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Theoretical Developments in the Field of Strongly Correlated Electrons:
Where we are and where we go
Workshop on Strongly Correlated Transition Metal Compounds IICologne, September 11-14 2006
Dieter Vollhardt
Supported by Deutsche Forschungsgemeinschaft through Collaborative Research Center 484
•DMFT•LDA+DMFT
Computationalschemes
•Models/Materials:
d=0: Quantum impurities/dots/nanoparticles
d=1: Chainsd=2: Surfaces/interfacesd=3: Bulk
•Cold atoms in optical lattices
Physical systems•ucSC•HTS•Kinks
•(Non)-Fermi liquids•Luttinger liquids
•Insulators
•Quantum phase transitions
•Non-equilibrium Phenomena
•Complex orderingCMRmultiferroicsnon-collinear magnetismorbital orderingfrustrationDMS
•Electron transfer in biological systems
Concepts and Phenomena
•QMC
•NRG•DMRG•fRG•Flow eqs.
•Bethe ansatz•Exact ground states
Theoretical methods
Strongly correlated electrons: Research topics (theory)
•DFT
Strongly correlated electrons: Research topics (theory)
•DMFT•LDA+DMFT
Computationalschemes
•Models/Materials:
d=0: Quantum impurities/dots/nanoparticles
d=1: Chainsd=2: Surfaces/interfacesd=3: Bulk
•Cold atoms in optical lattices
Physical systems
•QMC
•NRG•DMRG•fRG•Flow eqs.
•Bethe ansatz•Exact ground states
Theoretical methods
Recent reviews + books (theory)
•ucSC•HTS•Kinks
•(Non)-Fermi liquids•Luttinger liquids
•Insulators
•Quantum phase transitions
•Non-equilibrium Phenomena
•Complex orderingCMRmultiferroicsnon-collinear magnetismorbital orderingfrustrationDMS
•Electron transfer in biological systems
Concepts and Phenomena
•DFT
Proper time resolved treatment of local electronic interactions:
Dynamical Mean-Field Theory (DMFT)
Metzner, Vollhardt (1989)Müller-Hartmann (1989)
Georges, Kotliar (1992)Jarrell (1992)
• Best single-site MFT for correlated lattice fermions• Very good approximation for many 3D systems
Many advances in our understanding of, e.g.:
• Correlation phenomena at intermediate couplings• Mott-Hubbard metal-insulator transition
Georges, Kotliar, Krauth, Rozenberg (1996)Blümer (2003)
2D κ-organicsParis-Karlsruhe-Angers collaboration, Limelette et al. (2003)
Ca2−xSrxRuO4 Anisimov et al. (2002)
Orbital-selective Mott transitions Liebsch (2003, 2004)Koga, Kawakami, Rice Sigrist (2004, 2005)de’ Medici, Georges, Biermann (2005)Arita, Held (2005)Knecht, Blümer, van Dongen (2005)Liebsch, Costi (2006)
Hubbard IIPTEDNCAQMC NRG
Recent:PQMC Feldbacher, Held, Assaad (2004)DDMRG Hallberg (1995)
Ramasesha et al. (1997)Jeckelmann (2002)
CT-QMC Rubtsov, Savkin, Lichtenstein (2005)
DMFT: search for the “best” impurity solverDMFT: search for the “best” impurity solver
•Dynamical cluster approx. (DCA) Hettler et al. (1998, 2000)•Cluster DMFT (CDMFT) Kotliar et al. (2001)•Self-energy functional theory Potthoff (2003)
Σ(ω)
G( )ω
i ⇒
Beyond DMFT
Cluster Extensions
Dynamical vertex approximation (DΓA)
Local + non-local self-energy diagrams from local irred. vertexToschi, Katanin, Held (2006)
2D Hubbard model; 4-site cluster approach Lichtenstein, Katsnelson (2000)
Application 1: Coexistence of AF + dSC
2x2 CDMFT Capone, Kotliar (2006) Functional RG Metzner et al. (2005)
How to overcome cluster size limitations?How to overcome cluster size limitations?
Anisimov et al. (1997)Lichtenstein, Katsnelson (1998)
Kotliar, Vollhardt (2004)
Held (2005)Kotliar et al. (2006)“LDA+DMFT“
Material specific electronic structure (Density functional theory: LDA,GW, KKR, ...)
+Local electronic correlations
(Many-body theory: DMFT)
Electronic structure calculations with DMFT
VO2: a two-fluid incoherent metal?LDA+DMFT(IPT)Laad, Craco, Müller-Hartmann (2005)
-6 eVsatellite
LDSA
Ferromagnetic Ni LDA+DMFT(QMC)Lichtenstein, Katsnelson, Kotliar (2004)(Sr,Ca)VO3 LDA+DMFT(QMC)
Osaka – Augsburg – Ekaterinburg collaboration, Sekiyama et al. (2004, 2005)
XAS (at O K-edge)PES
Phonon spectrum of δ–PuLDA+DMFT(Hubb. I) Dai et al. (2003)
Application 2: Kinks in strongly correlated electron systems
Ekaterinburg – Augsburg – Stuttgart collaboration,Nekrasov et al. (2004, 2006)
Renormalization of LDA-bands by self-energy
SrVO3
0.2 eVω ≈Kinks at
Yoshida et al. (2005)
Origin of kinks in a purely electronic theory?
Strongly correlated paramagnetic metal
⇓KKT
•Meaning of ω* ? •Range of FL region ? •Consequences of ω* ?Byczuk, Kollar, Held, Yang, Nekrasov, Pruschke, DV (2006)
*linear flinear for
or
1
( )( ) ( [ ( ]) ( ))DMFT
GG G
ωω ω
ωω ω ω ωμ
≤Ω≤
⎯⎯⎯→Σ = − − Δ+ hybridization fct.
FL regime
FL regime
Byczuk, Kollar, Held, Yang, Nekrasov, Pruschke, DV (2006)
SrVO3
Kinks in effective dispersion: Generic features of strongly correlated electrons; Byczuk, Kollar, Held, Yang, Nekrasov, Pruschke, DV (2006) HTS: * 0.2 0.4 eVω ≈ −
analyt. given by Z + non-interact. quantities
0* FLZ Dω =
Charact. scale of non-interact. system
0.2 eV , / * 0.35; ' , ' 0.65; 1.5 eV 0.2 e V
LDA
LDAFL FLZ E Z m m
EZ E c Z
ωω
⎧ = =⎪= ⎨ ± =⎪⎩
≤≥ ≥
kk
k
FL regime
outside FL regime
SrVO3
Graf et al., cond-mat 0607319
Bi2212
• How to optimize/minimize material-specific input?
• Manageable, fully self-consistent schemes including clusters?
•Coupling to non-local phonons(q 0 fluctuations of collective modes)?
• Combination with Molecular Dynamics?
• How to optimize/minimize material-specific input?
• Manageable, fully self-consistent schemes including clusters?
•Coupling to non-local phonons(q 0 fluctuations of collective modes)?
• Combination with Molecular Dynamics?
Electronic structure calculations with DMFT
Insulators
Metal InsulatorBand insulator
Mott insulator
Fermi surfaceFermi liquid theory
“Luttinger surface“
Poles of ( )G ωk Zeros of ( )G ωk
Luttinger Theorem: valid not valid
Essler, Tsvelik (2002, 2005)Dzyaloshinskii (2003)Konik, Rice, Tsvelik (2005)Yang, Rice, Zhang (2006)Stanescu, Phillips, Choy (2006)Berthod, Giamarchi, Biermann,
Georges (2006)
Rosch (2006)
Precise conditions for breakdown of Luttinger theorem?Precise conditions for breakdown of Luttinger theorem?
Band insulator Mott insulator
qualitatively different?
Two-band Hubbard model (analytic): Smooth crossover Rosch (2006)
Two-plane Hubbard model (DMFT)
Smooth crossoverFuhrmann, Heilmann, Monien (2006)
Ionic Hubbard model (DMFT)
Garg, Krishnamurthy, Randeria (2006)
Phase transitions
Need better understanding of insulatorsNeed better understanding of insulators
Insulators: Interfaces and Heterostructures
Interfaces of correlated electronic systems: „Electronic reconstruction”Hesper, Tjeng, Heeres, Sawatzky (2000)
• Interfaces always present• Most devices interface driven
field doping
? Similar effect ?chemical doping
Mannhart, Schlom, Bednorz, Müller (1991)2. Phase transitions tunable by external (gate) fields, e.g., field doping
1. Surfaces: Atomic reconstruction
SrTiO3/LaTiO3 layers: Electronic reconstruction with metallic conductivityOhtomo, Müller, Grazul, Hwang (2002)
SrTiO3/LaAlO3 interfaces high mobility transistorsThiel, Hammerl, Schmehl, Schneider, Mannhart (2006)
Exp.:
Theory
Unrestricted Hartree-Fock: near-interface region metallic + ferromagnetic
Okamoto, Millis (2004)DMFT (2-site):Reduced carrier density Okamoto, Millis (2005)
Slave bosons: Similar Rüegg, Sigrist (unpubl.)
Hartree-Fock + DMFT Lee, MacDonald (2006)
LaAlO3layers in SrTiO3
Pavlenko, Elfimov, Kopp, Sawatzky (2006)
charge | layer type
Finite interface hole density affects superconducting properties,
YBa2Cu3O6 / SrTiO3 sandwich: LSDA + U
z
Confirmed: Metallic states at interfaces of bulk insulators
•How to construct realistic models?•Size of renormalized parameters?•Influence of correlations?•Stable configurations (atomic + electronic reconstruction)?•Tunable by electric field?•Role of impurities/defects?
•How to construct realistic models?•Size of renormalized parameters?•Influence of correlations?•Stable configurations (atomic + electronic reconstruction)?•Tunable by electric field?•Role of impurities/defects?
Engineering of devices with multifunctional properties
Surfaces in metals
Osaka – Augsburg – Ekaterinburg collaboration: Sekiyama et al. (2004)
Why are surface spectra of (Sr,Ca)VO3 so different?Why are surface spectra of (Sr,Ca)VO3 so different?
Transport beyond Real time evolution linear response
Nordlander et al. (1999)“How long does it take for the Kondo effect to develop?”
Goldhaber-Gordon et al., (1998)
Strongly correlated electrons in non-equilibrium
Tim
e-de
p. s
pect
ral d
ensi
ty
Time-resolved optical photoemission:Femto-sec pulse Tel≠Tlatt
Perfetti et al. (2006)
Kaindl et al. (2000)Iwai et al. (2003)Cavalleri et al. (2004)Chollet et al. (2005)
Recent advances in time-dependent RG-methods:
• time-dependent NRG Costi (1997), Anders, Schiller(2005)
• time-dependent DMRG Schollwöck, White (2006)
•frequency-dependent renormalization group Rosch, Kroha, Wölfle (2001)
• flow equation method Kehrein (2001)
Functional renormalization group (fRG)
fRG: Exact hierarchy of differential flow equations for the effective m-particle interactions with energy cutoff as flow parameter
New approximation scheme (presently: weak coupling, d=2,1,0)
Metzner et al. (2005)
Applicationsd=2: Unbiased stability analysis of Hubbard model
Coexistence of AFM + dSC (approx.)Full
d=1: Impurities in Luttinger liquidsaccurate description of complex low-energy behavior
d=0: Linear transport through quantum dots
Zanchi, Schulz (2000)Halboth, Metzner (2000)Honerkamp et al. (2001)
( )ωΣ Katanin, Kampf (2004)
Meden et al. (2002)Andergassen et al. (2004) Enss et al. (2005)
Meden, Marquardt (2005)
•How to build in fluctuating OP fields ( spont. symm. breaking, quantum criticality)?•Non-equilibrium?
Difficulty:Non-equilibrium beyond linear response
Transport through a Kondo impurity
Kondo impurity with applied voltage bias V>>TK
voltage bias current shot noise
decoherence
→→↓
→temperature
Kehrein (2005)Quantitative result Tequ(V)
Fermi searight
Fermi sealeft
Spin-1/2��������������������������������������������������������������������������������
��������������������������������������������������������������������������������
��������������������������������
��������������������������������
Paaske, Rosch, Kroha, Wölfle (2004)
Flow equations non-equilibrium spin dynamics Kehrein (2006)
Non-equilibrium steady state ≠ thermal equilibrium state
0 2.5 5 7.5 10
ω / TK
0
0.025
0.05
0.075
0.1
C(ω
) x
TK
V/TK
= 32V/T
K = 16
V/TK
= 8
0 2.5 5 7.5 10
ω / TK
0
0.005
0.01
0.015
0.02
0.025
χ’’(ω
) x
TK
V/TK
= 32V/T
K = 16
V/TK
= 8
Spin
-spi
n co
rrel
atio
n fu
ncti
on
0T =≠
Kondo model with voltage bias
Fluctuation-dissipation theorem not valid
• Properties of ac-driven correlated impurity models? • Inclusion of dissipation?• Non-equilibrium/transport beyond linear response in bulk materials?⇒ non-equilibrium DMFT Freericks, Turkowski, Zlatic (2006)
• Time-development of Mott-Hubbard gap? • Field-driven correlated systems: New collective behavior? • Quantum criticality and non-equilibrium?
• Properties of ac-driven correlated impurity models? • Inclusion of dissipation?• Non-equilibrium/transport beyond linear response in bulk materials?⇒ non-equilibrium DMFT Freericks, Turkowski, Zlatic (2006)
• Time-development of Mott-Hubbard gap? • Field-driven correlated systems: New collective behavior? • Quantum criticality and non-equilibrium?
SC
Strongly correlated electrons in non-equilibrium
Correlated matter in optical lattices
Fermionic atoms in optical lattices Modugno et al. (2003)Köhl et al. (2005)
Observation of Fermi surface
Köhl, Esslinger (2006)
Jaksch, Bruder, Cirac, Gardiner, Zoller (1998)Experimental realization and test of itinerant quantum models
Hubbard model with ultracold atoms
SU(N) Hubbard models Honerkamp, Hofstetter (2004)
S= 1/2 N=2 spin states ElectronsLtot = F N=2F+1 hyperfine states Atoms
N=3, e.g. 6Li, U<0: Color superconductivity, baryon formation (QCD)Rapp, Zarand, Honerkamp, Hofstetter (2006)
“From neutron stars tobaryonic matter in the lab”
Theoretically desired experiments:
Tune/observe • Mott-Hubbard MIT• magnetic states• non-homogeneous matter• disorder (Anderson localization, glassy states,…)• Correlated, disordered electron systems
Byczuk, Hofstetter, DV (2005)
•Fermions/Bosons with higher spin• …
• Mott-Hubbard MIT• magnetic states• non-homogeneous matter• disorder (Anderson localization, glassy states,…)• Correlated, disordered electron systems
Byczuk, Hofstetter, DV (2005)
•Fermions/Bosons with higher spin• …
Correlated electron physics:More fascinating than ever!