Electronic Correlation in Solids what is it and how to tackle it?

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1 Electronic Correlation in Solids what is it and how to tackle it? Silke Biermann Centre de Physique Théorique Ecole Polytechnique, Palaiseau, France p. 1

2 Correlations? What is it? p. 2

3 Correlations? Ashcroft-Mermin, Solid state physics gives the beyond Hartree-Fock definition : The correlation energy of an electronic system is the difference between the exact energy and its Hartree-Fock energy. p. 3

4 Correlations? Correlatio (lat.): mutual relationship The behavior of a given electron is not independent of the behavior of the others! p. 4

5 The standard model of solids: Cu atom 1d size.3. E F Energy (ev) F. Bloch W L Λ Γ X Z W K Electrons in a periodic potential occupy one-particle (Bloch) states, delocalised over the solid. feel each other only through an effective mean potential (and the Pauli principle). independent particle picture p. 5

6 Correlations? Correlatio (lat.): mutual relationship The behavior of a given electron is not independent of the behavior of the others! Mathematically: AB A B (1) p. 6

7 Correlations? 5 % have blue eyes 5 % have yellow eyes p. 7

8 Correlations? 5 % are left-handed 5 % are right-handed p. 8

9 Correlations? What s the probability for a left-handed yellow-eyed kangaroo??? p. 9

10 Correlations? probability for a left-handed yellow-eyed kangaroo = 1/2 1/2 = 1/4 only if the two properties are uncorrelated Otherwise: anything can happen... p. 1

11 Correlations? Correlatio (lat.): mutual relationship The behavior of a given electron is not independent of the behavior of the others! Mathematically: AB A B (2) For electrons (in a given atomic orbital): n n n n n σ = number operator for electrons with spin σ. p. 11

12 Correlations? Count electrons on a given atom in a given orbital: n σ = counts electrons with spin σ n n counts double-occupations n n = n n only if the second electron does not care about the orbital being already occupied or not p. 12

13 Exercise (!): Does n n = n n hold? 1. Hamiltonian: H = ǫ(n +n ) 2. Hamiltonian: H = ǫ(n +n )+Un n p. 13

14 Correlations n n = n n? (1) Hamiltonian: H = ǫ(n +n ) Operatorsn and n have eigenvalues and 1. p. 14

15 Correlations n n = n n? (1) Hamiltonian: H = ǫ(n +n ) Operatorsn and n have eigenvalues and 1. n n = 1 Z = 1 Z n =,1, n =,1 n =,1 = n n n e βǫn n n e βǫ(n +n ) n =,1 n e βǫn p. 14

16 Correlations n n = n n? (1) Hamiltonian: H = ǫ(n +n ) Operatorsn and n have eigenvalues and 1. n n = 1 Z = 1 Z n =,1, n =,1 n =,1 = n n n e βǫn n n e βǫ(n +n ) n =,1 No correlations! (Hamiltonian separable) n e βǫn p. 14

17 Correlations n n = n n? (2) Hamiltonian: H = ǫ(n +n )+Un n Operatorsn and n have eigenvalues and 1. p. 15

18 Correlations n n = n n? (2) Hamiltonian: H = ǫ(n +n )+Un n Operatorsn and n have eigenvalues and 1. n n = 1 Z n =,1, n =,1 n n n n e βǫ(n +n ) βun n p. 15

19 Correlations n n = n n? (2) Hamiltonian: H = ǫ(n +n )+Un n Operatorsn and n have eigenvalues and 1. n n = 1 Z n =,1, n =,1 n n n n e βǫ(n +n ) βun n Correlations! (Hamiltonian not separable) p. 15

20 Periodic array of sites with one orbital We can have n +n = 1 for each site, but yet n n = (insulator!) Is this possible within a one-particle picture? p. 16

21 Periodic array of sites with one orbital n +n = 1 for each site, and n n = only possible in a one-particle picture if we allow for symmetry breaking (e.g. magnetic), such that n n = p. 17

22 Mott s ficticious H-solid: Hydrogen atoms with lattice spacing 1 m H H H H H H H H H H H H H H H H H H H H H H H H (not to scale...) H H H H H H H H H H H H H H H H Metal or insulator? p. 18

23 Mott s ficticious H-solid: Hydrogen atoms with lattice spacing 1 m H H H H H H H H H H H H H H H H H H H H H H H H (not to scale...) H H H H H H H H H H H H H H H H Metal or insulator? Band structure: metal Reality: Mott insulator! p. 18

24 Mott s ficticious H-solid: Hydrogen atoms with lattice spacing 1 m H H H H H H H H H H H H H H H H H H H H H H H H (not to scale...) H H H H H H H H H H H H H H H H Metal or insulator? Band structure: metal Reality: Mott insulator! Coulomb repulsion dominates over kinetic energy! p. 18

25 Why does band theory work at all? p. 19

26 Band structure from photoemission Example: Copper p. 2

27 Why does band theory work at all? Band structure relies on one-electron picture But: electrons interact! Several answers...: Pauli principle Screening } reduce effects of interactions Landau s Fermi liquid theory: quasi-particles p. 21

28 The standard model (contd.) Landau theory of quasiparticles: one-particle picture as a low-energy theory with renormalized parameters k y (π/a) (a) (b) 2. SrVO3 atom size Intensity (arb. units) Energy (ev). -1. E F -2. R Λ Γ X Z M Σ Γ (c) SrVO 3 hν=9 ev (d) CaVO 3 hν=8 ev Energy relative to E F (ev) (cf. Patrick Rinke s lecture!) p. 22

29 Why does the standard model work? Band structure relies on one-electron picture But: electrons interact! Several answers...: Pauli principle Screening } reduce effects of interactions Landau s Fermi liquid theory: quasi-particles It does not always work... p. 23

30 YTiO 3 in LDA YTiO 3 : a distorted perovskite compound with d 1 configuration (i.e. 1 electron in t 2g orbitals), paramagnetic above 3 K. DFT-LDA results: 1. YTiO 3 DOS E E Fermi [ev] ( ) DFT-LDA = Density Functional Theory within the local density approximation p. 24

31 YTiO 3 : in reality... Photoemission reveals a (Mott) insulator: (Fujimori et al.) 1. LDA YTiO 3 DOS E E Fermi [ev] p. 25

32 YTiO 3 : in reality... Photoemission reveals a (Mott) insulator: (Fujimori et al.) 1. LDA YTiO 3 DOS E E Fermi [ev] How to produce a paramagnetic insulating state with 1 electron in 3 bands? not possible in band theory breakdown of independent particle picture p. 25

33 Further outline From N-particle to 1-particle Hamiltonians Problems of DFT-LDA Modelling correlated behavior: the Hubbard model The Mott metal-insulator transition Dynamical mean field theory Dynamical mean field theory within electronic structure calculations ( LDA+DMFT ) Examples: Typical examples: d 1 perovskites SrVO 3, CaVO 3, LaTiO 3, YTiO 3 A not typical one...: VO 2 Conclusions and perspectives p. 26

34 The N particle problem... and its mean-field solution: N-electron Schrödinger equation H N Ψ(r 1,r 2,...,r N ) = E N Ψ(r 1,r 2,...,r N ) with H N = H kinetic N +H external N i j e 2 r i r j becomes separable in mean-field theory: H N = i h i p. 27

35 For example, using the Hartree(-Fock) mean field: h i = h kinetic i +h external i +e 2 dr n(r) r i r Solutions are Slater determinants of one-particle states, fulfilling h i φ(r i ) = ǫφ(r i ) Bloch s theorem => use quantum numbers k, n for 1-particle states 1-particle energiesǫ kn => band structure of the solid p. 28

36 Density functional theory achieves a mapping onto a separable system (mapping of interacting system onto non-interacting system of the same density in an effective potential) for the ground state. However: effective potential unknown => local density approximation strictly speaking: not for excited states In practice (and with the above caveats): DFT-LDA can be viewed as a specific choice for a mean field p. 29

37 Electronic Correlations General definition: Electronic correlations are those effects of the interactions between electrons that cannot be described by a mean field. More specific definitions: Electronic correlations are effects beyond... Hartree(-Fock)... DFT-LDA ( )... the best possible one-particle picture (from Fujimori et al., 1992) p. 3

38 Two regimes of failures of LDA 1. weak coupling : moderate correlations, perturbative approaches work (e.g. GW approximation ) (see Patrick Rinke s lecture) 2. strong coupling : strong correlations, non-perturbative approaches needed (e.g dynamical mean field theory) NB. Traditionally two communities, different techniques, but which in recent years have started to merge... NB. Correlation effects can show up in some quantities more than in others! p. 31

39 Problems of DFT-LDA... 3% error in volume ofδ-pu by DFT-LDA ( ) α-γ transition in Ce not described by LDA correlation effects in Ni, Fe, Mn... LDA misses insulating phases of certain oxides (VO 2, V 2 O 3, LaTiO 3, YTiO 3, Ti 2 O 3...) bad description of spectra of some metallic compounds (SrVO 3, CaVO 3...) E.g. photoemission of YTiO 3 : (Fujimori et al.) p. 32

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41 Metal-Insulator Transitions Metal-insulator transition in VO 2 at T c =34 K Morin et al., 1959 p. 34

42 SrVO 3 : a correlated metal SrVO 3 within DFT-LDA Photoemission N i,i (E) SrVO3 N i,j (E) -1 1 E (ev) hν = 9 ev hν = 275 ev hν = 4.8 ev hν = 21.2 ev E (ev) Intensity (arb. units) SrVO 3 (x = ) Sr.5 Ca.5 VO 3 CaVO 3 (x = 1) Binding Energy (ev) (Sekiyama et al. 23) p. 35

43 The Hubbard model H = D 2 ( ) c iσ c jσ +c jσ c iσ +U i n i n i <ij>σ (Hubbard, 1963) Ground state at half-filling and finite U: antiferromagnetic Frustrated model paramagnetic solution? p. 36

44 Spectra for one atom Electron removal and addition spectra 1 ell.dat E = ǫ E = ǫ+u U=Coulomb interaction between two 1s electrons p. 37

45 Atomic limit: D= H = U i n i n i atomic eigenstates, localized in real space Spectral function = discrete peaks separated by U 1 ell.dat p. 38

46 Non-interacting limit: U= H = D 2 ( ) c iσ c jσ +c jσ c iσ = kσ ǫ k c kσ c kσ <ij>σ with e.g. ǫ k = D[cos(k x )+cos(k y )+cos(k z )) on a 3D square lattice (lattice constant 1) with nearest neighbor hopping. Spectral function = non-interacting DOS.6 DOS E E Fermi p. 39

47 Atomic and band-like spectra.5.4 U.6 DOS E E Fermi E E Fermi Spectral function ρ(ω) probes possibility of adding/removing an electron at energy ω. In non-interacting case: ρ(ω)= DOS. In general case: relaxation effects! In atomic limit : probe local Coulomb interaction p. 4

48 Hubbard model within DMFT ( ) H = D 2 ( ) c iσ c jσ +c jσ c iσ +U i n i n i <ij>σ (Hubbard, 1963) ρ(ω) 2 2 U/D=1 U/D=2 Quasi-particle peak Hubbard bands ImG 2 2 U/D=2.5 U/D=3 2 U/D=4 Georges&Kotliar ω /D ( ) DMFT = Dynamical Mean Field Theory, paramagnetic solution p. 41

49 Spectra of perovskites Photoemission hν = 9 ev hν = 275 ev hν = 4.8 ev hν = 21.2 ev Intensity (arb. units) SrVO 3 (x = ) Sr.5 Ca.5 VO 3 CaVO 3 (x = 1) Binding Energy (ev) (Sekiyama et al. 23) p. 42

50 Spectra of perovskites Photoemission 2 U/D=1 Intensity (arb. units) hν = 9 ev hν = 275 ev hν = 4.8 ev hν = 21.2 ev SrVO 3 (x = ) Sr.5 Ca.5 VO 3 ImG U/D=2 U/D=2.5 U/D=3 2 U/D=4 CaVO 3 (x = 1) Binding Energy (ev) ω /D (Sekiyama et al. 23) p. 43

51 Green s function survival kit ρ(ω) = 1 π IG ii(ω) Definition of Green s function: G ij (t) = ˆTc i (t)c j () Quasi-particles are poles of G(k,ω) = 1 ω +µ ǫ o (k) Σ(k,ω) All correlations are hidden in the self-energy: Σ(k,ω) = G 1 (k,ω) G 1 (k,ω) p. 44

52 Hubbard model within DMFT ( ) H = D 2 ( ) c iσ c jσ +c jσ c iσ +U i n i n i <ij>σ (Hubbard, 1963) ρ(ω) 2 2 U/D=1 U/D=2 Quasi-particle peak Hubbard bands ImG 2 2 U/D=2.5 U/D=3 2 U/D=4 Georges&Kotliar ω /D ( ) DMFT = Dynamical Mean Field Theory, paramagnetic solution p. 45

53 The mechanism? Σ (ω) diverges forω (i.e. at the Fermi level): Quasi-particle lifetime ( 1/Σ (ω = )) vanishes! Opening of a gap at the Fermi level ω = A(k,ω) = ImG(k,ω) 1 = Im ω +µ ǫ o (k) Σ(k,ω) = 1 Σ (k,ω) π(ω +µ ǫ o (k) Σ (k,ω)) 2 +Σ (k,ω) 2 Here (particle-hole symetry and local self-energy): A(k,ω) = 1 π Σ (ω) (ω ǫ o (k)) 2 +Σ (ω) 2 p. 46

54 What about the metal? In a Fermi liquid (local self-energy, for simplicity...): ImΣ(ω) = Γω 2 +O(ω 3 ) ReΣ(ω) = ReΣ()+(1 Z 1 )ω +O(ω 2 ) A(k,ω) = Z2 π IΣ(ω) (ω Zǫ (k)) 2 +( ZIΣ(ω)) 2+A inkoh For small ImΣ(i.e. well-defined quasi-particles): Lorentzian of width ZIm Σ, poles at renormalized quasi-particle bands Zǫ (k), weight Z (instead of 1 in non-interacting case) p. 47

55 Hubbard model within DMFT ( ) H = D 2 ( ) c iσ c jσ +c jσ c iσ +U i n i n i <ij>σ (Hubbard, 1963) ρ(ω) 2 2 U/D=1 U/D=2 Quasi-particle peak Hubbard bands ImG 2 2 U/D=2.5 U/D=3 2 U/D=4 Georges&Kotliar ω /D ( ) DMFT = Dynamical Mean Field Theory, paramagnetic solution p. 48

56 Dynamical mean field theory maps a lattice problem onto a single-site (Anderson impurity) problem with a self-consistency condition (see e.g. Georges et al., Rev. Mod. Phys. 1996) p. 49

57 Effective dynamics for single-site problem S eff = β dτ β dτ σ c σ (τ)g 1 (τ τ )c σ (τ ) + U β dτn n with the dynamical mean fieldg 1 (τ τ ) G (τ τ ) p. 5

58 Déjà vu! p. 51

59 DMFT (contd.) Green s function: G imp (τ) = ˆTc(τ)c () Self-energy (k-independent): Σ imp (ω) = G 1 (ω) G 1 imp (ω) DMFT assumption : Σ imp = Σ lattice G imp = G lattice local Self-consistency condition forg 1 p. 52

60 The DMFT self-consistency cycle Anderson impurity model solver G 1 G(τ) = ˆTc(τ)c () G = ( Σ+G 1) 1 Σ = G 1 G 1 Self-consistency condition: G(ω) = k 1 ω +µ ǫ k Σ(ω) p. 53

61 Hubbard model again Phase diagram of half-filled model within DMFT: First order metal-insulator transition (ending in 2nd order critical points) p. 54

62 Real materials... : V 2 O 3 p. 55

63 Realistic Approach to Correlations Combine DMFT with band structure calculations (Anisimov et al. 1997, Lichtenstein et al. 1998) effective one-particle Hamiltonian within LDA represent in localized basis add Hubbard interaction term for correlated orbitals solve within Dynamical Mean Field Theory p. 56

64 LDA+DMFT H = {imσ} (H LDA im,i m H double counting im,i m Umm i n imσ n im σ imm σ (correl. orb.) im m σ (correl. orb.) solve withing DMFT )a + imσ a i m σ (U i mm J i mm )n imσ n im σ p. 57

65 LDA+DMFT the full scheme DFT part from charge density ρ(r) construct ˆV KS = ˆV ext + ˆV H + ˆV xc [ ˆV KS ] ψ kν = ε kν ψ kν update { χ Rm } build Ĝ KS = DMFT prelude [ iω n + µ ˆV KS ] 1 construct initial Ĝ DMFT loop ρ update compute new chemical potential µ ρ(r) = ρ KS (r) + ρ(r) (Appendix A) Ĝ 1 = Ĝ 1 loc + ˆΣ imp impurity solver G imp mm (τ τ ) = ˆT ˆd mσ (τ) ˆd m σ (τ ) Simp ˆΣ imp = Ĝ 1 Ĝ 1 imp self-consistency condition: construct Ĝloc Ĝ loc = ˆP (C) R [Ĝ 1 KS (ˆΣimp ˆΣ dc )] 1 ˆP (C) R F. Lechermann, A. Georges, A. Poteryaev, S. B., M. Posternak, A. Yamasaki, O. K. Andersen, Phys. Rev. B (26) p. 58

66 Some examples SrVO 3 : (correlated) metal CaVO 3 : (correlated) metal LaTiO 3 : at Mott transition YTiO 3 : insulator ) V O Sr Photoemission spectra: SrVO 3 hν = 9 ev hν = 275 ev hν = 4.8 ev hν = 21.2 ev Intensity (arb. units) SrVO 3 (x = ) Sr.5 Ca.5 VO 3 CaVO 3 (x = 1) Binding Energy (ev) Fujimori et al Sekiyama et al., 22 p. 59

67 LDA+DMFT: spectra of perovskites 1 DOS states/ev/spin/band J=.68 ev U=5 ev SrVO3 J=.68 ev U=5 ev CaVO3 1 DOS/states/eV/spin/band J=.64 ev U=5 ev LaTiO3 J=.64 ev U=5 ev YTiO E(eV) E(eV) (E. Pavarini, S. B. et al., Phys. Rev. Lett (24) ) p. 6

68 Spectra of perovskites Photoemission SrVO 3 I(ω) U = 3.5 ev U = 3.75 ev U = 4. ev LDA+DMFT Intensity (arb. units) hν = 9 ev hν = 275 ev hν = 4.8 ev hν = 21.2 ev SrVO 3 (x = ) Sr.5 Ca.5 VO binding energy CaVO 3 (x = 1) Binding Energy (ev) (see also Sekiyama et al. 23, Lechermann et al. 26) p. 61

69 Vanadium dioxide: VO 2 Metal-insulator transition accompanied by dimerization of V atoms: p. 62

70 VO 2 : Peierls or Mott? p. 63

71 How far do we get using Density Functional Theory for VO 2? metallic phase insulating phase (E - E F ) (ev) 2-2 (E - E F ) (ev) Γ X R Z Γ R A Γ M A Z -1-2 Γ Y C Z Γ A E Z Γ B D Z DFT-LDA : no incoherent weight not insulating (from V. Eyert) p. 64

72 VO 2 : the physical picture Charge transfer e π g a 1g and bonding-antibonding splitting metallic phase: insulating phase: ω [ev] 1 ω [ev] Γ Y C Z Γ.1-2 Γ Y C Z Γ.1 Spectral functions and band structure det ( ω k +µ H LDA (k) RΣ(ω k ) ) = J.M. Tomczak, S.B., J.Phys.:Cond.Mat. 27; J.M. Tomczak, F. Aryasetiawan, S.B., PRB 28 p. 65

73 VO 2 monoclinic phase quasi-particle poles (solutions of det[ω + µ H(k) Σ(ω)]=) and band structure from effective (orbital-dependent) potential ( for spectrum of insulating VO 2 : independent particle picture not so bad!! (but LDA is!)) p. 66

74 Optical Conductivity of VO 2 Re σ(ω) [1 3 (Ωcm) -1 ] Exp. (i) ω [ev] Exp. (iii) Exp. (ii) Theory Metal Re σ(ω) [1 3 (Ωcm) -1 ] Theory E [1] E [11] - E [aab] Experiments Verleur et al. E [1] Verleur et al. E [1] Okazaki et al. Qazilbash et al ω [ev] Insulator [Verleur et al.] : single crystals [Okazaki et al.] : thin filmse [1], T c =29 K [Qazilbash et al.] : polycrystalline films, preferential E [1], T c =34 K p. 67

75 Conclusions? p. 68

76 Not everything depends only on the average occupation! p. 69

77 Not everything depends only on the average occupation! p. 7

78 Not everything depends only on the average occupation! n n n n p. 71

79 Conclusion and perspectives There is a world beyond the one-electron approximation! Mott insulators Correlated metals (electrons become schizophrenic...) How to describe these phenomena on an equal footing? Hubbard model: kinetic energy Coulomb cost Hubbard goes realistic: LDA+DMFT correlated d- and f-electron materials accessible to first principles calculations! What s next? GW+DMFT (or on how to get rid off U and LDA...!) p. 72

80 Useful Reading (not complete) DMFT - Review: A. Georges et al., Rev. Mod. Phys., 1996 LDA+DMFT - Reviews: G. Kotliar et al., Rev. Mod. Phys. (27) D. Vollhardt et al., J. Phys. Soc. Jpn. 74, 136 (25) A. Georges, condmat43123 S. Biermann, in Encyclop. of Mat. Science. and Technol., Elsevier 25. F. Lechermann et al., Phys. Rev. B (26) p. 73

81 References, continued Some recent applications of LDA+DMFT: VO 2 : J. Tomczak, S.B., Psik-Newsletter, Aug. 28, J. Phys. Cond. Mat. 27; EPL 28, PRB 28, Phys. stat. solidi 29. J. Tomczak, F. Aryasetiawan, S.B., PRB 28; S.B., A. Poteryaev, A. Georges, A. Lichtenstein, PRL 25 V 2 O 3 : A. Poteryaev, J. Tomczak, S.B., A. Georges, A.I. Lichtenstein, A.N. Rubtsov, T. Saha-Dasgupta, O.K. Andersen, PRB 27. Cerium: Amadon, S. B., A. Georges, F. Aryasetiawan, PRL 26 d 1 Perovskites: E. Pavarini, S. B. et al., PRL 24 p. 74

82 You want to know still more? Send us your postdoc application! Join the crew p. 75