Replica approach to the generalized Rosenzweig-Porter model

Abstract

The generalized Rosenzweig-Porter model with real (GOE) off-diagonal entries arguably constitutes the simplest random matrix ensemble displaying a phase with fractal eigenstates, which we characterize here by using replica methods. We first derive analytical expressions for the average spectral density in the limit in which the size $N$ of the matrix is large but finite. We then focus on the number of eigenvalues in a finite interval and compute its cumulant generating function as well as the level compressibility, i.e., the ratio of the first two cumulants: these are useful tools to describe the local level statistics. In particular, the level compressibility is shown to be described by a universal scaling function, which we compute explicitly, when the system is probed over scales of the order of the Thouless energy. Interestingly, the same scaling function is found to describe the level compressibility of the complex (GUE) Rosenzweig-Porter model in this regime. We confirm our results with numerical tests.

arXiv commentSubmission to SciPost; 48 pages, 7 figures; accepted version

References in corpus (81)

• Chaos and Quantum Thermalization
• Thermalization and its mechanism for generic isolated quantum systems
• Metal-insulator transition in a weakly interacting many-electron system with localized single-particle states
• Many-body localization edge in the random-field Heisenberg chain
• Statistics of energy levels and eigenfunctions in disordered systems
• Quasiparticle Lifetime in a Finite System: A Non--Perturbative Approach
• Multifractal Scalings across the Many-Body Localization Transition
• Majorana Fermions in superconducting 1D systems having periodic, quasiperiodic, and disordered potentials
• A random matrix model with localization and ergodic transitions
• Introduction to Random Matrices - Theory and Practice
• Multifractal finite-size-scaling and universality at the Anderson transition
• Topological superconductor to Anderson localization transition in one-dimensional incommensurate lattices
• Localization and adiabatic pumping in a generalized Aubry-André-Harper model
• Thouless energy and multifractality across the many-body localization transition
• Sparse random matrices: the eigenvalue spectrum revisited
• Cavity Approach to the Spectral Density of Sparse Symmetric Random Matrices
• Correlation-induced localization
• Phase diagram of a non-Abelian Aubry-André-Harper model with $p$-wave superfluidity
• Correlations of Nearby Levels Induced by a Random Potential
• Rare thermal bubbles at the many-body localization transition from the Fock space point of view
• Non-ergodic delocalized phase in Anderson model on Bethe lattice and regular graph
• Statistics of eigenstates near the localization transition on random regular graphs
• Reverse annealing for the fully connected $p$-spin model
• Non-ergodic metallic and insulating phases of Josephson junction chains
• Two-eigenfunction correlation in a multifractal metal and insulator
• From non-ergodic eigenvectors to local resolvent statistics and back: a random matrix perspective
• Many-body localization transition with power-law interactions: Statistics of eigenstates
• The Index Distribution of Gaussian Random Matrices
• The Many-Body localization transition in the Hilbert space
• Phase transitions and edge scaling of number variance in Gaussian random matrices
• Dynamical phases in a "multifractal" Rosenzweig-Porter model
• Eigenfunction distribution for the Rosenzweig-Porter model
• How many eigenvalues of a Gaussian random matrix are positive?
• Fragile ergodic phases in logarithmically-normal Rosenzweig-Porter model
• Survival probability in Generalized Rosenzweig-Porter random matrix ensemble
• Law of addition in random matrix theory
• The Lévy-Rosenzweig-Porter random matrix ensemble
• Multifractality of eigenstates in the delocalized non-ergodic phase of some random matrix models : Wigner-Weisskopf approach
• Non-Ergodic Delocalization in the Rosenzweig-Porter Model
• Multifractality and its role in anomalous transport in the disordered XXZ spin-chain
• Multifractality without fine-tuning in a Floquet quasiperiodic chain
• Eigenvectors under a generic perturbation: non-perturbative results from the random matrix approach
• Index Distribution of Random Matrices with an Application to Disordered Systems
• Localization transition on the Random Regular Graph as an unstable tricritical point in a log-normal Rosenzweig-Porter random matrix ensemble
• Mobility edge and multifractality in a periodically driven Aubry-André model
• Multifractal metal in a disordered Josephson Junction Array
• Non-ergodic extended states in the SYK model
• From ergodic to non-ergodic chaos in Rosenzweig-Porter model
• Spectral diffusion and scaling of many-body delocalization transitions
• Transition from Poisson to gaussian unitary statistics: The two-point correlation function
• Non-ergodic delocalized states for efficient population transfer within a narrow band of the energy landscape
• Emergent fractal phase in energy stratified random models
• Finite size correction to the spectrum of regular random graphs: an analytical solution
• On super-Poissonian behavior of the Rosenzweig-Porter model in the non-ergodic extended regime
• Non-ergodic delocalized phase with Poisson level statistics
• Number statistics for $\beta$-ensembles of random matrices: applications to trapped fermions at zero temperature
• Non-Hermitian Rosenzweig-Porter random-matrix ensemble: Obstruction to the fractal phase
• Quantum algorithm for energy matching in hard optimization problems
• A minimal model of many body localization
• Perturbation Theory for the Rosenzweig-Porter Matrix Model
• Non-ergodic extended phase of the Quantum Random Energy model
• Large deviation function for the number of eigenvalues of sparse random graphs inside an interval
• An exactly solvable record model for rainfall
• Critical-to-Insulator Transitions and Fractality Edges in Perturbed Flatbands
• Spectral correlations in the crossover between GUE and Poisson regularity: on the identification of scales
• Full counting statistics for interacting trapped fermions
• Flat-band-based multifractality in the all-band-flat diamond chain
• Anomalous multifractality in quantum chains with strongly correlated disorder
• Spread of wave packets in disordered hierarchical lattices
• Spectral order statistics of Gaussian random matrices: large deviations for trapped fermions and associated phase transitions
• Out of equilibrium Phase Diagram of the Quantum Random Energy Model
• Tilted elastic lines with columnar and point disorder, non-Hermitian quantum mechanics and spiked random matrices: pinning and localization
• Sensitivity of (multi)fractal eigenstates to a perturbation of the Hamiltonian
• Propagation of singular behavior for Gaussian perturbations of random matrices
• Fully localized and partially delocalized states in the tails of Erdös-Rényi graphs in the critical regime
• Stability of large complex systems with heterogeneous relaxation dynamics
• Large deviation theory for diluted Wishart random matrices
• Universal record statistics for random walks and Lévy flights with a nonzero staying probability
• Bose-Einstein-Like condensation of deformed random matrix: A replica approach
• Replica-symmetric approach to the typical eigenvalue fluctuations of Gaussian random matrices
• Index of a matrix, complex logarithms, and multidimensional Fresnel integrals