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

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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!