they are believed to be present in the Many-Body Localized (MBL) phase and are of high demand in quantum annealing and machine learning. Inspired by the success of the Rosenzweig-Porter (RP) model with Gaussian-distributed hopping elements, several RP-like ensembles with the fat-tailed distributed hopping terms have been proposed, with claims that they host the desired multifractal phase. In the present work, we develop a general (graphical) approach allowing a self-consistent analytical calculation of fractal dimensions for a generic RP model and investigate what features of the RP Hamiltonians can be responsible for the multifractal phase emergence. We conclude that the only feature contributing to a genuine multifractality is the on-site energies’ distribution, meaning that no random matrix model with a statistically homogeneous distribution of diagonal disorder and uncorrelated off-diagonal terms can host a multifractal phase.
Motivated by a series of recent works, an interest in multifractal phases has risen as they are believed to be present in the Many-Body Localized (MBL) phase and are of high demand in quantum annealing and machine learning. Inspired by the success of the Rosenzweig-Porter (RP) model with Gaussian-distributed hopping elements, several RP-like ensembles with the fat-tailed distributed hopping terms have been proposed, with claims that they host the desired multifractal phase. In the present work, we develop a general (graphical) approach allowing a self-consistent analytical calculation of fractal dimensions for a generic RP model and investigate what features of the RP Hamiltonians can be responsible for the multifractal phase emergence. We conclude that the only feature contributing to a genuine multifractality is the on-site energies’ distribution, meaning that no random matrix model with a statistically homogeneous distribution of diagonal disorder and uncorrelated off-diagonal terms can host a multifractal phase.
Abstract In recent years, the Rosenzweig–Porter (RP) ensemble, obtained by adding a diagonal matrix with independent and identically distributed elements to a Gaussian random matrix, has been widely used as a minimal model for the emergence of fractal eigenstates in complex many-body systems. A key open question concerns the robustness of its phase diagram when the assumption of independent and uncorrelated entries is relaxed—an assumption that simplifies its analysis but is generally violated in realistic quantum systems. In this work, we take a first step in this direction by considering a deformed Wishart (rather than Gaussian) random matrix, which we dub the ‘Wishart–RP’ ensemble. Using perturbation theory, as well as the cavity and replica methods and the Dyson Brownian motion approach, we characterize its phase diagram and localization properties. Remarkably, we show that the level compressibility, which quantifies spectral correlations in the fractal phase, coincides with that of the Gaussian RP model, thereby extending the universality conjectured in Venturelli et al (2023 SciPost Phys. 14 110) beyond the fully uncorrelated setting. We confirm our results with numerical tests.
In this paper, we consider an extension of the Rosenzweig-Porter model, the L\'evy-RP (L-RP) model, in which the off-diagonal matrix elements are broadly distributed, providing a more realistic benchmark to develop an effective description of nonergodic extended (NEE) states in interacting many-body disordered systems. We put forward a simple, general, and intuitive argument that allows one to unveil the multifractal structure of the minibands in the local spectrum when hybridization is due to anomalously large transition amplitudes in the tails of the distribution. The idea is that the energy spreading of the minibands can be determined self-consistently by requiring that the maximal hybridization rate Hij between a site i and the other ND1 sites of the support set is of the same order of the Thouless energy itself ${N}^{{D}_{1}\ensuremath{-}1}$. This argument yields the fractal dimensions that characterize the statistics of the multifractal wave functions in the NEE phase, as well as the whole phase diagram of the L-RP ensemble. Its predictions are confirmed both analytically, by a thorough investigation of the self-consistent equation for the local density of states obtained using the cavity approach, and numerically, via extensive exact diagonalizations.
Eigenstate multifractality is of significant interest with potential applications in various fields of quantum physics. Most of the previous studies concentrated on fine-tuned quantum models to realize multifractality, which is generally believed to be a critical phenomenon and fragile to random perturbations. In this work, we propose a set of generic principles based on the power-law decay of the eigenstates that allow us to distinguish a fractal phase from a genuine multifractal phase. We demonstrate the above principles in a 1D tight-binding model with inhomogeneous nearest-neighbor hopping that can be mapped to the standard quantum harmonic oscillator via energy-coordinate duality. We analytically calculate the fractal dimensions and the spectrum of fractal dimensions, which are in agreement with numerical simulations.
The Rosenzweig-Porter (RP) model has garnered much attention in the last decade, as it is a simple analytically tractable model showing both ergodic-nonergodic extended and Anderson localization transitions. Thus, it is a good toy model to understand the Hilbert-space structure of many-body localization phenomenon. In our Letter, we present analytical evidence, supported by exact numerics, that demonstrates the controllable tuning of the phase diagram in the RP model by employing on-site potentials with a nontrivial fractal dimension instead of the conventional random disorder. We demonstrate that such disorder extends the fractal phase and creates an unusual dependence of fractal dimensions of the eigenfunctions. Furthermore, we study the fate of level statistics in such a system to understand how these changes are reflected in the eigenvalue statistics.
Two established results have never been composed. The fidelity susceptibility of a quantum state to a Hamiltonian parameter equals one quarter of its quantum Fisher information, F_Q = 4·chi_f; and an exact calculation in the Rosenzweig-Porter (RP) ensemble shows that eigenfunction fidelity susceptibility is enhanced in the fractal non-ergodic-extended phase (Skvortsov, Amini and Kravtsov, PRB 106, 054208, 2022). Composed, they make the eigenvector fractal dimension D2 — until now a diagnostic of localization physics — an operational metrological figure of merit: along the RP fractal phase, at fixed distance from criticality and for field-like (local) couplings to the sensed parameter, the finite-size enhancement exponent of the quantum Fisher information over the ergodic reference is 1 − D2. We confirm this in-house under preregistered criteria on the generalized-RP ensemble with D2 measured from inverse-participation-ratio scaling (not assumed): enhancement = 1 − D2 within 0.15 at 4/4 interior points of the phase (deviations 0.009–0.078; correlation 0.996). Two further preregistered experiments fix the criterion's boundaries: geometric fractality alone — a disorder-free Sierpinski-gasket lattice with certified eigenpairs and degenerate-multiplet treatment — shows no effect (enhancement −0.12; the RP formula does not extend), while critical power-law banded random matrices extend the law at all three tested bandwidths (deviations 0.016–0.064). The criterion is therefore: multifractal eigenstates of random ensembles, sensed through local couplings — randomness, not fractal geometry, is the missing ingredient, and an operator-structure scope law (diagonal inherits; short-range inherits once its range fits the fractal support; collective and dense couplings are D2-blind) gives the design rule. On the platform side we construct — because it cannot be cited — the minimal model that carries the physics onto cold atoms: an interacting tilted+disordered lattice in which the tilt is only the sensed parameter (the bare Wannier-Stark ladder is integrable, so sourcing RP statistics from gravity would be immediately refutable) and the dense RP-like spectrum comes from interactions and disorder. A preregistered coexistence test on the interacting t–V chain finds a textbook ergodic-to-localized crossover whose typical susceptibility peaks at the crossover's ergodic edge — not mid-crossover — with the predicted heavy x^−2 tail (Hill index 1.96–2.04) in the chaotic regime and estimator-dependent rare-event gains inside the crossover. The platform statement is scoped accordingly, and one instrument caveat is stated up front: this attaches to lattice/Bloch gravimetry, not to free-fall light-pulse interferometry. Tier-2 throughout: a shared random-matrix description with preregistered numerical evidence, not a hardware claim.
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 ET, 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 ET, 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.