Diagrammatic Theory of Strongly
Correlated Electron Systems
Outline
• 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
Outline
• 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
Use of HTc
Magnetic levitation (Japan 1999, 343 m.p.h)
Magnetic resonance imaging
Fault current limiters of 6.4MVA, response time ms
E-bombs (strong EM pulse)
5000-horsepower motor made with sc wire
(July 2001)
Electric generators, 99% efficiency
Energy storage 3MW
Use of HTc
Underground cable in Copenhagen (for 150000
citizens,30 meters long, May 2001)
Researching the possibility to build petaflop
computers
Market $200 billion by the year 2010
Materials undergoing MIT
High temperature superconductors (2D systems, transition with doping)
Other 3d transition metal oxides (Nickel,Vanadium,Titanium,…)
2D and 3D, transition with doping or pressure
Many f-electron systems
Hubbard model –
generic model for materials undergoing MIT
E= -2t2/U
E= 0
Dynamical mean-field theory & MIT
mapping
fermionic bath
U
Zhang, Rozenberg and Kotliar 1992
Doping Mott insulator –
DMFT perspective
Metallic system always Fermi liquid ImS(w)w2
Fermi surface unchanged (volume and shape)
Narrow quasiparticle peak of width ZeFd at the Fermi level
Effective mass (m*/m1/Z) diverges at the transition
High-temperature (T>> ZeF) almost free spin
LHB
UHB
quasip. peak
Georges, Kotliar, Krauth and Rozenberg 1996
d
Outline
• 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
Nonlocal interaction in DMFT?
Local quantum fluctuations (between states
)
completely taken into account within DMFT
Nonlocal quantum fluctuations are mostly lost in DMFT (nonlocal RKKY inter.)
(residual ground-state entropy of par. Mott insulator is ln2  2N deg. states)
Why?
Metzner Vollhardt 89
mean-field description of the exchange term is exact within DMFT
J disappears completely in the paramagnetic phase !
How does intersite exchange J
change Mott transition?
Hubbard model
For simplicity, take the infinite U limit  t-J model:
Outline
• 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
Extended DMFT
J and t equally important:
Si & Smith 96,
Kajuter & Kotliar 96
mapping
fermionic bath
bosonic bath
fluctuating magnetic field
Source of the inelasting scattering
Still local and conserving theory
Long range fluctuations frozen
Strong inelasting scattering
due to local magnetic fluctuations
Local quantities can be calculated from the corresponding impurity problem
Fermion bubble is zero in
the paramagnetic state
Outline
• 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
Im
Pseudogap – Incoherent metal
highly incoherent response
Pseudogap due to strong inelasting scattering from local magnetic fluctuations
Not due to finite ranged fluctuating antiferromagnetic (superconducting) domains
Local spectral function
Outline
• 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
(m-ReS(0))/zt
Luttinger’s theorem?
A(k,w) d=0.02
A(k,0)
A(k,w)
ky
k
kx
White lines corresponds to noninteracting system
A(k,w) d=0.04
A(k,0)
A(k,w)
ky
k
kx
White lines corresponds to noninteracting system
A(k,w) d=0.06
A(k,0)
A(k,w)
ky
k
kx
White lines corresponds to noninteracting system
A(k,w) d=0.08
A(k,0)
A(k,w)
ky
k
kx
White lines corresponds to noninteracting system
A(k,w) d=0.10
A(k,0)
A(k,w)
ky
k
kx
White lines corresponds to noninteracting system
A(k,w) d=0.12
A(k,0)
A(k,w)
ky
k
kx
White lines corresponds to noninteracting system
A(k,w) d=0.14
A(k,0)
A(k,w)
ky
k
kx
White lines corresponds to noninteracting system
A(k,w) d=0.16
A(k,0)
A(k,w)
ky
k
kx
White lines corresponds to noninteracting system
A(k,w) d=0.18
A(k,0)
A(k,w)
ky
k
kx
White lines corresponds to noninteracting system
A(k,w) d=0.20
A(k,0)
A(k,w)
ky
k
kx
White lines corresponds to noninteracting system
A(k,w) d=0.22
A(k,0)
A(k,w)
ky
k
kx
White lines corresponds to noninteracting system
A(k,w) d=0.24
A(k,0)
A(k,w)
ky
k
kx
White lines corresponds to noninteracting system
Outline
• 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
Entropy
ED:
Jaklič & Prelovšek, 1995
Experiment:
LSCO (T/t0.07)
Cooper & Loram
EMDT+NCA
ED 20 sites
d&m
EMDT+NCA
ED 20 sites
Outline
• 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
Hall coefficient
T~1000K
LSCO: Nishikawa, Takeda & Sato (1994)
Outline
• 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
Outline
• 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
Motivation
•A need to solve the DMFT impurity problem
for real materials with orbital degeneracy
•Quantum dots in mesoscopic structures
Several methods available to solve AIM:
Numerical renormalization group (NRG)
Quantum Monte Carlo simulation (QMC)
Exact diagonalization (ED)
Iterated perturbation theory (IPT)
Resummations of perturbation theory (NCA, CTMA)
Either slow or less flexible
Auxiliary particle technique
NCA
Simple fast and flexible method
Works for T>0.2 TK
Works only in the case of U=
Naive extension very badly fails
TK several orders of magnitude too small
Outline
• 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
Luttinger-Ward functional for SUNCA
Outline
• 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
Scaling of TK
Comparison with NRG
Outline
• 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
Summary
EDMFT
• Purely local magnetic fluctuations can



induce pseudogap
suppress large entropy at low doping
induce strongly growing RH with decreasing T and d
• Luttinger’s theorem is not applicable in the
incoherent regime (d<0.20)
• Fermi liquid is recovered only when e*>J
SUNCA
• Infinite series of skeleton diagrams is needed to
recover correct low energy scale of the AIM at
finite Coulomb interaction U
Extended Dynamical Mean Field
Metal-insulator transition
el-el correlations not important:
band insulator:
•the lowest conduction band is full
(possible only for even number of electrons)
•gap due to the periodic potential – few eV
simple metal
•Conduction band partially occupied
semiconductor
zt
el-el correlations important:
Mott insulator despite the odd number of
electrons
Cannot be explained within the
independent-electron picture (many body
effect)
U
eF*
Several competing mechanisms and
several energy scales
Zhang, Rozenberg and Kotliar 1992
Doping Mott insulator –
DMFT perspective
Metallic system always Fermi liquid ImS(w)w2
Fermi surface unchanged (volume and shape)
Narrow quasiparticle peak of width ZeFd at the Fermi level
Effective mass (m*/m1/Z) diverges at the transition
High-temperature (T>> ZeF) almost free spin
LHB
UHB
quasip. peak
Georges, Kotliar, Krauth and Rozenberg 1996
d
Independent electron picture not adequate
Yields both bandlike and localized behaviour
Favor local magnetic moments
Lead to a conventional band spectrum