Dynamical Signatures of Chaos to Integrability Crossover in $2\times 2$ Generalized Random Matrix Ensembles

Abstract

We introduce a two-parameter ensemble of generalized $2\times 2$ real symmetric random matrices called the $\beta$-Rosenzweig-Porter ensemble (\brpe), parameterized by $\beta$, a fictitious inverse temperature of the analogous Coulomb gas model, and $\gamma$, controlling the relative strength of disorder. \brpe\ encompasses RPE from all of the Dyson's threefold symmetry classes: orthogonal, unitary and symplectic for $\beta =1,2,4$. Firstly, we study the energy correlations by calculating the density and 2nd moment of the Nearest Neighbor Spacing (NNS) and robustly quantify the crossover among various degrees of level repulsions. Secondly, the dynamical properties are determined from an exact calculation of the temporal evolution of the fidelity enabling an identification of the characteristic Thouless and the equilibration timescales. The relative depth of the correlation hole in the average fidelity serves as a dynamical signature of the crossover from chaos to integrability and enables us to construct the phase diagram of \brpe\ in the $\gamma$-$\beta$ plane. Our results are in qualitative agreement with numerically computed fidelity for $N\gg 2$ matrix ensembles. Furthermore, we observe that for large $N$ the 2nd moment of NNS and the relative depth of the correlation hole exhibit a second order phase transition at $\gamma =2$.

References in corpus (54)

• Novel Symmetry Classes in Mesoscopic Normal-Superconducting Hybrid Structures
• Random Matrix Theories in Quantum Physics: Common Concepts
• Observation of many-body localization of interacting fermions in a quasi-random optical lattice
• Perfect state transfer in quantum spin networks
• Quasiparticle Lifetime in a Finite System: A Non--Perturbative Approach
• Dynamics of Loschmidt echoes and fidelity decay
• Perfect Transfer of Arbitrary States in Quantum Spin Networks
• Matrix Models for Beta Ensembles
• Anderson localization on the Bethe lattice: non-ergodicity of extended states
• A random matrix model with localization and ergodic transitions
• Thouless and relaxation time scales in many-body quantum systems
• Scaling theory of the Anderson transition in random graphs: ergodicity and universality
• Quantum two-level systems in Josephson junctions as naturally formed qubits
• Non-ergodic metallic and insulating phases of Josephson junction chains
• From non-ergodic eigenvectors to local resolvent statistics and back: a random matrix perspective
• Generic dynamical features of quenched interacting quantum systems: Survival probability, density imbalance and out-of-time-ordered correlator
• Goos-Hänchen and Imbert-Fedorov shifts from a quantum-mechanical perspective
• Dynamical phases in a "multifractal" Rosenzweig-Porter model
• Eigenfunction distribution for the Rosenzweig-Porter model
• Fragile ergodic phases in logarithmically-normal Rosenzweig-Porter model
• Quantum phase transitions in photonic cavities with two-level systems
• Survival probability in Generalized Rosenzweig-Porter random matrix ensemble
• Dynamical Manifestations of Quantum Chaos: Correlation Hole and Bulge
• 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
• Extreme statistics and spacing distribution in a Brownian gas correlated by resetting
• Eigenpolarizations for Giant Transverse Optical Beam Shifts
• Drive Induced Delocalization in Aubry-André Model
• From ergodic to non-ergodic chaos in Rosenzweig-Porter model
• Multifractal metal in a disordered Josephson Junction Array
• Realistic many-body quantum systems vs full random matrices: static and dynamical properties
• Quantum phase with coexisting localized, extended, and critical zones
• Number statistics for $\beta$-ensembles of random matrices: applications to trapped fermions at zero temperature
• Exact Simulation of Pigment-Protein Complexes Unveils Vibronic Renormalization of Electronic Parameters in Ultrafast Spectroscopy
• From closed to open 1D Anderson model: Transport versus spectral statistics
• Invariant $\beta$-Wishart ensembles, crossover densities and asymptotic corrections to the Marchenko-Pastur law
• Quench, thermalization and residual entropy across a non-Fermi liquid to Fermi liquid transition
• Level-statistics in Disordered Systems: A single parametric scaling and Connection to Brownian Ensembles
• Transport, multifractality, and the breakdown of single-parameter scaling at the localization transition in quasiperiodic systems
• Scaling of Fock-space propagator and multifractality across the many-body localization transition
• Spread of wave packets in disordered hierarchical lattices
• Nearest-neigbor spacing distributions of the beta-Hermite ensemble of random matrices
• Absence of Mobility Edge in Short-range Uncorrelated Disordered Model: Coexistence of Localized and Extended States
• Diagnostics of nonergodic extended states and many body localization proximity effect through real-space and Fock-space excitations
• Deformed Gaussian Orthogonal Ensemble description of Small-World networks
• Stability of large complex systems with heterogeneous relaxation dynamics
• Chaos due to symmetry-breaking in deformed Poisson ensemble
• Eigenvalue Statistics for Generalized Symmetric and Hermitian Matrices
• Experimental test of the Rosenzweig-Porter model for the transition from Poisson to Gaussian unitary ensemble statistics
• Critical Level Statistics at the Many-Body Localization Transition Region
• A unified fluctuation formula for one-cut $\beta$-ensembles of random matrices
• On invariant 2x2 β-ensembles of random matrices
• Transport in deformed centrosymmetric networks
• Dynamical Detection of Level Repulsion in the One-Particle Aubry-André Model

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