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Spectral and Wavefunction Statistics (I)
V.E.Kravtsov,Abdus Salam ICTP
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Level Statistics of non-interacting electrons in quantum dots Quantum dots and random matrices
Level repulsion
Introduction to Random Matrix Theory
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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
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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
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Gaussian Invariant Random Matrix Ensemble
)ˆexp()ˆ( 2HTrHP
)ˆ()ˆ( HPUHUP
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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
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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
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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
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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
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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
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Random interaction in chaotic dot is responsible for transformation of the Wigner surmise into a Gaussian distribution.
Interaction effects