#21 From Quantum Chaos to Anderson Localization
Notes for # 21
Classical dynamical systems can
be separated into two classes – Integrable and Chaotic. For quantum systems
this distinction manifests itself e.g., in spectral statistics. Roughly
speaking, integrability leads to Poisson distribution of the eigen-energies,
while chaos implies Wigner-Dyson statistics of the levels, which are
characteristic for the ensembles of Random Matrices. The latter statistics is realized in a variety of physical
systems.
Phenomenon of Anderson
localization of the eigenfunctions of quantum particles in a random potential,
which was originally discovered in connection with charge transport in
disordered conductors, provides a nice laboratory to study this Poisson versus
Wigner-Dyson crossover, i.e., crossover from quantum integrable to quantum
chaotic behaviour. We will briefly discuss the problem of Anderson Localization
from the spectral statistics point of view and demonstrate that the onset of
the chaotic behaviour for a rather broad class of systems can be understood as
a delocalization in the space of quantum numbers that characterize the original
integrable system. We will illustrate this conclusion by a number of examples
and make an attempt to describe the chaotic nature of the nuclear spectra
within the framework of these ideas.