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Alexander Ossipov

Yan Fyodorov

School of Mathematical Sciences,

University of Nottingham

Multifractality indelay times statistics

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Outline

1. Definitions and basic relation

2. Derivation of the basic relation

3. Distributions of delay times

4. Discussion and conclusions

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S-matrix and Wigner delay time

sample

lead

incomingoutgoing

S-matrix:

Wigner delay time:

One-channel scattering:

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Basic relation

A.O. and Y. V. Fyodorov, Phys. Rev. B 71, 125133(2005)

Scaled delay time:

Eigenfunction intensity:

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Outline

1. Definitions and basic relation

2. Derivation of the basic relation

3. Distributions of delay times

4. Discussion and conclusions

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Two representations of the S-matrix

S-matrix

Modulus and Phase:

Modulus and Phase are independent:

K-matrix:

Green‘s function:

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Delay time and reflection coefficient

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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)

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Eigenfunction intensities

Eigenfunction intensity:

Green‘s function:

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Outline

1. Defenitions and basic relation

2. Derivation of the basic relation

3. Distributions of delay times

4. Discussion and conclusions

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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

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Distribution of delay times:metallic regime

Y. V. Fyodorov and A. D. Mirlin, JETP Letters 60, 790 (1994)

Conductance

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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

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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:

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Outline

1. Defenitions and basic relation

2. Derivation of the basic relation

3. Distributions of delay times

4. Discussion and conclusions

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Non-perfect coupling

Transmission coefficient:

Perfect coupling:

Non-perfect coupling:

Phase density:

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Numerical test

Power-law banded random matrices:

J. A. Mendez-Bermudez and T. Kottos, Phys. Rev. B 72, 064108 (2005)

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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.

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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