Universal energy-level statistics for disordered metals

Anderson localization and multifractal spectrum at the transition point in a two-dimensional non-Hermitian AII†system

The Anderson localization transition in a two-dimensional AII^†system is studied by eigenvalue statistics and then confirmed by the multifractal analysis of the wave functions at the transition point. The system is modeled by a two-dimensional lattice structure with real-quaternion off-diagonal elements and complex on-site energies, whose real and imaginary parts are two independent random variables. Via finite-size scaling analysis of eigenvalue spacing ratios, we find the non-Hermiticity reduces the critical disorder and give an estimate of the critical exponentν= 1.89, showing the system belongs to a new universal class other than the AII class and probably shares the same exponent with two-dimensional Hermitian DIII systems although they have different symmetries. The Anderson localization transition is further confirmed by checking the linearity in the parametric representation of the singularity strength and by checking the universality of the forms of the singularity spectra of different system sizes. The generalized dimensions are obtained asD1=1.80andD2=1.62.

Level Statistics and Time Evolution at the Mobility Edge

We present results for the statistics of the eigenvalues in random matrix ensembles characterized by an Anderson delocalization-localization transition. The nearest-level-spacing distribution function P(S) and the number variance 〈(δN(E))2〉 are shown at the mobility edge where we obtain universal curves interpolating between Wigner-Dyson and Poisson statistics valid for delocalized and for localized eigenfunctions, respectively. We also discuss the connection of level statistics with dynamics by considering the time evolution of a quantum wavepacket in a quasirandom matrix model. The critical quantum dynamics is characterized by anomalous diffusion, described via continuous sets of multifractal exponents.