Surprising Effects of the Interaction between Electrons in Solids. Dieter Vollhardt

Transcription

1 Surprising Effects of the Interaction between Electrons in Solids Dieter Vollhardt Shanghai Jiao Tong University; April 6, 2016

2 Augsburg: founded in 15 BC Roman Emperor Augustus 63 BC 14 AD

3 Center founded in 1996 Center for Correlated Matter, Hangzhou 浙江大学关联物质研究中心 founded in 2012 Augsburg: founded in 15 BC

4 Surprising Effects of Electronic Correlations in Solids Outline: Peculiarities of quantum many-particle systems Electronic correlations in solids Dynamical mean-field theory: Models vs. materials Other developments & perspectives

5 Properties of solids Magnetite (Fe 3 O 4 ) tse-lan Microscopic explanation of the physical properties? O(10 23 ) interacting electrons + ions quantum many-particle problem

6 Peculiarities of Interacting Many-Particle Systems

7 Interacting many-particle systems Elementary ( bare ) particles + fundamental interactions # particles N effective ( quasi ) particles + effective interactions

8 Non-interacting electrons Spin = 1 2 Fermion N, Pauli exclusion principle Fermi-Dirac statistics Ground state: Fermi body/surface

9 Fermi gas: Ground state k z Fermi surface k y k x Fermi sphere

10 Fermi gas: Excited state k z Particle Fermi surface Hole k y k x Fermi sphere k-eigenstates: infinite life time Switch on repulsive interaction unsolvable many-body problem

11 Fermi liquid Standard model of condensed matter physics Fermi surface k z (Quasi-) Hole Landau (1956/58) 1-1 correspondence between one-particle states (k,σ) (Quasi-) Particle = elementary excitation k y k x Fermi sphere Well-defined k-states ("quasiparticles") with - finite life time - effective mass m* - effective interaction

12 Simple metals "Heavy Fermions" Steglich et al. (1979) C/T (mj/mole K 2 ) lim T 0 c T V T 2 (K 2 ) = γ m* m Potassium from Kittel s book m * m C/T (mj/mole K 2 ) m* 1000 m Result of elementary excitations (quasiparticles) T 2 (K 2 ) Stewart et al. (1983) CeCu 2 Si 2, UBe 13 : very heavy quasiparticles (Kondo impurity physics)

13 Interacting many-particle systems Elementary ( bare ) particles + fundamental interactions # particles N effective ( quasi ) particles + effective interactions Vacuum e Q r Q r Coulomb potential

14 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/ ξ Effective Yukawa potential

15 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

16 Interacting many-particle systems Elementary ( bare ) particles + fundamental interactions # particles N effective ( quasi ) particles + effective interactions Electrons in real solids e Held (2004) Q r Strong effective interaction of electrons in narrow orbitals

17 Die Festkörperphysik ist eine Schmutzphysik (Pauli) One shouldn t wallow in dirt (Pauli to Peierls, 1929) Fe 3 O 4 but dirt physics can be fundamental and universal Magnetic impurity coupled (J) to non-interacting (mobile) electrons J J T>T K (high energies) Asymptotically free local moment T<T K (low energies) Screening of moment (confinement) Kondo effect K F 1/ F T Ee Λ J ( ) N( E ) Prototypical interaction problem with running coupling constant J(Λ) QED, QCD

18 Interacting many-particle systems # particles N Entirely new phenomena, e.g., phase transitions Unpredicted behavior emerges We used to think that if we knew one, we knew two, because one and one are two. We are finding out that we must learn a great deal more about 'and'. Eddington ( ) More is different Anderson (1972)

19 Interacting many-particle systems # particles N Entirely new phenomena, e.g., phase transitions Unpredicted behavior emerges Examples: Superconductivity Magnetism Metal-insulator transition Traffic Weather Stock market

20 Interacting many-particle systems # particles N Entirely new phenomena, e.g., phase transitions Fe 3 O 4 Metal-insulator transition Why? Strong repulsion between electrons: Correlation effect

21 Correlations

22 Correlation [lat.]: con + relatio ("with relation") Correlations in mathematics, natural sciences: AB A B e.g., densities: n( r) n( r') n( r) n( r') = n 2 Definition of Correlations (I): Effects beyond factorization approximations (e.g., Hartree-Fock)

23 Temporal/spatial correlations in everyday life Correlation effect?

24 Temporal/spatial correlations in everyday life But: External periodic potential long-range order enforced no genuine correlation effect

25 Temporal/spatial correlations in everyday life Strong correlations! Time/space average (Hartree approximation) inappropriate

26 Electronic Correlations in Solids

27 Electronic Energy Bands Property Energy levels Character, Representation Example Insulators Atomic levels Spatially localized electrons: n σ i NaCl, solid Ne overlap of wave functions: matrix element t? Simple metals Narrow bands Broad bands n iσ Extended waves: n σ k n kσ Transition + rare earth metals (Fe, Ce, V 2 O 3 ) Na, Al ε k band overlap t band width W v lattice spacing: a 1 = = ε average time spent on atom: τ k k k 1 aw τ W

28 Electronic Energy Bands Property Energy levels Character, Representation Example Insulators Atomic levels Spatially localized electrons: n σ i NaCl, solid Ne overlap of wave functions: matrix element t Correlated metals Simple metals Narrow bands Broad bands n iσ Extended waves: n σ k n kσ Transition + rare earth metals (Fe, Ce, V 2 O 3 ) Na, Al v ε k band overlap t band width W lattice spacing: a 1 = = ε average time spent on atom: τ k k k 1 aw τ W Small W: Strong electronic correlations

29 Narrow d,f-bands electronic correlations important

30 Correlated electron materials have unusual properties huge resistivity changes gigantic volume anomalies colossal magnetoresistance high-t c superconductivity metallic behavior at interfaces of insulators With potential for technological applications: sensors, switches } spintronics thermoelectrics How high-t c superconductor devices functional materials: oxide heterostructures to study correlated systems theoretically?

31 Understand materials modeling Þ realistic model Þ maximal reduction for fermions: Hubbard model

32 Hubbard model Gutzwiller, 1963 Hubbard, 1963 Kanamori, 1963 time H = t c c + U n n ij,, σ iσ jσ i i i Dimension of Hilbert space L: # lattice sites L O(4 ) n n n n i i i i Static (Hartree-Fock) mean-field theories generally insufficient Computational time for N 2 molecule: ca. 1 year with compute nodes

33 Hubbard model Gutzwiller, 1963 Hubbard, 1963 Kanamori, 1963 time H = t c c + U n n ij,, σ iσ jσ i i i Purely numerical approaches (d=2,3): hopeless Theoretical challenge of many-fermion problems: Construct reliable, comprehensive, non-perturbative approximation schemes n n n n i i i i Static (Hartree-Fock) mean-field theories generally insufficient

34 Non-perturbative approximation schemes for real materials, GGA How to combine? time Held (2004)

35 Dynamical Mean-Field Theory (DMFT) of Correlated Electrons

36 Limit d or Z of lattice electrons H = t c c + U n n ij,, σ iσ jσ i i i Hubbard model Face-centered cubic lattice (d=3) time Metzner, DV (1989) dz, dynamical mean-field Z=12 Self-consistent single-impurity Anderson model Georges, Kotliar (1992))

37 Dynamical mean-field theory (DMFT) of correlated electrons Σ( k, ω) Σ( ω) U Metzner, DV (1989) Georges, Kotliar (1992) Kotliar, DV (2004) Dynamics of local electronic interaction described exactly Spectral function Landau quasiparticles, not electrons Mott insulator

38 Dynamical mean-field theory (DMFT) of correlated electrons Σ( k, ω) Σ( ω) U Metzner, DV (1989) Georges, Kotliar (1992) Kotliar, DV (2004) Dynamics of local electronic interaction described exactly Spectral function Definition of Correlations (II): Transfer of spectral weight Experimentally detectable!

39 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 et al. (1997) Lichtenstein, Katsnelson (1998) Held et al. (2003) Kotliar et al. (2006)

40 Computational scheme for correlated electron materials: Material specific electronic structure (Density functional theory: LDA, GGA,...) or GW + Local electronic correlations (Many-body theory: DMFT) = X+DMFT X=LDA, GGA; GW, Solve self-consistently with an impurity solver, e.g., QMC, NRG, ED,...

41 Goal: Dynamical mean-field approach with predictive power for strongly correlated materials Research Unit FOR

42 LDA+DMFT: Application 1 (Sr,Ca)VO 3 : 3d 1 system

43 Theory Electronic structure V O V = 180 V O V 162 No correlation effects/spectral transfer

44 LDA+DMFT results Osaka Augsburg Ekaterinburg collaboration, (2004, 2005) / CaVO 3 spectral function constrained LDA: U=5.55 ev, J=1.0 ev Measure by spectroscopy

45 Comparison with experiment Bulk sensitive photoemission spectroscopy occupied states Osaka Augsburg Ekaterinburg collaboration, (2004, 2005) X-ray absorption spectroscopy unoccupied states T. Yoshida et al. (arxiv: ) 3-peak structure detected

46 LDA+DMFT: Application 2 Fe Most abundant element by mass on Earth Ferromagnetism: Exceptionally high Curie temperature (T C = 1043 K) Still most widely used metal in modern day industry ( iron age )

47 Fe T struct austenite ferrite hexaferrum DFT/GGA: Paramagnetic α-phase unstable (i) Why is the paramagnetic α-phase stable? (ii) How to compute T struct?

48 Until recently: LDA+DMFT investigations of correlated materials for given lattice structure How do electrons + ions influence each other? Which lattice structure is stabilized?

49 Investigation of the structural stability of Fe Collaborators: Ivan Leonov (Augsburg) Vladimir Anisimov (Ekaterinburg) Alexander Poteryaev (Ekaterinburg)

50 Fe T struct austenite ferrite hexaferrum GGA+DMFT: Electronic correlations (local repulsion) - increase unit cell volume correct density & compressibility - stabilize paramagnetic α-phase T struct > T C - cause the bcc-fcc structural phase transition Leonov, Poteryaev, Anisimov, DV (2011)

51 Lattice dynamics of paramagnetic α-fe Non-magnetic GGA phonon dispersion Pressure, GPa 1. Brillouin zone Dynamically unstable + elastically unstable (C 11, C < 0) Experiment: Neuhaus, Petry, Krimmel (1997)

52 Lattice dynamics of paramagnetic α-fe phonon frequencies calculated with frozen-phonon method harmonic approximation Pressure, GPa GGA+DMFT phonon dispersion at 1.2 T C Calculated: equilibrium lattice constant a~2.883 Å (a exp ~2.897 Å) Debye temperature Θ~458 K Theory: Leonov, Poteryaev, Anisimov, DV (2012) Experiment: Neuhaus, Petry, Krimmel (1997) Recently: Correlation-driven topological Fermi surface transition in FeSe Leonov et al., PRL (2015)

53 Other Developments & Perspectives

54 Correlated electrons in non-equilibrium Real-time evolution of correlation phenomena, e.g., time-resolved photoemission spectroscopy spectral function insulator ± 1T-TaS 2 metal Perfetti et al. (2006) Required: Theory of non-equilibrium in correlated bulk materials

55 Correlated electrons in non-equilibrium Non-equilibrium DMFT Freericks, Turkowski (2006) Eckstein (2009) Sudden quench in Hubbard model from U=0 to U>0 Momentum distribution (U=3.3) Eckstein, Kollar, Werner (2010) Thermalization time after quench Extension & application, e.g., to time resolved PES pump-probe experiments

56 ± Cu Correlated many-particle systems: More surprising than ever