Introduction to SDFunctional and C-DMFT

Transcription

1 Introduction to SDFunctional and C-DMFT A. Lichtenstein University of Hamburg In collaborations with: M. Katsnelson, V. Savkin, L. Chioncel, L. Pourovskii (Nijmegen) A. Poteryaev, S. Biermann, M. Rozenberg, A. Georges (Paris) G. Kotliar, S. Savrasov (Rutgers) A. Rubtsov (Moscow) O.K. Andersen, E. Pavarini, T. Saha-Dasgupta (Stuttgart)

2 Spectral function: Correlations effects

3 From Atom to Solids

4 Theory: interactions vs. hopping Multiband Hubbard model (<im jm 0 >=δ ij δ mm0 ) Coulomb inraatomic interaction Matrix elements of electron-electron interactions: Exact diagonalization of atom: t ij =0 gives multiplets! Solution with hoppings t ij 0 in solids is unknown!

5 Strong correlations in real system Real U? Multiplets in solids U =< m m V m m > mm 1 2mm ee 3 4 Local moments above Tc E

6 Functionals: MFT- DFT- DMFT G. Kotliar et. al. (2002) A. Georges (2004) Weiss Mean-Field Theory (MFT) of classical magnets Kohn Density Functional Theory (DFT) of inhomogeneous electron gas in solids Dynamical Mean-Field Theory (DMFT) of strongly correlated electgron systems

7 The Euclidian Action x=(r,τ ) Many-body System

8 Functionals: general consideration The one-electron Green's function Introduction of the source (constraining field) Functional derivative:

9 Functionals: Legendre Transformation The Functionals of Green's function The partition function Z λ =0 Constraining field J=Σ : the inverse of the exact Green's function

10 Baym-Kadanoff Functional Exact representation of Φ Different Functionals: DFT: G=ρ SDF: G=G(iω) BKF: G=G(k,iω)

11 Mean Field Theory (MFT) Classical Ising Magnets: Start from the free-energy functional of H α =α H with additional constrain: m i =<S i >: Invert magnetisation and find the Lagrange parameters: Legandre transformation (physical system : Γ [m i ]=Γ α =1 [m i ])

12 MFT: reference system Non-interacting limit (α =0): Ω is stationary with respect to the λ : This gives Γ α =0 as independent spins functional: Relation to the Weiss effective filelds: h i eff =λ i α =0

13 MFT-Functional Coupling constant integrations: Because of constraint <(S i -m i )>=0 Introduce the connected correlation function: which is equal to: βg=(δ 2 Γ/δ m δ m) -1, we obtain: Final expression for the functional: Γ =Γ 0 +Γ MF +Γ corr

14 DFT-Density Functional Theory Inhomogeneous electron gas in solids ( U=e 2 / r-r 0 ): Energy functional with constrained density <n (r) > =ρ (r) Stationarity in λ insures that: Construct a functional of ρ (r) only:

15 DFT: reference system Non-interacting electrons in effective potential (t=- 2 /2): Minimization with respect to λ (r): Coupling constant trick:

16 DFT - Functional where density-density correlations defined as: Exact relations for density functional: Local density approximation (LDA):

17 DMFT - spectral density functional Hubbard-model on a lattice (n=c + c): Constrain local Green's function: G iiσ (τ -τ 0 )=<Tc iσ (τ )c iσ + (τ 0 )> Inverting the relation: G=G α [ ] yeld for Lagrange multipliers: = α [G] and Functional of a local Green's functions: Γ α [G]=Ω α [G, α [G]]. In the Baym-Kadanoff approach we have functional of a total lattice GF: G ij and not local G ii

18 DMFT reference system Consider α =0 case of local action: Green function comes from solution of quantum impurity problem: Lagrange function 0 play role of a Weiss field: Important aspect of DMFT: The Weiss function 0 (iω ) is dynamical Equivalent local problem is NOT a one-body problem Iterative solution:

19 Exact functional of local Green function coupling constat integrations, starting from: here F imp - free enrgy of impurity model, and invering: 0 = 0 [G]: Using stationary condition: Final expression for DMFT-functional:

20 DMFT on infinite connectivity Bethe Lattice For the Bethe lattice with z and hoppings t 2 ij =t2 /z non zero term for expantion of functional in powers of α: Integration over α gives Bethe-lattice DMFT functional: Then stability and self-consistentcy conditions are:

21 Exchange and Functionals

22 Exchange interactions from DFT Heisenberg exchagne: Magnetic torque: Exchange interactions: Spin wave spectrum: Non-collinear magnetism : M. Katsnelson and A. Lichtenstein Phys. Rev. 61, 8906 (2000)

23 From Atoms to Solids Atomic physics (LDA+U) Bands effects (LDA) N(E) N(E) n d SL> E F n+1 d E E F E N(E) LDA+DMFT LHB QP UHB E F E

24 Static limit: LDA+U Rotationally invariant LDA+U functional Local screend Coulomb correlations LDA-double counting term (n σ =Tr(n mm0 σ ) and n=n +n ): Occupation matrix for correlated electrons:

25 Slater parametrization of U Multipole expansion: Coulomb matrix elements in Y lm basis: Angular part 3j symbols Slater integrals:

26 Average interaction: U and J Average Coulomb parameter: Average Exchange parameter: For d-electrons: Coulomb and exchange interactions:

27 DMFT: Self-Consistent Set of Equations Σ Σ Σ Gˆ i BZ 1 Ω ( ω ) = Gˆ ( k, i ) n k ω n Σ U Σ ˆ ˆ 1 1 G 0 n ( iω ) = G ( iω ) + Σ( iω ) n ˆ n Σ Σ Σ QMC ED τ U τ Quantum Impurity Solver DMRG FLEX IPT G( τ τ ) Σˆ new ( ) 1( ) 1 iω = G iω G ( iω ) n ˆ 0 n ˆ n

28 exp{ τu Multi-band QMC-scheme Discrete HS-transformation (Hirsch, 1983) mm' [ n n m m' cosh 1 2 ( n m + n m' )]} = 1 2 S mm ' = ± 1 ( λ ) exp τu mm ' 1 2 = mm ' exp{ λ S mm' mm' ( n m n Number of Ising fields: N = M( 2M 1), M = { m, σ} Green Functions: 1 Gmm' ( τ, τ ') = Gmm' ( τ, τ ', S) detg Z S mm ' ( τ ) 1 1 G ( τ, τ ', S) = G ( τ, τ ') + V ( τ ) δ δ V σ mm' mm ' m mm' ττ ' ( τ ) = λ S ( τ ) σ m mm ' mm ' mm ' m ' mm' = + 1, m < m ' 1, m > m ' H = t c c + U n n ij σ + i ij iσ jσ mm' mm' m m' 1 τ m m' U )} mm m τ

29 Multiorbital CT-QMC: general U-vertex 5 orbitals, full U-vertex DOS (1/eV) W=2 U=2 J= Energy (ev)

30 Choice of double counting in LDA+DMFT Shift of chemical potential for correlated state G 1 ( k, ) ( i ) H ( k) [ E ] [ ( ) ] ω = ω + µ LDA + dc δµ c Σ ω δµ c Natural choice Edc = δµ c : G 1 ( k, ω) = ( iω + µ ) H ( k) Σc( ω) G Transformations: 1 1 c0 G0 = δµ LDA 1 c( ) c0 1 c( ) c Σ ω = G G =Σ ω δµ Condition for δµ c (in QMC) Tr G [ ] = Tr[ G0] c G(τ) Ni Ferro E g -up G τ G

31 Spectral function ARPES and DMFT ARPES (A. Damascelli, et al PRL2000) LDA+DMFT, A. Liebsch,et al PRL(2000) DMFT LDA Sr 2 RuO 4 Van Hove=10 mev m * /m=

32 DMFT for a general lattice Bath Green function for multiband case: from the cavity construction: G ij (0) is the Green function with eliminated 0-site: Using the Fourier transform:

33 DMFT for a general lattice: GF GF in DMFT: Here: Taking into account that t(k)=0 we have: and Using this formulas we obtained: Finally for a bath GF we have: A. K. and M. Katsnelson, PRB (1998)

34 DMFT-SCF LDA+DMFT (orthogonal NMTO): ) ( ˆ ), ˆ ( ) ˆ ( ) ˆ ( ), ( ) ( ) ( ) ( ), ( 1 ' ' ' ' 1 ' α ω α ω ω ω ω µ ω ω α + = = Σ + = U i k G U i G i k G i G i k H i i k G IBZ k n O n BZ k n LL n LL n DMFT LL LDA LL n n LL h Energy DOS E F sp d dd dd pd d dp pp p d ps s H k +Σ = +Σ H H H H H H H H H s s s s ) ˆ( ) ( ˆ ω Correlated d-states:

35 2-band DMFT Marcelo Rozenberg (1997)

36 2-band DMFT: effect of J

37 ARPES: electronic structure of 3d-metal C. Carbone et. al. (Trieste, FZJ and BESY)

38 LDA+DMFT calculation for fcc-mn DOS Spectral Function S. Biermann et al. JETP Lett. (2004)

39 NiMnSb: Half-metallic ferromagnetism (HMF) NiMnSb semi Heusler alloy (fcc F-43m) - HMF (µ=4µ B ), T c =730K De Groot et.al. PRL 50, 2024 (1983)

40 Half-Metallic Ferromagnets at finite temperature N(E) E E F Spin-spirals (P. Dowben) Spin-orbit (P. Dederichs) NQP D. Edwards and J. Hertz (1973)

41 Non-Quasiparticle states in HMF systems Semicircular DMFT-model L. Chioncel, PRB (2004)

42 W σ 12 NQP: LDA+DMFT for NiMnAs σσ ' σ ' ( τ ) = W ( τ ) G ( τ ) σσ ' W ++ + W W ( i ) = ω n 34 13,42 + W 34 U=3eV,J=0.9eV, T=300K Im Im ( ) loc E ( E) k, σ σ d d ( E) ( E E ) ( E) NQP F 2

43 Free cluster 1 = G +Σ Cluster DMFT scheme G = [ iω + µ H( k) Σ( ω)] -1 G k 1-1 Σ=G G 1 Periodic cluster G= [ iω + µ H( k) Σ( k, ω)] -1 G k = iω + µ t Gt ab ai ij jb ij G = G G G G 1 ij ij ia ab bj ab 1

44 Different cluster-dmft scheme 4-atoms Ring U=4 W=4 Cluster DMFT Dynamical Cluster Approx. Density of states, ev -1 0,16 0,12 0,08 0,04 CLS DCA ED4 E F Exact Diagonalization 0, Energy, ev

45 AFM and d-wave in HTSC A.L. and M.Katsnelson PRB(2000)

46 AFM+d-wave in CDMFT In superconducting state:

47 Coexistence of AFM and d-wave

48 1d-case M. Capone et. al.

49 Dimensional Crossover: chain- DMFT TMTTF2(X) Effective 1d-problem in the bath (QMC) with self-consistency condition

50 Chain-DMFT for quasi-1d system A. Georges, et al. PRB (2000)

51 Incommensurate FL-LL crossover FL T * LL T * T 0. 5 t π S. Biermann et. al. PRL (2002).

52 Cluster-DMFT for two-leg ladder NaV 2 O 5 The C-DMFT map the original lattice model to self-consistent solution of two-site cluster in effective medium

53 Cluster-DMFT scheme

54 Cluster-DMFT : Exact Diagonalization

55 NMTO-tight-binding model Valence two-bands for bonding-antibonding t 2g A-orbital: Original LMTO Simple TB-model NMTO-TB (OKA) T. Saha-Dasgupta (2002)

56 Metal-Insulator phase diagram Old TB New TB LMTO U,V x V. Mazurenko et. al. PRB(2002)

57 DMFT-DOS for insulating phase LMTO-values: U=2.8, V=0.17 ev U=2.8, V=0.5 ev

58 The Goodenough diagram E e g e ga1g E F Ti 2 O 3 d 1 state 1-st order MIT Around 470K Width 250 K a 1g e g LDA gives Metallic state O 2p O 2p Ti 2 O 3 V 2 O 3 Problem: Bonding vs. U

59 V-V pair in V 2 O 3 Gij ( ω) i j a e 1

60 Different Crystal Structures VO 2 Ti 2 O 3 O1 O2 V O Ti Rutile structure for high temperature Monoclinic for low temperature Corundum structure for high and low temperature phases

61 Non-local U-V Cluster DMFT for Ti 2 O 3 A. Poteryaev et. al. PRL (2004) U=2 ev, J=0.5 ev, Ve g -a1 g =0.5 ev

62 Phase diagram of VO 2 : singlet formation M. Marezio et al., (1972) Temperature (K) Metal Insulator Rutile structure Monoclinic distortion in the insulating phase G ( ω) ij i U UH t ij j U ε i a b LH ε j Correlation vs. Bonding U/t

63 MIT in VO 2 S. Biermann et al. PRL (2004)

64 Conclusions LMTO+DMFT is a perfect scheme for realistic description of electronic structure and magnetism of correlation electron materials Non-quasiparticle states are an important ingredient of the half-metallic ferromagnetism and spintronics Cluster LDA+DMFT method can be useful for the short-range non-local Coulomb correlations in solids: metal-insulator transition in Ti 2 O 3 and VO 2