Power-law banded random matrices: a testing ground for the Anderson transition

Imre VargaElméleti Fizika TanszékBudapesti Műszaki és Gazdaságtudományi Egyetem, Magyarország

Imre VargaDepartment of Theoretical PhysicsBudapest University of Technology and Economics, Hungarycollaborators: Daniel Braun (Toulouse)

Tsampikos Kottos (Middletown, CT) José Antonio Méndez Bermúdez

(Puebla) Stefan Kettemann (Bremen, Pohang), Eduardo Mucciolo (Orlando, FL)thanks to: V.Kravtsov, B.Shapiro, A.Ossipov, A.Garcia-Garcia, Th.Seligman,

A.Mirlin, I.Lerner, Y.Fyodorov, F.Evers, B.Eckhardt, U.Smilansky, etc. also to AvH, OTKA, CiC, Conacyt, DFG, etc.

Power-law banded random matrices: a testing ground for the Anderson transition

Outline Introduction

Anderson transition Intermediate statistics PBRM and the MIT Spectral statistics, multifractal states

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

Summary

Hamiltonian:

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

Nearest-neighbor hopping V (symmetry: , , ) 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 (tD and tH) g = ETh/D = tH/tD

One-parameter scaling (1979)

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

Gell-Mann – Low function

Mobility edge (d=3)De

nsity

of s

tate

s

Cond

uctiv

ity

Localized wave functions

A non-interacting electron moving in random potential

Quantum interference of scattering waves

Anderson localization of electrons

E

extended

localizedlocalized

extended

localized

critical

Ec

Spectral statistics (d=3)

MITZharekeshev ‘96

Spectral statistics (d=3)

W < Wc• extended states• RMT-like

W > Wc• localized states• Poisson-like

W = Wc• multifractal states• intermediate

‘mermaid’

Anderson - MIT Dependence on symmetry parameter

superscaling relationthru parameter g

with and are the RMT limit

IV, Hofstetter, Pipek ’99

Eigenstates for weak and strong W

extended stateweak disorder, band center

localized statestrong disorder, band edge

(L=240) R.A.Römer

Multifractality at the MIT (3d) Inverse participation numbers

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

• higher accuracy• scaling with L

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

Multifraktál állapotok a valóságban

LDOS fluktuációk a fém-szigetelő átalakulás közelében Ga1-xMnxAs-ban

Multifractality: scaling behavior of moments of (critical) wave functions

Critical wave function at a metal-insulator transition point

Continuous set of independent and universal critical exponents

multifractal exponents

: anomalous scaling dimensions

singularity spectrum

fractal dimension

: measure of r where

In a metal

Unusual features of the MIT (3d) Interplay of eigenvector and spectral

statistics Chalker et al. ‘95

Anomalous diffusion at the MIT Huckestein et al. ‘97

Correlation dimension strong probability overlap (Chalker ’88)

LDOS vs wave function fluctuations Huckestein et al. ‘97

2D

Unusual features of the MIT (3d)

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

PBRM for RMT, as if

for 1/2 < a < 1 similar to metal with d=1/(a-1/2)

for BRM Poisson, as if for a > 3/2 power law localization with exponent a

(cf. Yeung-Oono ‘87)

for criticality (cf. Levitov ‘90) no mobility edge! continuous line of transitions: b

PBRM transitionCuevas et al. ‘01

• asymmetric transition• Kosterlitz-Thouless

Kottos and IV ‘01 (unpub.)

PBRM at criticality ( ) for b 1 non-linear s-model RG, SUSY (Mirlin ‘00)

• large conductance: g*=4bb for b 1 real-space RG, virial expansion, SUSY

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

Mirlin ‘00

PBRM at criticality – DOS ( )b=0.1

b=1.0

b=10.0

L=1024

PBRM at criticality (b=1)semi-Poisson statistics is qualitatively valid only

IV and Braun ‘00

β = 1

β = 2

joint distribution state-to-state fluctuation

IV ‘02

How does multifractality show up?

Scattering (1 lead)

LDOS vs wave function fluctuations

Anomalous diffusion at the MIT

Nature of entanglement

Screening of magnetic impurities

Open system: 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!

Scattering: PBRM + 1 lead JA Méndez-Bermúdez – Kottos ‘05 Ossipov –

Fyodorov ‘05:

JA Méndez-Bermúdez – IV 06:

Wave function and LDOSWave functions LDOS

J.A. Méndez-Bermúdez and IV ‘08 (in prep.)

Wave function and LDOS

J.A. Méndez-Bermúdez and IV ‘08 (in prep.)

Wave packet dynamicsasymptotic wave packet profile survival probability

J.A. Méndez-Bermúdez and IV ‘08 (in prep.)

Wave packet dynamicseffective dimensionality changes

J. A. Méndez-Bermúdez and IV ‘08 (in prep.)

AB

A

1 qubit in a tight-binding lattice site i with or without an electron: A

2 qubits in a tight-binding lattice site i and j with or without an electron: A

Entanglement at criticality

concurrence [Wootters (1997)] (bipartite systems)

tangle [Meyer and Wallach (2002)]

(multipartite)

AB

ji

i

4321 0,max)( AC

)()( yyAyyAAR

)Tr1(2)( 2AAQ

BA Tr

IV and JA Méndez-Bermúdez ‘08

Entanglement at criticalityAverage concurrence in an eigenstate

||2 jiijC

1||112

ii

jiij M

CM

C

Average tangle

114 P

NQ 12 12 P

MC

where M=N(N-1)/2 and41

iiP

IPR of state

b=0.3

by 1

IV and JA Méndez-Bermúdez ‘08

?)1( 2DLC

Entanglement at criticality

)(1 bNfNQ )(1 bNfNC

IV and JA Méndez-Bermúdez ‘08(cf. Kopp et al. ’07; Jia et al. ’08)

T < TK alatt spin-flip szórás,szinglet alapállapot,Kondo-árnyékolás

Kondo effektus fémben (1964)

Kondo effektus rendezetlen fémben

TK helyfüggő P(TK) széles, bimodális

1-hurok (Nagaoka – Suhl):

Árnyékolatlan (szabad) mágneses momentumok,

ha

Kissé rendezetlen vezető:Szigetelő:

Kondo effektus a kritikus pontban

lognormálishullámfüggvény eloszlás

hullámfüggvény intenzitások együttes eloszlása

hullámfüggvényekenergiakorrelációja

Kondo effektus a kritikus pont körülA mágneses momentumokközül pontosan egy szabad:

A kritikus pontban nincsenek szabad momentumok

A szigetelő oldalon:

A kritikus ponttól távolodva léteznek szabad momentumok

A fém-szigetelő átalakulás szimmetriája

Kritikus pontszimmetria függő:

esetén

Magnetic impurity

S Kettemann, E Mucciolo, IV ‘09

Summary PBRM: a good testing ground for the Anderson transition

d=1 → scaling with L no mobility edge (!) features similar to Anderson MIT → deviations found tunable transition → b serve as 1/d or g multifractal states induce unusual behavior

Scattering Wave packet dynamics Entanglement Interplay with magnetic impurities

Outlook Effect of interactions on the HF level Dynamical stability versus chaotic environment

Thank you for your attention

Outlook: Current and future problems

free magnetic moments + e-e interactionso S. Kettemann (Hamburg)o E. Mucciolo (Orlando)

interplay of multifractality and interaction

decoherence of qubits in critical environmento Th. Seligman (Cuernavaca)o J.A. Méndez-Bermúdez (Puebla)