Universal Fluctuations in Spectra of Disordered Systems at the Anderson Transition

Abstract

Using the level--spacing distribution and the total probability function of the numbers of levels in a given energy interval we analyze the crossover of the level statistics between the delocalized and the localized regimes. By numerically calculating the electron spectra of systems of up to $32^3$ lattice sites described by the Anderson Hamiltonian it is shown that the distribution $P(s)$ of neighboring spacings is {\em scale- independent} at the metal-insulator transition. For large spacings it has a Poisson-like asymptotic form $P(s)\propto \exp (-As/\Delta )$, where $A\approx 1.9$. At the critical point we obtain a linear relationship between the variance of the number of levels $⟨[\delta n(\epsilon ){]}^2⟩$ and their average number $⟨n(\epsilon )⟩$ within the interval $\epsilon$. The constant of proportionality is less than unity due to the repulsion of the levels. Both $P(s)$ and $⟨[\delta n(\epsilon ){]}^2⟩$ are determined by the probability density $Q_n(\epsilon )$ of having exactly $n$ levels in the energy interval $\epsilon$. The distribution $Q_n(\epsilon )$ at the critical point is found to be size--independent and to obey a Gaussian law near its maximum, where $n\sim ⟨n⟩$.

9 pages of Revtex, 3 figures included (tar-compressed and uuencoded using script UUFILES), to appear in Jpn.J.Appl.Phys

References in corpus (4)

• Universal Spectral Correlations at the Mobility Edge
• Relation between Energy Level Statistics and Phase Transition and its Application to the Anderson Model
• Scaling of Level Statistics at the Disorder-Induced Metal-Insulator Transition
• The Probability of an Eigenvalue Number Fluctuation in an Interval of a Random Matrix Spectrum

Cited by in corpus (25)

• Random Matrix Theories in Quantum Physics: Common Concepts
• Statistics and Scaling in Disordered Mesoscopic Electron Systems
• Asymptotics of Universal Probability of Neighboring Level Spacings at the Anderson Transition
• Spectral Rigidity and Eigenfunction Correlations at the Anderson Transition
• Universality classes of the Anderson Transitions Driven by non-Hermitian Disorder
• Anomalous diffusion at the Anderson transitions
• Statistics of Resonances and Delay Times in Random Media: Beyond Random Matrix Theory
• Level spacings at the metal-insulator transition in the Anderson Hamiltonians and multifractal random matrix ensembles
• Anderson transition in three-dimensional disordered systems with symplectic symmetry
• Crossover from critical orthogonal to critical unitary statistics at the Anderson transition
• Spectral statistics near the quantum percolation threshold
• Random Matrix Theory of the Energy-Level Statistics of Disordered Systems at the Anderson Transition
• Wave function statistics at the symplectic 2D Anderson transition: bulk properties
• Resonance width distribution for high-dimensional random media
• Coherent forward scattering as a signature of Anderson metal-insulator transitions
• Three-Dimensional Quantum Percolation Studied by Level Statistics
• Renormalization group approach to energy level statistics at the integer quantum Hall transition
• Metal-insulator transition in two-dimensional disordered systems with power-law transfer terms
• Scaling of level-statistics and critical exponent of disordered two-dimensional symplectic systems
• Dependence of critical level statistics on the sample shape
• Influence of boundary conditions on level statistics and eigenstates at the metal insulator transition
• Critical behaviour at the metal-insulator transition in 3-dimensional disordered systems in a strong magnetic field
• Real-space renormalization-group approach to the integer quantum Hall effect
• Scaling Behavior of Level Statistics in Quantum Hall Regime
• Spectral Properties of Three Dimensional Layered Quantum Hall Systems