Alexander Ossipov Yan Fyodorov School of Mathematical Sciences, University of Nottingham

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. lead sample S-matrix and Wigner delay time outgoing incoming S-matrix: Wigner delay time: One-channel scattering:

4. Eigenfunction intensity: Scaled delay time: Basic relation A.O. and Y. V. Fyodorov, Phys. Rev. B 71, 125133(2005)

5. Outline 1. Definitions and basic relation 2. Derivation of the basic relation 3. Distributions of delay times 4. Discussion and conclusions

6. K-matrix: Modulus and Phase: Modulus and Phase are independent: Green‘s function: Two representations of the S-matrix S-matrix

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. Green‘s function: Eigenfunction intensity: Eigenfunction intensities

11. Outline 1. Defenitions and basic relation 2. Derivation of the basic relation 3. Distributions of delay times 4. Discussion and conclusions

12. Eigenfunctions: Crossover between unitary and orthogonal symmetry classes Distribution of delay times: RMT Delay times: 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. Conductance Distribution of delay times:metallic regime Y. V. Fyodorov and A. D. Mirlin, JETP Letters 60, 790 (1994)

14. anomalously localized states 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)

15. fractal dimension of the eigenfunctions Power-law banded random matrices: Distribution of delay times: criticality Weak multifractality in the metallic regime in 2D: 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. Transmission coefficient: Perfect coupling: Non-perfect coupling: Phase density: Non-perfect coupling

18. Power-law banded random matrices: Numerical test J. A. Mendez-Bermudez and T. Kottos, Phys. Rev. B 72, 064108 (2005)

19. 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. Related works

20. 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 Summary