Diagrammatic Theory of Strongly Correlated Electron Systems

Diagrammatic Theory of Strongly Diagrammatic Theory of Strongly Correlated Electron SystemsCorrelated Electron Systems

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• Introduction Metal-insulator transition Intersite interactions in DMFT

• Extended DMFT Pseudogap – Incoherent metal Luttinger’s theorem? Thermodynamics Transport

• Anderson impurity model at finite U Motivation Definition of the SUNCA approximation Results of SUNCA

• Summary

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• Summary

Use of HTcUse of HTc

Magnetic levitation (Japan 1999, 343 m.p.h)Magnetic levitation (Japan 1999, 343 m.p.h)Magnetic resonance imagingMagnetic resonance imagingFault current limiters of 6.4MVA, response time msFault current limiters of 6.4MVA, response time msE-bombs (strong EM pulse)E-bombs (strong EM pulse)

5000-horsepower motor made with sc wire5000-horsepower motor made with sc wire (July 2001)(July 2001)Electric generators, 99% efficiencyElectric generators, 99% efficiencyEnergy storage 3MWEnergy storage 3MW

Use of HTcUse of HTc

Underground cable in Copenhagen (for 150000 Underground cable in Copenhagen (for 150000 citizens,30 meters long, May 2001)citizens,30 meters long, May 2001)

Researching the possibility to build petaflop computers

Market $200 billion by the year 2010Market $200 billion by the year 2010

Materials undergoing MITMaterials undergoing MIT

High temperature superconductors (2D systems, transition with doping)High temperature superconductors (2D systems, transition with doping)Other 3d transition metal oxides (Nickel,Vanadium,Titanium,…)Other 3d transition metal oxides (Nickel,Vanadium,Titanium,…)

2D and 3D, transition with doping or pressure2D and 3D, transition with doping or pressureMany f-electron systemsMany f-electron systems

Hubbard model –Hubbard model – generic model for materials undergoing MITgeneric model for materials undergoing MIT

E= -2tE= -2t22/U/U

E= 0E= 0

Dynamical mean-field theory & MITDynamical mean-field theory & MIT

mappingmapping

fermionic bathfermionic bath

Zhang, Rozenberg and Kotliar 1992Zhang, Rozenberg and Kotliar 1992

UU

Doping Mott insulator – Doping Mott insulator – DMFT perspectiveDMFT perspective

Metallic system always Fermi liquid Metallic system always Fermi liquid ImIm

Fermi surface unchanged (volume and shape)Fermi surface unchanged (volume and shape)

Narrow quasiparticle peak of width Narrow quasiparticle peak of width ZZFF at the Fermi level at the Fermi level

Effective mass (m*/mEffective mass (m*/m1/Z) diverges at the transition1/Z) diverges at the transition

High-temperature (T>> High-temperature (T>> ZZFF) almost free spin) almost free spin

Georges, Kotliar, Krauth and Rozenberg 1996Georges, Kotliar, Krauth and Rozenberg 1996

LHBLHB UHBUHB

quasip. peakquasip. peak

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• Summary

Nonlocal interaction in DMFT?Nonlocal interaction in DMFT?Local quantum fluctuationsLocal quantum fluctuations (between states ) (between states )

completely completely taken into accounttaken into account within DMFT within DMFTNonlocal quantum fluctuationsNonlocal quantum fluctuations are mostly are mostly lostlost in DMFT (nonlocal RKKY inter.) in DMFT (nonlocal RKKY inter.) (residual ground-state entropy of par. Mott insulator is ln2 (residual ground-state entropy of par. Mott insulator is ln2 2 2NN deg. states) deg. states)

Why?Why?

Metzner Vollhardt 89Metzner Vollhardt 89

mean-field description of the exchange term is exact within DMFTmean-field description of the exchange term is exact within DMFT

JJ disappears completely in the paramagnetic phase disappears completely in the paramagnetic phase !!

For simplicity, take the infinite U limit For simplicity, take the infinite U limit t-J model: t-J model:

How does intersite exchange How does intersite exchange JJ change Mott transition?change Mott transition?

Hubbard modelHubbard model

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• Summary

Extended DMFTExtended DMFT

JJ and and tt equally important: equally important:

fermionic bathfermionic bath

mappingmapping

bosonic bathbosonic bathfluctuating magnetic fieldfluctuating magnetic field

Si & Smith 96, Si & Smith 96, Kajuter & Kotliar 96Kajuter & Kotliar 96

Source of the inelasting scatteringSource of the inelasting scattering

Still local and conserving theoryStill local and conserving theory

Local quantities can be calculated from the corresponding impurity problemLocal quantities can be calculated from the corresponding impurity problem

Long range fluctuations frozenLong range fluctuations frozen

Strong inelasting scattering Strong inelasting scattering due to local magnetic fluctuationsdue to local magnetic fluctuations

Fermion bubble is Fermion bubble is zerozero in in the paramagnetic statethe paramagnetic state

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• Summary

Pseudogap – Incoherent metalPseudogap – Incoherent metal

ImIm

Pseudogap due to strong Pseudogap due to strong inelasting scatteringinelasting scattering from from local magnetic fluctuationslocal magnetic fluctuations

Not Not due to finite ranged fluctuating antiferromagnetic (superconducting) due to finite ranged fluctuating antiferromagnetic (superconducting) domainsdomains

highly incoherent responsehighly incoherent response

Local spectral functionLocal spectral function

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• Summary

Luttinger’s theorem?Luttinger’s theorem?

Re

Re

ztzt

A(A(kk,,) ) =0.02=0.02

kx

ky

k

A(k,0) A(k,)

White lines corresponds to noninteracting systemWhite lines corresponds to noninteracting system

A(A(kk,,) ) =0.04=0.04

kx

ky

k

A(k,0) A(k,)


A(A(kk,,) ) =0.06=0.06

kx

ky

k

A(k,0) A(k,)


A(A(kk,,) ) =0.08=0.08

kx

ky

k

A(k,0) A(k,)


A(A(kk,,) ) =0.10=0.10

kx

ky

k

A(k,0) A(k,)


A(A(kk,,) ) =0.12=0.12

kx

ky

k

A(k,0) A(k,)


A(A(kk,,) ) =0.14=0.14

kx

ky

k

A(k,0) A(k,)


A(A(kk,,) ) =0.16=0.16

kx

ky

k

A(k,0) A(k,)


A(A(kk,,) ) =0.18=0.18

kx

ky

k

A(k,0) A(k,)


A(A(kk,,) ) =0.20=0.20

kx

ky

k

A(k,0) A(k,)


A(A(kk,,) ) =0.22=0.22

kx

ky

k

A(k,0) A(k,)


A(A(kk,,) ) =0.24=0.24

kx

ky

k

A(k,0) A(k,)


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• Summary

EntropyEntropy

EMDT+NCAEMDT+NCA ED 20 sitesED 20 sites

ED:ED:JakliJaklič č & & Prelovšek, 1995Prelovšek, 1995Experiment:Experiment:LSCO (T/tLSCO (T/t0.07)0.07)Cooper & LoramCooper & Loram

& &

EMDT+NCAEMDT+NCA ED 20 sitesED 20 sites

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Hall coefficientHall coefficient

TT~1000K~1000K

LSCO: Nishikawa, Takeda & Sato (1994)LSCO: Nishikawa, Takeda & Sato (1994)

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MotivationMotivation

Numerical renormalization group (NRG)Numerical renormalization group (NRG)Quantum Monte Carlo simulation (QMC)Quantum Monte Carlo simulation (QMC)Exact diagonalization (ED)Exact diagonalization (ED)Iterated perturbation theory (IPT)Iterated perturbation theory (IPT)Resummations of perturbation theory (NCA, CTMA)Resummations of perturbation theory (NCA, CTMA)

•A need to solve the DMFT impurity problem A need to solve the DMFT impurity problem for real materials with orbital degeneracyfor real materials with orbital degeneracy

•Quantum dots in mesoscopic structuresQuantum dots in mesoscopic structures

Several methods available to solve AIM:Several methods available to solve AIM:

Either slow or less flexibleEither slow or less flexible

Auxiliary particle techniqueAuxiliary particle technique

NCANCA

Simple fast and flexible methodSimple fast and flexible methodWorks for T>0.2 TWorks for T>0.2 TKK

Works only in the case of U=Works only in the case of U=

Naive extension very badly failsNaive extension very badly failsTTKK several orders of magnitude too small several orders of magnitude too small

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Luttinger-Ward functional for SUNCALuttinger-Ward functional for SUNCA

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Scaling of TScaling of TKK

Comparison with NRGComparison with NRG

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• Summary

SummarySummary

EDMFT• Purely local magnetic fluctuations can

induce pseudogap suppress large entropy at low doping induce strongly growing RH with decreasing T and

• Luttinger’s theorem is not applicable in the incoherent regime (<0.20)

• Fermi liquid is recovered only when *>JSUNCA• Infinite series of skeleton diagrams is needed to

recover correct low energy scale of the AIM at finite Coulomb interaction U

Extended Dynamical Mean FieldExtended Dynamical Mean Field

Metal-insulator transitionMetal-insulator transition

el-el correlations el-el correlations not importantnot important::band insulator: band insulator:

•the lowest conduction band is fullthe lowest conduction band is full ((possible possible only for even number of electrons)only for even number of electrons)•gap due to the periodic potential – few eVgap due to the periodic potential – few eV

simple simple metalmetal•Conduction band partially occupiedConduction band partially occupied

semiconductorsemiconductor

el-el correlations el-el correlations importantimportant::

Mott insulator despite the odd number of Mott insulator despite the odd number of electronselectrons

Cannot be explained within the Cannot be explained within the independent-electron picture (many body independent-electron picture (many body effect)effect)

Several competing mechanisms and Several competing mechanisms and several energy scaleseveral energy scaless

ztzt

FF**

Zhang, Rozenberg and Kotliar 1992Zhang, Rozenberg and Kotliar 1992

UU

Doping Mott insulator – Doping Mott insulator – DMFT perspectiveDMFT perspective

Metallic system always Fermi liquid Metallic system always Fermi liquid ImIm

Fermi surface unchanged (volume and shape)Fermi surface unchanged (volume and shape)

Narrow quasiparticle peak of width Narrow quasiparticle peak of width ZZFF at the Fermi level at the Fermi level

Effective mass (m*/mEffective mass (m*/m1/Z) diverges at the transition1/Z) diverges at the transition

High-temperature (T>> High-temperature (T>> ZZFF) almost free spin) almost free spin

Georges, Kotliar, Krauth and Rozenberg 1996Georges, Kotliar, Krauth and Rozenberg 1996

LHBLHB UHBUHB

quasip. peakquasip. peak

Independent electron picture not adequateIndependent electron picture not adequateYields both bandlike and localized behaviourYields both bandlike and localized behaviourFavor local magnetic momentsFavor local magnetic momentsLead to a conventional band spectrumLead to a conventional band spectrum

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Diagrammatic Theory of Strongly Correlated Electron Systems