Center for Electronic Correlations and Magnetism University of Augsburg Surprising Effects of Electronic Correlations in Solids Dieter Vollhardt Supported by TRR 80 FOR 1346 University of Florida, Gainesville; February 17, 2011
Outline: Peculiarities of quantum many-particle systems Correlations Electronic correlations in solids Dynamical Mean-Field Theory: Models vs. materials Other Developments & Perspectives
Peculiarities of Interacting Many-Particle Systems
Interacting many-particle systems Elementary ( bare ) particles + fundamental interactions # particles N effective ( quasi ) particles + effective interactions
Non-interacting electrons Spin = 1 2 Fermion N Pauli exclusion principle of many fermions Fermi-Dirac statistics Ground state: Fermi body/surface
Fermi gas: Ground state k z Fermi surface k y k x Fermi body
Fermi gas: Excited state k z Particle Fermi surface (k F ) Hole k y k x Fermi body k-eigenstates: infinite life time Switch on repulsive interaction
Fermi liquid Standard model of condensed matter physics Fermi surface (k F ) k z (Quasi-) Hole (Quasi-) Particle Landau (1956/58) 1-1 correspondence between one-particle states (k,σ) = elementary excitation k y k x Fermi body Well-defined k-states ("quasiparticles") with - finite life time - effective mass - effective interaction
Simple metals "Heavy Fermions" Steglich et al. (1979) C/T (mj/mole K 2 ) T 2 (K 2 ) cv m * lim, v T 0 T m m* m Potassium F k m F * C/T (mj/mole K 2 ) m* 1000 m Result of of elementary excitations (quasiparticles) T 2 (K 2 ) CeCu 2 Si 2, UBe 13 : very heavy quasiparticles Stewart et al. (1983)
Interacting many-particle systems Elementary ( bare ) particles + fundamental interactions # particles N effective ( quasi ) particles + effective interactions
Interacting many-particle systems Elementary ( bare ) particles + fundamental interactions # particles N effective ( quasi ) particles + effective interactions Vacuum e Q r Q r Coulomb potential
Interacting many-particle systems Elementary ( bare ) particles + fundamental interactions # particles N effective ( quasi ) particles + effective interactions Electron gas: Screening Simplest approximation: Thomas-Fermi e Q r Q e r r/ Effective Yukawa potential
Interacting many-particle systems Elementary ( bare ) particles + fundamental interactions # particles N effective ( quasi ) particles + effective interactions Electron gas: Screening Better approximation: Lindhard e Q r Q cos(2 kr ) 3 F r Friedel oscillations
Interacting many-particle systems Elementary ( bare ) particles + fundamental interactions # particles N effective ( quasi ) particles + effective interactions Electrons in real solids Q r e Held (2004) Strong effective interaction of electrons in localized orbitals
Interacting many-particle systems # particles N Entirely new phenomena, e.g., phase transitions Unpredicted behavior emerges We We used to to think that that if if we we knew one, one, we we knew two, two, because one one and and one one are are two. two. We We are are finding out out that that we we must learn a great deal deal more about 'and'. Arthur Arthur Eddington Eddington (1882-1944) (1882-1944) More is different Anderson (1972)
Correlations
Correlation [lat.]: con + relatio ("with relation") Grammar: either... or Correlations in mathematics, natural sciences: AB A B e.g., densities: () r (') r () r (') r Correlations: Effects beyond factorization approximations (e.g., Hartree-Fock)
Temporal/spatial correlations in everyday life Beware: External periodic potential
Temporal/spatial correlations in everyday life Time/space average insufficient
Electronic Correlations in Solids
Partially filled d-orbitals Partially filled f-orbitals Narrow d,f-orbitals/bands strong electronic correlations
Correlated electron materials Fascinating topics for fundamental research large resistivity changes gigantic volume changes high-t c superconductivity strong thermoelectric response colossal magnetoresistance huge multiferroic effects large susceptibilities with Technological applications: sensors, switches magnetic storage thermoelectrics functional materials,...
Electronic Correlations: Models
Hubbard model Gutzwiller, 1963 Hubbard, 1963 Kanamori, 1963 time H t c c U n n ij,, i j i i i n n n n i i i i Purely numerical approaches (d=2,3): hopeless Theoretical challenge: Construct reliable, comprehensive non-perturbative approximation scheme Static (Hartree-Fock-type) mean-field theories generally insufficient
Dynamical Mean-Field Theory (DMFT) of Correlated Electrons
Theory of correlated electrons H t c c U n n ij,, i j i i i Hubbard model Face-centered cubic lattice (d=3) time Metzner, DV (1989) dz, Strong simplifications dynamical mean-field Selfconsistency problem Dynamical single-site mean-field theory Z=12 Müller-Hartmann (1989); Brandt, Mielsch (1989); Janiš (1991); Janiš, DV (1992)
Theory of correlated electrons H t c c U n n ij,, i j i i i Hubbard model Face-centered cubic lattice (d=3) time Metzner, DV (1989) dz, Strong simplifications dynamical mean-field Selfconsistency problem Self-consistent single-impurity Anderson model Z=12 Georges, Kotliar (1992); Jarrell (1992)
DMFT self-consistency equations for G and Σ i G( ) Σ(ω): mean field ( dynamical potential ) (i) Effective single impurity problem: local propagator,... single-site ("impurity") action A (ii) k-integrated Dyson equation ( lattice Green function : lattice enters) G 0 ( ( )) free electrons in a dynamic potential Σ(ω) Solve with an impurity solver, e.g., QMC, NRG, ED,...
Dynamical mean-field theory (DMFT) of correlated electrons U Proper time resolved treatment of local electronic interactions Spectral function U Spectral transfer Experimentally detectable?
Correlated Electron: Materials
Held (2004) How to combine? time Held (2004)
Computational scheme for correlated electron materials: Material specific electronic structure (Density functional theory: LDA, GGA,...) or GW + Local electronic correlations (Many-body theory: DMFT) LDA+DMFT Anisimov, Poteryaev, Korotin, Anokhin, Kotliar (1997) Lichtenstein, Katsnelson (1998) Nekrasov, Held, Blümer, Poteryaev, Anisimov, DV (2000)
Application of LDA+DMFT (Sr,Ca)VO 3 : 3d 1 system
Theory Electronic structure V O V 180 V O V 162 No correlation effects/spectral transfer
LDA+DMFT results k-integrated spectral function / CaVO 3 constrained LDA: U=5.55 ev, J=1.0 ev Osaka Augsburg Ekaterinburg collaboration: Sekiyama, Fujiwara, Imada, Suga, Eisaki, Uchida, Takegahara, Harima, Saitoh, Nekrasov, Keller, Kondakov, Kozhevnikov, Pruschke, Held, DV, Anisimov (2004) Measure by spectroscopy
Comparison with experiment Bulk sensitive photoemission spectroscopy occupied states Osaka Augsburg Ekaterinburg collaboration, (2004, 2005) X-ray absorption spectroscopy unoccupied states 3-peak structure detected
State-of-the-art LDA+DMFT: Electronic correlations & structural transformations Electron correlations can induce structural transformations (2011) Fe Pressure, GPa
State-of-the-art LDA+DMFT arxiv:1012.2412
Perspective of the LDA+DMFT approach Explain and predict properties of complex correlated materials Phase diagram of light actinide series Temperature ( 0 C) Element Phase diagram connecting individual binary alloy diagrams Black: two-phase regions; Brown : details unknown Boring and Smith (2000)
Perspective of the LDA+DMFT approach Explain and predict properties of complex correlated materials Cu Phase diagram of La 1-x Sr x MnO 3 Hemberger et al. (2002) 1, 2, multi-electron transfer in metalloprotein complexes Photosynthesis
Developments & Perspectives
1. Quantum phase transitions Non-Fermi liquid Quantum critical ordered non-ordered Quantum critical point DFG Research Unit Augsburg-Dresden- Göttingen-Karlsruhe-Köln-München Driven by quantum fluctuations S. Sachdev, Quantum Phase Transitions, 2 nd ed. (Cambridge, 2011)
Field induced quantum phase transition magnetic transition Non-Fermi liquid antiferromagnetic Quantum critical non-magnetic Quantum critical point H DFG Research Unit Augsburg-Dresden- Göttingen-Karlsruhe-Köln-München Driven by quantum fluctuations Non-Fermi liquid behavior Emergence of novel degrees of freedom New phases of matter
Field induced quantum phase transition Driven by quantum fluctuations Custers, Gegenwart, Wilhelm, Neumaier, Tokiwa, Trovarelli, Geibel, Steglich, Pepin, Coleman (2003) Non-Fermi liquid behavior Emergence of novel degrees of freedom New phases of matter Si, Steglich (2010)
2. Correlated electrons in non-equilibrium Real-time evolution of correlation phenomena, e.g., time-resolved photoemission spectroscopy Photoelectron intensity in Mott insulating phase insulator metal Perfetti et al. (2006) Required: Theory of non-equilibrium beyond linear response in correlated bulk materials
2. Correlated electrons in non-equilibrium Non-equilibrium DMFT Quench in Hubbard model from U=0 to U>0 Freericks, Turkowski (2006) Eckstein, Kollar, Werner (2009) Momentum distribution (U=3.3) Thermalization Application, e.g. time resolved PES pump-probe experiments time after quench Eckstein, Kollar (2008) Freericks, Krishnamurthy, Pruschke (2008)
3. Correlated cold atoms in optical lattices Bosonic/fermionic atoms in optical lattices: Exp. realization of models High degree of tunability: quantum simulator Greiner et al. (2002) Observation of Fermi surface ( 40 K atoms) Köhl, Esslinger (2006)
3. Correlated cold atoms in optical lattices Hubbard model with ultracold atoms Jaksch, Bruder, Cirac, Gardiner, Zoller (1998) Atomic total angular momentum L tot = F N=2F+1 hyperfine states SU(N) Hubbard models Honerkamp, Hofstetter (2004) N=3, e.g. 6 Li, U<0: Color superconductivity, baryon formation (QCD) Rapp, Zarand, Honerkamp, Hofstetter (2006) Cooper pairs
Cu Non-Fermi liquid Quantum critical point Correlated many-particle systems: More fascinating than ever