1. Multifractality in delay times statistics
Alexander Ossipov
Yan Fyodorov
School of Mathematical Sciences,
University of Nottingham

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
outgoing
incoming
S-matrix:
Wigner delay time:
One-channel scattering:

4. Eigenfunction intensity:
Basic relation
Eigenfunction intensity:
Scaled delay time:
A.O. and Y. V. Fyodorov, Phys. Rev. B 71, (2005)

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
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
Green‘s function:
Eigenfunction intensity:

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
Eigenfunctions:
Delay times:
Crossover between unitary and orthogonal symmetry classes
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)

13. Distribution of delay times: metallic regime
Conductance
Y. V. Fyodorov and A. D. Mirlin, JETP Letters 60, 790 (1994)

14. Distribution of delay times: metallic regime
anomalously localized states
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)

15. Distribution of delay times: criticality
fractal dimension of the eigenfunctions
Weak multifractality in the metallic regime in 2D:
Power-law banded random matrices:
A. D. Mirlin et. al. Phys. Rev. E 54, 3221 (1996)
A. D. Mirlin and F. Evers, Phys. Rev. B 62, 7920 (2000)

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:
Phase density:

18. Numerical test Power-law banded random matrices:
J. A. Mendez-Bermudez and T. Kottos, Phys. Rev. B 72, (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