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

)exp()( rkiark

Rotation of basis:

)exp()exp( , rqiurkiq

qk

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

)ˆexp()ˆ( 2HTrHP

)ˆ()ˆ( HPUHUP

Distribution of eigenvalues and eigenvectors

dEdUUEUHJEdHHPn

n )ˆ()exp()( 2

UEUH

Old variables: Hnm New variables: eigenvalues En, eigenvectors ),...,(),...,( ,2,1

)()(2

)(1 Nnnn

nN

nn UUU

Jacobian J and the effective level interaction

222

2222

)(||)(2

])()ˆ(2ˆˆ2[)ˆ(

nnnm

mnmn dEdfEE

dEfdEEfEdfdTrHdTr

mnnmmn

dxdxDHdTr ,

2)ˆ( ]det[ nmDJ

fdUdUdUUfd ˆˆ

mnmn

mnmnnm EEEED 1||lnexp)(]det[

Effective logarithmic interactionDouble set

for complex U

222 )()'()( dxydy

Dyson symmetry classesdEdUUEUHJEdHHP

nn )ˆ()exp()( 2

mnmn EEUEUHJ 1||lnexp)ˆ(

E

||ln mn EE

is a temperature of the effective plasma

model

for complex H=1 for real H

Level repulsion: a pedestrian derivation

2*1

V

VH

When two levels are much closer than the average spacing: 2X2

block:

222121 ||)( VEE

21

,,

)(2

R

The Wigner surmise

)(

16exp 2

2

o

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.

]exp[

Independently fluctuating energy

levels

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

Interaction effects