Spectral and Wavefunction Statistics (I) V.E.Kravtsov, Abdus Salam ICTP

Level Statistics of non-interacting electrons in quantum dots  Quantum dots and random matrices  Level repulsion  Introduction to Random Matrix Theory

Quantum dots and Quantum billiards Integrable circular billiard Chaotic billiard Wavefunction amplitude in a chaotic billiard Random superposition of plane waves Simple statistics of wavefunctions in a chaotic billiard

Invariance under basis rotation Random superposition of plane waves on an energy shell E=const Rotation of basis: Does not change statistics of wavefunction (basis invariance) Large number of terms in the sum over k (large energy E) CLT Gaussian statistics

Gaussian Invariant Random Matrix Ensemble

Distribution of eigenvalues and eigenvectors Old variables: H nm New variables: eigenvalues E n, eigenvectors

Jacobian J and the effective level interaction Effective logarithmic interaction Double set for complex U 

Dyson symmetry classes E  is a temperature of the effective plasma model  for complex H =1 for real H

Level repulsion: a pedestrian derivation When two levels are much closer than the average spacing: 2X2 block:

The Wigner surmise   Was expected for Coulomb blockade peaks of conductance in small quantum dots but was significantly modified by interaction effects. Observed for level statistics in microwave cavities. Independently fluctuating energy levels

Random interaction in chaotic dot is responsible for transformation of the Wigner surmise into a Gaussian distribution. Interaction effects