Collaborators: José-Antonio Méndez-Bermúdez (Puebla)Stefan Kettemann (Bremen, Pohang)Eduardo Mucciolo (Orlando)Victor Gopar (Zaragoza)Andrei Jesús Martinez Mendoza (Puebla)

thanks to: V.Kravtsov, B.Shapiro, A.Ossipov, Th.Seligman, A.Mirlin, F.Evers,O.Yevtushenko, E.Cuevas, B.Eckhardt, U.Smilansky, G.Zarand

AvH, OTKA, CiC, Conacyt, DFG, POSTECH etc.

Imre VargaElméleti Fizika TanszékBudapesti Műszaki és Gazdaságtudományi Egyetem, H-1111 Budapest, Magyarország

Imre VargaElméleti Fizika TanszékBudapesti Műszaki és Gazdaságtudományi Egyetem, H-1111 Budapest, Hungary

Magnetic impurities, wave packet dynamics, transportin the presence of multi-fractal states

Imre VargaDepartment of Theoretical PhysicsBudapest University of Technology and Economics, H-1111 Budapest, Hungary

Outline Introduction Anderson transition Random matrix modell of the MIT: PBRM Spectral statistics, multifractal states

New results with PBRM at criticality Scattering, transport Wave packet dynamics Magnetic impurities

Summary and outlook

Hamiltonian:

Energies n are uncorrelated, random numbers from uniform (or bimodal, Gaussian, Cauchy, etc.) distribution W

Nearest-neighbor hopping V

Bloch states for W V, localized states for W V

W V ??

Anderson model (1958)

WV <<WV >> WV ∝

Two energy (time) scales: ETh and (D and H) g = ETh/ = H/D

One-parameter scaling (1979)

Metal – insulator transition (MIT) for d>2.

Gell-Mann – Low function

Spectral statistics (d=3)

MIT

Spectral statistics (d=3)

W < Wc• extended states• RMT-like

W > Wc• localized states• Poisson-like

W = Wc• multifractal states• intermediate

‘mermaid’

Superscaling Dependence on symmetry parameter β

superscaling relationthru parameter g

with andare the RMT limit

Varga, Hofstetter, Pipek ’99

Eigenstates

extended stateweak disorder, band center

localized statestrong disorder, band edge

(L=240) R.A.Römer

Multifractal eigenstate at the MIT

Inverse participation numbers

Box counting technique• fixed L• state-to-state fluctuations

PDF analysis

• higher accuracy• scaling with L

http://en.wikipedia.org/wiki/Metal-insulator_transition (L=240) R.A.Römer

Multifractal statesdetected in reality

Unusual features of the MIT Interplay of eigenvector and spectral

statistics Chalker et al. ‘95

Anomalous diffusion at the MIT Huckestein et al. ‘97

Correlation dimension Kravtsov and Cuevas ‘07

LDOS vs wave function fluctuations Huckestein et al. ‘97

2D

Tested for “weak” multifractality (3dO, 2dU, 2dS), i.e. when D2d.

Scattering at the MIT

Kottos and Weiss ’02; Weiss, et al. ‘06

Detect the MIT using a stopwatch!

PBRM: Power-law Band Random Matrix

Model: matrix with and

asymptotically

parameters: and

Mirlin, et al. ‘96, Mirlin ‘00

PBRM: Power-law Band Random Matrix

for RMT, as if for similar to metal with

d

for BRM Poisson, as if for power law localization with exponent (cf. Yeung-Oono 87)

for criticality (cf. Levitov ‘90)

continuous line of transitions: b

PBRM at criticality ( ) for b non-linear -model RG, SUSY (Mirlin ‘00)

• large conductance: g*=4b for b real-space RG, virial expansion, SUSY

(Levitov 99, Yevtushenko-Kravtsov ‘03, Yevtushenko-Ossipov ‘07)

Mirlin ‘00

Multifractality universal distribution of participation numbers

with

state-to-state fluctuations of correlation dimensions

and

connection between the two: fixed point

so

Cuevas ‘02 and Varga ‘02

β = 1

β = 2

joint distribution state-to-state fluctuation

Varga ‘02

How does multifractality show up?

Scattering, conductance

LDOS vs wave function fluctuations

Anomalous diffusion at the MIT

Screening of magnetic impurities

Scattering: PBRM + 1 lead scattering matrix

Wigner delay time

resonance width, eigenvalues of poles of

Perfect coupling distribution of phases for

b > 1:

with

perfect coupling achieved:

Measure multifractality using a stopwatch!

PBRM + 1 lead

JA Méndez-Bermúdez – Kottos ‘05 Ossipov – Fyodorov ‘05:

b = 0.1 b = 0.1

b = 1.0 b = 1.0

b = 10 b = 10

JAMB and Varga ‘06

b = 0.1b = 0.1

b = 1.0 b = 1.0

b = 10b = 10

JAMB and Varga ‘06

Scaling of typical values

JAMB and Varga ‘06

Scattering, transport geometry

Geometry A

Geometry B

JAMB, Gopar, and Varga ’10, (Monthus and Garel ‘09, ’10)

Scattering matrix elements

A

B

Transport quantities: T, P

Conductance

Shot noise

w(T) for M=1

A B

A Bw(ln T) for small b

w(T) for M=2

A B

AB

universal behavior

ConductanceA B

Shot noiseA B

AConductance and shot noise

Unusual features of the MIT

LDOS vs wave function fluctuations

Anomalous diffusion at the MIT

Tested for weak multifractality (3dO, 2dU, 2dS)

Wave function and LDOS

Wave functions LDOS

JAMB and Varga ’10 (in preparation)

Wave function and LDOS

Wave packet dynamicsasymptotic wave packet profile survival probability

Wave packet dynamics (summary)

effective dimensionality changes with b

JAMB and Varga ’10 (in preparation)

Kondo effect in disordered metals

TK depends on r P(TK) wide, bimodal

Nagaoka – Suhl:

Unscreened (free) magnetic moments exist,

if

Weakly disordered metal: Insulator:

D the bandwidthS. Kettemann, E. Mucciolo and I. Varga ’09

Kondo effect at the critical pointlognormaldistribution ofwave function components

Joined probability of wave functions

enhanced energy correlationof wave functions

Kondo effect around the critical point

Free magnetic moments exist away from the critical point

Symmetry class of the metal-insulator transition

Critical point symmetry dependent:

In case of

Summary and outlook

Summary new details of scattering, transport at criticality; wave packet dynamics understood for weak

multifractality only (b → ∞); new details found about the MIT

as an interplay of Kondo physics and disorder;

Outlook spectral correlation for strong multifractality RKKY + Kondo effect of many body physics other ???

Thank you for your attention