Correlations&Disorder in Metals Vollhardt 2009

7/27/2019 Correlations&Disorder in Metals Vollhardt 2009

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Supported by Deutsche Forschungsgemeinschaft through SFB 484

XIV. Training Course in the Physics of Strongly Correlated SystemsSalerno, October 8, 2009

Dieter Vollhardt

Theory of correlated fermionic condensed matter

3. Correlation-induced phenomena in

electronic systemsb. Electronic correlations and disorder


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Metal-Insulator transitions: Examples

Disorder and averaging

Mott-Hubbard transition vs. Anderson localization

In collaboration with:

Krzysztof ByczukWalter Hofstetter

Outline:


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Metal-Insulator Transitionsin the Presence of Disorder:

Examples


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d=3Anderson metal-insulator transition:

(disorder induced)


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Kravchenko, Mason, Bowker,Furneaux, Pudalov,

DIorio (1995)

Metal-insulator transition in a dilute,low-disordered Si MOSFET

d=2


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Disorder(quenched)


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, ,

iijH c c nt

i j ii j i

Random local potential

Anderson disorder model on the lattice

Random hopping


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i

, ,

iH c c nt

i j ii j i

Disorder Scattering of a (quantum) particle

1( 0)

Scattering time

Weak scattering (d=3)

Drude/Boltzmann conductivity


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i

, ,

iH c c nt

i j ii j i

: disorder strength

Disorder distributions, e.g.,:

1 /

Box disorder

arith( )i i i iO d P O

e.g., local DOSarith

( )i


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i

, ,

iH c c nt

i j ii j i

: disorder strength

Disorder distributions, e.g.,:

1 /

Box disorder

.1-xx

Binary alloy disorder (alloys A1-xBx, e.g., Fe1-xCox)

arith( )i i i iO d P O

e.g., local DOSarith

( )i


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( )( ) ( ) ie rr rDisorder affects wave fct.

Anderson localization

Localization of a particle, , due to, e.g.,(0) 0


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( )( ) ( ) ie rr rDisorder affects wave fct.

Anderson localization

Alloy band splitting

max( , ), for 1c Ut d

Binary alloy disorder, bounded Hamiltonian

Upper alloySubband (UAB)

Lower alloySubband (LAB):

Localization of a particle, , due to, e.g.,(0) 0


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DMFT for disordered systems


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, ,

iH c c nt

i j ii j i

Soven (1967)Taylor (1967)

Coherent potential approximation (CPA)(best single-site approximation)

robust results for cannot describe Anderson localizationarith( )i


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Example: CPA results for phonon DOS for disordered cubic crystal

Elliot, Krumhansl, Leath (RMP, 1974)


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Coherent Potential Approximation and d

Vlaming, DV (1992)

Jani, DV (1992)

G( )

iv

Generalization of DMFT to disordered and interacting lattice electrons

Local potential operator:

CPA = exact solution of the Anderson disorder model in d

DMFT witharith

( )i CPAMore precisely:


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Mott-Hubbard Transitionvs.

Anderson Localization


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, ,

it UH c c n n n

i j ii ii j i i

Anderson-Hubbard Hamiltonian

: disorder strength

1 /

Box disorder

n=1

1. Can both transitions be characterized by the average local DOS?2. Further destabilization of correlated metallic phase by disorder?3. Are the Mott insulator and Anderson insulator separated by

another (metallic) phase?

1. Can both transitions be characterized by the average local DOS?2. Further destabilization of correlated metallic phase by disorder?3. Are the Mott insulator and Anderson insulator separated by

another (metallic) phase?

=0: Mott-Hubbard metal-insulator transition for U>UcU=0: Anderson localization for in d>20c


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Usually unknown

Anderson (1958)


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Approximation of PDF: calculate averages + moments

Why? Because arithmetic averagedoes not yield the max. of the PDF!

PDF of disordered systems: very broad/long tails

arithx

Variance

medianxgeomx

typicalx


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Approximation of PDF: calculate averages + moments

Why? Because arithmetic averagedoes not yield the max. of the PDF!

PDF of disordered systems: very broad/long tails

Localized(strong disorder):Very asymmetric;Long tails

(Log-normaldistribution)

Metallic (weak disorder):Rather symmetric; peak close toarithmetic mean

Local DOS

H

istogram

of

LDOS(d=3)

arithx


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ln ( )

typical geometric( ) ( ) i

E

i iE E e

Anderson localization:

Anderson (1958)


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lattice Green function

DMFT for Anderson-Hubbard model

Dobrosavljevic, Pastor, Nikolic (2003)


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?metal

Andersoninsulator

Mott-Hubbardinsulator

Mott-Hubbard Transition vs. Anderson Localization

c

cU


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Non-magnetic phase diagram; n=1,T=0

Byczuk, Hofstetter, DV (2005)

Local DOS

, ,

it Uc c n n n

i j ii ii j i i


: disorder strength

1 /

Critical behavior atlocalization transition

Solution by DMFT(NRG)

1

1

2

2

3

3


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Hubbard

subbands

vanish

, ,

it Uc c n n n

i j ii ii j i i


: disorder strength

1 /

Anderson- andMott insulatorneighboring


Disorder increases

Interaction in/decreases

cU

Ac

Interactions may increasemetallicity

d=2:Denteneer, Scalettar, Trivedi (1999)


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Hubbard

subbands

vanish

, ,

it Uc c n n n

i j ii ii j i i


: disorder strength

1 /

Anderson- andMott insulatorneighboring


DMRG for disordered bosons in d=1

Rapsch, Schollwck, Zwerger (1999)


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Antiferromagnetismvs.

Anderson Localization


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NN hopping, bipartite lattice, n=1:

Take into account antiferromagnetic order

, ,

it Uc c n n n

i j ii ii j i i


: disorder strength

1 /

Unrestricted Hartree-Fock, d=3

Tusch, Logan (1993)


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, ,

it Uc c n n n

i j ii ii j i i


: disorder strength

1 /


DMFT: Non-magnetic phase diagram Magnetic phase diagram

(small) (large)Noboundary

NN hopping, bipartite lattice, n=1:

Take into account antiferromagnetic order

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Correlations&Disorder in Metals Vollhardt 2009