Post on 03-Jan-2016
description
1
Alexander Ossipov
Yan Fyodorov
School of Mathematical Sciences,
University of Nottingham
Multifractality indelay times statistics
2
Outline
1. Definitions and basic relation
2. Derivation of the basic relation
3. Distributions of delay times
4. Discussion and conclusions
3
S-matrix and Wigner delay time
sample
lead
incomingoutgoing
S-matrix:
Wigner delay time:
One-channel scattering:
4
Basic relation
A.O. and Y. V. Fyodorov, Phys. Rev. B 71, 125133(2005)
Scaled delay time:
Eigenfunction intensity:
5
Outline
1. Definitions and basic relation
2. Derivation of the basic relation
3. Distributions of delay times
4. Discussion and conclusions
6
Two representations of the S-matrix
S-matrix
Modulus and Phase:
Modulus and Phase are independent:
K-matrix:
Green‘s function:
7
Delay time and reflection coefficient
8
K-matrix
A. D. Mirlin and Y. V. Fyodorov, Phys. Rev. Lett. 72, 526 (1994)
Y. V. Fyodorov and D. V. Savin, JETP Letters 80, 725 (2004)
9
Eigenfunction intensities
Eigenfunction intensity:
Green‘s function:
10
11
Outline
1. Defenitions and basic relation
2. Derivation of the basic relation
3. Distributions of delay times
4. Discussion and conclusions
12
Distribution of delay times: RMT
Y. V. Fyodorov and H.-J. Sommers, Phys. Rev. Lett. 76, 4709 (1996)
V. A. Gopar, P. A. Mello, and M. Büttiker, Phys. Rev. Lett. 77, 3005 (1996)
Eigenfunctions:
Delay times:
Crossover between unitary and orthogonal symmetry classes
13
Distribution of delay times:metallic regime
Y. V. Fyodorov and A. D. Mirlin, JETP Letters 60, 790 (1994)
Conductance
14
Distribution of delay times:metallic regime
B. L. Altshuler, V. E. Kravtsov, I. V. Lerner, Mesoscopic Phenomena in Solids, (1991)
V. I. Falko and K. B. Efetov, Europhys. Lett. 32, 627 (1995)
A. D. Mirlin, Phys. Rep. 326, 259 (2000)
anomalously localized states
15
Distribution of delay times: criticality
A. D. Mirlin et. al. Phys. Rev. E 54, 3221 (1996)
A. D. Mirlin and F. Evers, Phys. Rev. B 62, 7920 (2000)
fractal dimension of the eigenfunctions
Weak multifractality in the metallic regime in 2D:
Power-law banded random matrices:
16
Outline
1. Defenitions and basic relation
2. Derivation of the basic relation
3. Distributions of delay times
4. Discussion and conclusions
17
Non-perfect coupling
Transmission coefficient:
Perfect coupling:
Non-perfect coupling:
Phase density:
18
Numerical test
Power-law banded random matrices:
J. A. Mendez-Bermudez and T. Kottos, Phys. Rev. B 72, 064108 (2005)
19
Related works
V. A. Gopar, P. A. Mello, and M. Büttiker, Phys. Rev. Lett. 77, 3005 (1996)
Distribution of the Wigner delay times in the RMT regime, using residues of the K-matrix and the Wigner conjecture.
J. T. Chalker and S. Siak, J. Phys.: Condens. Matter 2, 2671 (1990)
Anderson localization on the Cayley tree. Relation between the current density in a link and the energy derivative of the total phaseshift in the one-dimensional version of the network model.
20
Summary
• Exact relation between statistics of delay times and eigenfunctions in all regimes
• Properties of the eigenfunctions can be accessed by measuring scattering characteristics
• Anomalous scaling of the Wigner delay time moments at criticality