Statistics of energy levels and eigenfunctions in disordered systems

Abstract

The article reviews recent developments in the theory of fluctuations and correlations of energy levels and eigenfunction amplitudes in diffusive mesoscopic samples. Various spatial geometries are considered, with emphasis on low-dimensional (quasi-1D and 2D) systems. Calculations are based on the supermatrix sigma-model approach. The method reproduces, in so-called zero-mode approximation, the universal random matrix theory (RMT) results for the energy-level and eigenfunction fluctuations. Going beyond this approximation allows us to study system-specific deviations from universality, which are determined by the diffusive classical dynamics in the system. These deviations are especially strong in the far ``tails'' of the distribution function of the eigenfunction amplitudes (as well as of some related quantities, such as local density of states, relaxation time, etc.). These asymptotic ``tails'' are governed by anomalously localized states which are formed in rare realizations of the random potential. The deviations of the level and eigenfunction statistics from their RMT form strengthen with increasing disorder and become especially pronounced at the Anderson metal-insulator transition. In this regime, the wave functions are multifractal, while the level statistics acquires a scale-independent form with distinct critical features. Fluctuations of the conductance and of the local intensity of a classical wave radiated by a point-like source in the quasi-1D geometry are also studied within the sigma-model approach. For a ballistic system with rough surface an appropriately modified (``ballistic'') sigma-model is used. Finally, the interplay of the fluctuations and the electron-electron interaction in small samples is discussed, with application to the Coulomb blockade spectra.

review article, 127 pages. Small changes (typos corrected, references added). To be published in Physics Reports (March 2000)

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• Interaction Matrix Element Fluctuations in Ballistic Quantum Dots: Dynamical Effects
• Statistical properties of speckle patterns for a random number of scatterers and nonuniform phase distributions
• The Wishart--Rosenzweig--Porter random matrix ensemble
• Effect of surface disorder on the chiral surface states of a three-dimensional quantum Hall system
• Anderson localization of a Majorana fermion
• Wave localization in number-theoretic landscapes
• Cross-Over between universality classes in a magnetically disordered metallic wire
• Field Theory for the Global Density of States Distribution Function in Disordered Conductors
• Quantum birthmarks: Ergodicity breaking beyond scarring
• Magnetic instability and spin-glass order beyond the Anderson-Mott transition in interacting power-law random banded matrix fermions
• Density of states correlations in Lévy Rosenzweig-Porter model via supersymmetry approach
• Dynamic effect of phase conjugation on wave localization
• Analytic and numerical toolkit for the Anderson model in one dimension
• Entanglement and localization of wavefunctions
• Localization properties of the sparse Barrat-Mézard trap model
• Enhancement in Breaking of Time-reversal Invariance in the Quantum Kicked Rotor
• Fluctuations of Wigner-type random matrices associated with symmetric spaces of class DIII and CI
• Inverted pendulum driven by a horizontal random force: statistics of the never-falling trajectory and supersymmetry
• Virial expansion of the non-linear sigma model in the strong coupling limit
• Many-body spectral transitions through the lens of the variable-range SYK2 model
• Universal fluctuations of localized two interacting particles in one dimension
• The renormalization group for interacting fermions: from Fermi liquids to quantum dots
• Enhanced spin Hall effect in strong magnetic disorder
• A nearly closed ballistic billiard with random boundary transmission
• Anderson localization: a density matrix approach
• Exploring the role of diffusive coupling in spatiotemporal chaos
• Coexistence of localized and extended states in the Anderson model with long-range hopping
• 2D Anderson Localization and KPZ sub-Universality Classes : sensitivity to boundary conditions and insensitivity to symmetry classes
• Multifractal correlations of the local density of states in dirty superconducting films
• Replica approach to the generalized Rosenzweig-Porter model
• Exact generating function of a zero-dimensional supersymmetric non-linear sigma model