Critical statistics in quantum chaos and Calogero-Sutherland model at finite temperature

Abstract

We investigate the spectral properties of a generalized GOE (Gaussian Orthogonal Ensemble) capable of describing critical statistics. The joint distribution of eigenvalues of this model is expressed as the diagonal element of the density matrix of a gas of particles governed by the Calogero-Sutherland Hamiltonian (C-S). Taking advantage of the correspondence between C-S particles and eigenvalues, we show that the number variance of our random matrix model is asymptotically linear with a slope depending on the parameters of the model. Such linear behavior is a signature of critical statistics. This random matrix model may be relevant for the description of spectral correlations of complex quantum systems with a self-similar/fractal Poincaré section of its classical counterpart. This is shown in detail for two examples: the anisotropic Kepler problem and a kicked particle in a well potential. In both cases the number variance and the ${\Delta }_3$-statistic is accurately described by our analytical results.

27 pages, 4 figures, modified conclusions in section four

References in corpus (1)

• Spectral properties of a generalized chGUE

Cited by in corpus (34)

• Anderson Transitions
• Spectral statistics across the many-body localization transition
• Anomalous Thouless energy and critical statistics on the metallic side of the many-body localization transition
• Non-interacting fermions at finite temperature in a $d$-dimensional trap: universal correlations
• Many-body critical phase: extended and nonthermal
• Finite temperature free fermions and the Kardar-Parisi-Zhang equation at finite time
• Noninteracting fermions in a trap and random matrix theory
• Randomly incomplete spectra and intermediate statistics
• Dimensional dependence of the metal-insulator transition
• Random matrix ensembles associated with Lax matrices
• The Anderson transition in quantum chaos
• Chiral phase transition and Anderson localization in the Instanton Liquid Model for QCD
• Family of generalized random matrix ensembles
• $2\delta$-Kicked Quantum Rotors: Localization and `Critical' Statistics
• Semi-Poisson statistics in quantum chaos
• Level compressibility in a critical random matrix ensemble: The second virial coefficient
• Density of states for almost diagonal random matrices
• Power spectrum for critical statistics: A novel spectral characterization of the Anderson transition
• Theory of 2$\delta$-kicked Quantum Rotors
• Emerging spectra of singular correlation matrices under small power-map deformations
• Kinetic energy of a trapped Fermi gas at finite temperature
• Fluctuations of observables for free fermions in a harmonic trap at finite temperature
• Long range disorder and Anderson transition in systems with chiral symmetry
• Classical intermittency and quantum Anderson transition
• Classical singularities and Semi-Poisson statistics in quantum chaos and disordered systems
• Two-level correlation function of critical random-matrix ensembles
• Is it possible to observe experimentally a metal-insulator transition in ultra cold atoms?
• Power law eigenvalue density, scaling and critical random matrix ensembles
• Metal or Insulator? Dirac operator spectrum in holographic QCD
• Sixfold Way of Traversable Wormholes in the Sachdev-Ye-Kitaev Model
• Effect of a magnetic flux on the critical behavior of a system with long range hopping
• Circulant $L$-ensembles in the thermodynamic limit
• Creation Mechanism of Devil's Staircase Surface and Unstable and Stable Periodic Orbit in the Anisotropic Kepler Problem
• Nonlinear sigma model approach for level correlations in chiral disordered systems