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

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

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

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

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

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

Delay time and reflection coefficient

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)

Eigenfunction intensities Green‘s function: Eigenfunction intensity:

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

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)

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

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)

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)

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

Non-perfect coupling Transmission coefficient: Perfect coupling: Phase density:

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

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.

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