Dynamical Mean Field Theory

Transcription

1 Dynamical Mean Field Theory Review articles Antoine Georges, Gabriel Kotliar, Werner Krauth, and Marcelo J. Rozenberg, Rev. Mod. Phys. 68, 13 (1996) Thomas Maier, Mark Jarrell, Thomas Pruschke, and Matthias H. Hettler, Rev. Mod. Phys. 77, 1027 (2005) K. Held, II. A. Nekrasov, G. Keller, V. Eyert, N. Bluemer, A. K. McMahan, R. T Scalettar, T.h. Pruschke, V. I. Anisimov, and D. Vollhardt, D.; Phys. Status Solidi 243, (2006) G. Kotliar, S. Y. Savrasov, K. Haule, V. S. Oudovenko, O. Parcollet, and C. A. Marianetti, Rev. Mod. Phys. 78, 865 (2006)

2 Dynamical Mean Field Theory Many-body formalism: analogous to Hohenberg-Kohn H = α,β E αβ ψ αψ β + αβγδ I αβγδ ψ αψ β ψ γψ δ +... =>Luttinger-Ward functional F[{Σ}] =F univ [{Σ}] Trln[G 1 0 Σ] G0: Green function of noninteracting reference problem (contains atomic positions) F univ : determined (formally) from sum of diagrams. Depends only on interactions I αβγδ

3 Example Φ LW [{G}] defined as sum of all vacuum to vacuum diagrams (with symmetry factors) Second order term Hubbard model 1 2 δφ δg = Σ F univ = Φ LW [{G}] Tr [ΣG]

4 More formalities F[{Σ}] =F univ [{Σ}] Trln[G 1 0 Σ] Diagrammatic definition of Funiv => δf univ δσ = G Thus stationarity condition δf δσ = 0 => G = G 1 0 Σ 1 Difficulties with this formulation: --dont know Funiv (exc. perturbatively) --cant do extremization

5 Key first step: infinite d limit Metzner and Vollhardt, PRL (1987) 1992 Breakthrough Kotliar and Georges found analogue of Kohn-Sham steps: useful approximation for F and way to carry out minimization via auxiliary problem

6 Modern interpretation of Kotliar and Georges idea Parametrize self energy in terms of small number N of functions of frequency Σ αβ (ω) = ab f αβ ab Σab DMFT(ω) α = 1...M; a = 1...N << M parametrization function f determines flavor of DMFT (DFT analogue: LDA, GGA, B3LYP,...) Also must truncate interaction I αβγδ appropriately

7 Approximation to self energy + truncated interactions imply approximation to Funiv Approximated functional F approx univ is functional of finite (small) number of functions of frequency Σ ab (ω), thus is the universal functional of some 0 (space) +1 (time) dimensional quantum field theory. Derivative gives Green function of this model: δf approx univ δσ ba (ω) = Gab QI(ω)

8 Specifying the quantum impurity model Need --Interactions. These are the appropriate truncation of the interactions in the original model --a noninteracting ( bare ) Green function G 0 Then can compute the Green function and self energy G QI = G 1 0 Σ 1

9 Analogy: Density functional <=> Luttinger Ward functional Kohn-Sham equations <=> quantum impurity model Particle density <=> electron Green function

10 Useful to view auxiliary problem as quantum impurity model (cluster of sites coupled to noninteracting bath) Quantum impurity model is in principle nothing more than a machine for generating self energies (as Kohn-Sham eigenstates are artifice for generating electron density) As with Kohn Sham eigenstates, it is tempting (and maybe reasonable) to ascribe physical signficance to it

11 In Hamiltonian representation H QI = ab d ae ab QId b + Interactions Impurity Hamiltonian + p,ab V p ab d ac pb + H.c. + H bath [{c pac pa }] Coupling to bath Important part of bath: hybridization function ab (z) = p V p ac 1 z ε bath p G 1 0 = ω E ab QI ab (ω) V p, cb

12 Thus F F approx [{Σ ab }] Trln[G 1 univ 0 ab f αβ ab Σab ] and stationarity implies δf = Gab δσba QI Tr αβ f αβ ab G 1 0 cd f αβ cd Σcd 1 = 0 or, using G QI = G0 1 Σ ab 1 from impurity solver G 1 ab 0 = Σ ab + so G 0 is fixed Tr αβ f αβ ab G 1 0 cd f αβ cd Σcd 1 1 ab

13 In practice Guess hybridization function Solve QI model; find self energy Use extremum condition to update hybridization function Continue until convergence is reached. This actually works

14 Technical note From your impurity solver you need G at all interesting frequencies. Solution ~uniformly accurate over whole relevant frequency range. This is challenging. ˆ, Ê Severe sign problem if = 0 =>reasonably high symmetry desirable so hybridization function commutes with impurity energies For models with rotationally invariant multiplet interactions, cost grows exponentially with number of orbitals. 5 do-able with great effort more with Ising approximation (typically bad) For simple 1 band Hubbard model, can access up to 16 orbitals at interesting temperatures.

15 advantages of method * Moving part Tr p [φ a (p)g lattice (Σ approx )] some sort of spatial average over electron spectral function--but still a function of frequency *Computational task: solve quantum impurity model: not necessarily easy, but do-able =>releases many-body physics from twin tyrannies of --focus on coherent quasiparticles/expansion about well understood broken symmetry state --emphasis on particle density and ground state properties

16 In formal terms: --Approximation to full M x M self energy matrix in terms of N(N+1)/2 functions determined from solution of auxiliary problem specified by self-consistency condition. --Auxiliary problem: find (at all frequencies) Green functions of N-orbital quantum impurity model. ``Exponential wall remains: Question (practical): for feasible N, can you get the physics information you want?

17 Questions (not addressed here) What choices of parametrization function f are admissible What kinds of impurity models What can be solved Where is the exponential wall (how large can N be?) what kinds of interactions can be included What else (besides electron self energy) can be calculated Quality of approximation

18 Single-site (N=1) DMFT of Hubbard model impurity model: One non-degenerate d orbital per unit cell H = t i j c iσ c jσ + U ij i DMFT impurity model n i n i A = dτdτ d σ(τ)g0 1 (τ τ )d σ (τ )+U dτn (τ)n (τ) Impurity model is just the Anderson model, with conduction band density of states fixed by selfconsistency condition. Regimes of Anderson model correspond to different physical behavior

19 Phase diagram and spectral function at n=1 X. Y. Zhang, M. J. Rozenberg, and G. Kotliar Phys. Rev. Lett. 70, 1666 (1993

20 interaction driven MIT via pole splitting Σ(z) 2 1 z ω z ω 2 ω 1,2 0 as U U c2 Σ(z 0) = 2 1ω ω 1 z 2 1ω ω1 2 ω 1 ω 2 ω1 2ω2 2 µ renormalization mass renormalization 0 if p-h symmetry diverges as approach insulator 5 1 U=0.85U c2 U=0.85U c2 0 Im Σ -5 Im G Energy P. Cornaglia data U=0.85 Uc Energy

21 doping driven transition: also coexistence regime; 2 pole structure ImΣ(ω) δ= δ= δ= δ= ω

22 Many regimes of behavior Key Point: theory captures incoherent dynamics at high frequency and crossover to coherent low frequency behavior From A Georges College de France lecture spring June 2010

23 Regimes of behavior configurations of impurity 0>, >, >, 2> Strong correlations generic occupancy resonance forms fermi liquid transport 0>, 2> freeze out local moment incoherent transport All configs; equal prob. T Strong correlations integer occupancy Mott insulator 0>, 2> freeze out local moment gap opens All configs; equal prob T

24 Orbital degeneracy: t2g orbitals J=U/6 Mott, magnetic and orbitally ordered phases. Also ``spin freezing Chan, Werner, Millis, Phys. Rev. B 80, (2009)

25 Spin freezing line: associated with larger spin. =>power law self energy Werner, Gull, Troyer, Millis, Phys. Rev. Lett. 101, (2008)

26 The power law self energy does not extend to very lowest energies Georges, Medici, Mravlje, Ann Rev Cond Mat Phys (2012 Higher spin: extraordinarily low quasiparticle scale

27 Third example: nickelate superlattices Chaloupka-Khalliulin Idea: Bulk LaNiO3 Ni [d] 7 (1 electron in two degenerate eg bands). Phys. Rev. Lett (2008) In correctly chosen superlattice structure, split eg bands, get 1 electron in 1 band-- like high-tc

28 Pseudocubic LaNiO3 Relevant orbitals: eg symmetry Ni- O antibonding combinations 3z 2 -r 2 Hybridizes strongly along z Hybridizes weakly in x-y x 2 -y 2 Hybridizes strongly along x-y Hybridizes very weakly in z 2 orbitals transform as doublet in cubic symmetry

29 One electron (band theory) physics Pseudocubic LaNiO3 Hamada s result: Two-fold degenerate eg band complex Hamada J Phys. Chem Sol

30 Superlattice Superlattice: La2AlNiO6 LaAlO3 layer: insulating barrier Question (Khalliulin): how much orbital polarization can we get? P = n x 2 y 2 n 3z2 r 2 n x2 y 2 + n 3z2 r 2 MJ Han, X Wang, C. Marianetti and A. J. Millis, arxiv:

31 Band theory of La2NiAlO6 heterostructure: La2NiAlO6 LaNiO3 Hansmann et al PRL degeneracy of eg bands lifted: 3z 2 -r 2 band moves up

32 Band theory of La2NiAlO6 heterostructure: La2NiAlO6 2d fermi surface LaNiO3 3d fermi surface Hamada J Phys. Chem Sol Hansmann et al PRL No qualitative difference

33 DFT+DMFT: Correlate frontier orbitals Map conduction bands to (multiple-orbital) Hubbard model. Solve with single-site DMFT Disperson: fit band theory Interaction (phenomenological) P. Hansmann, X. Yang, A. Toschi, G. Khaliullin, O. K. Andersen,and K. Held, Phys. Rev. Lett. 103, and Phys. Rev. B 82,

34 Result: Proximity to Mott insulator=>enhanced polarization Interactions enhance orbital polarization Interactions drive orbitally selective metal-insulator transition U=0 U=6eV Phys. Rev. Lett. 103, Phys. Rev. B 82,

35 Summary Dynamical mean field theory: --approximation to electron self energy obtained from solution of auxiliary quantum impurity problem --can treat high temperature and incoherent phases; metal insulator transition and low scales --limits of accuracy of approximation: subject of different set of lectures --meshes localized and k-space physics?how to go beyond phenomenologically defined models?

36 General formalism Green function: G(r, r ; ω) = = {ψ(r),ψ (r )} ω ω1 Ĥ0 Σ(r, r ; ω) 1 H 0 = i P 2 i 2m e + V ext (r i ) Bare Hamiltonian: kinetic energy and lattice potential DFT: static approximation to self energy G DFT (r, r ; ω) = 1 ω1 Ĥ0 Σ DFT (r, r ) Many body theory: supplement DFT self energy with additional terms expressing ``beyond DFT physics

37 Express beyond DFT physics as dynamical self energy Σ(r, r ; ω) =Σ DFT (r, r )+Σ dyn (r, r ; ω) Cannot treat many body physics of all electronic degrees of freedom =>must select subset of states for which dynamical self energy is computed *different ways to do this

38 Frontier Orbital Approach Express G in band basis. DFT part is diagonal. Many-body (``dynamical ) self energy is in general a matrix Ĝ DFT (k, ω) = Σ nm dyn(k, ω) = ω ε KS n (k) δ nm ˆΣ 1 dyn (k, ω) drdr ψ KS nk (r)σ dyn (r, r ; ω)ψ KS mk (r ) *Keep only self-energy and interaction matrix elements within frontier orbital basis. (Interaction: phenomenological U,J or screened coulomb interaction projected onto these states)

39 Issues: Cubic LaNiO3 Fat bands Red: Ni-d Blue : O 2pσ relevant (mainly d) bands entangled with p bands. Practical question: how to separate bands. Principle question: how important are the O states J. Rondinelli prvt comm ``d orbital corresponding to the p-d antibonding bands has large spatial extent. Is it reasonable to ascribe just an on-site interaction U OK Andersen JPhys. Chem. Solids 56, 1573 ~1995

40 Atomic Orbital Approach alternative: define atomic like orbital centered on sites i project self energy onto this basis ψ i at(r) Σ dyn (r, r ; ω) ij ψ i at(r) > Σ ij dyn (ω) <ψj at(r ) Ĝ(r, r,ω)= 1 ω ĤKS Σ dyn (r, r ; ω) Interaction: intra-d interactions already discussed

41 In real materials applications retain only on-site (but orbitally dependent) terms in self energy and only on-site Slater-Kanamori terms in interaction Σ dyn (r, r ; ω) iab ψ ia d (r) > Σ ab dyn(ω) <ψ ib d (r ) Σ double count (r, r ) iab ψ ia d (r) > Σ ab dc <ψ ib d (r ) Ĝ(r, r,ω)= 1 ω Ĥ KS Σ dc (r, r ) Σ dyn (r, r ; ω)

42 Additional step (not often taken in practice): Full charge self consistency Recall Kohn-Sham potential VKS[{n(r)}] is a functional of density. =>need to use density computed from interacting G µ dω n(r) = π ImG(r, r,ω) to construct VKS.

43 Two ways to define atomic orbital *A priori: simply choose a wave function you think is appropriate (e.g. free atom hartree fock d-wave function) *Wannier function φ i (r) = BZ If isolated band (index n) with wave function ψ nk (r) then real space wave function centered in unit cell i (pos Ri) (dk)e ik R i ψ nk (r)

44 Maximally localized Wannier function LaNiO3 If many bands in a energy window, then many ways to construct Wannier functions-- parameterized by family of unitary transformations depending on k J. Rondinelli ANL prvt comm φ a i (r) = BZ (dk)e ik R i n Û an (k)ψ nk (r) Marzari-Vanderbilt: showed how to choose unitary transformation to make wannier functions correspond (as closely as Fourier s theorem allows) to atomic-like functions centered on particular atoms (e.g. O 2p and Ni 3d)

45 If you have a set of bands reasonably well separated from other bands (e.g. p-d complex in many transition metal oxides) Write G H KS and self energy as matrices in Wannier basis again retain only terms in dynamical self energy corresponding to orbitals where correlations are important Ĝ(ω) = ω1 ĤKS n ˆΣ dyn (ω) 1 H KS ab = drdr φ a(r) 2 2m + V latt + Σ DFT (r, r ) φ b (r ) H KS ab is formal definition of tight-binding model corresponding to ``ab-initio wave functions

46 Note! All of these procedures assume that the wave functions of band theory have some meaning in the actual correlated material

47 Note! atomic orbital approach Σ dyn (r, r ; ω) ij ψ i at(r) > Σ ij dyn (ω) <ψj at(r ) will in general lead to hartree shift which moves atomic (here, d) orbitals relative to the p orbitals

48 What do people do in practice *DFT+U (Anisimov 1998) Identify atomic-like d-orbital. Keep on-site d-d self energy arising from Hartree approx to Slater interactions V a = U < n a > +(U 2J) b=a < n b > +(U 3J) b=a < n b > This calculation ``knows about correlations in the d-multiplet only via ordering. In non-ordered state, shift is same for all orbitals As with most Hartree approximations, tendency to order is overestimated Crucial physical effect: hartree shift of entire d-multiplet by amount of order (U-2.5J)*n/spin/orbital

49 Hartree term: shifts d-multiplet relative to p-states. this is important to the physics!! d-d transition: 2d N ->d N-1 d N+1 E E d-p transition: d N p 2 ->d N+1 p 1 Energy U Energy Lower energy controls the interesting physics > U: Mott-Hubbard < U: Charge Transfer Shift of d relative to p changes the physics

50 thus the crucial question: In a beyond-dft calculation: where is the renormalized d energy, relative to the ligands

51 thus the crucial question: In a beyond-dft calculation: where is the renormalized d energy, relative to the ligands This is sometimes called ``the double counting problem, based on the idea that DFT includes some of the intra-d interactions and one does not want to count them twice. But the real question is--how much does the d-level shift relative to band theory. a large literature exists (see, e.g. Karolak et al, Journal of Electron Spectroscopy and related phenomena, 181 (2010) 11 15) but there is as yet no generally agreed upon prescription. Results are sensitive to choice of double counting correction.

52 p-d covalency key issue: renormalized d energy, relative to the ligands. Question: what does renormalized d energy mean My point of view (not universally accepted): parametrize the renormalized d energy by d occupancy Unhybridized bands: upper one is pure d. Increase hybridization, increase d- character of occupied states

53 d-occupancy: * Intuitive notion: e.g. La 3+ Ni 3+ O3 2- =>Ni d 7 (Nd=6+1) *Theoretically needed (if you want to put correlations on d-orbital you need to know what this orbital is and how much it is occupied) *definition: In terms of exact Green function G(r, r ; ω) and predefined d-wave function φ d N d = dω π f(ω) d 3 rd 3 r Im (φ a d(r)) G σ (r, r,ω)φ a d(r ) a,σ

54 Notes *d occupancy depends on how orbital is defined (as does the entire edifice of DFT+...) Expect: all reasonable definitions give consistent answers (more on this later) *should focus only on occupancy of relevant orbitals

55 Notes *Density functional theory only gives you n(r) = a,σ dω π f(ω) d 3 r Im [ G σ (r, r,ω)] Thus even exact density functional, solved exactly, is not guaranteed to give you the exact Nd But: seems plausible that in most cases DFT is a reasonably good approximation (more on this later)

56 Notes *Nd: theoretically precisely defined; experimental measures are indirect --Intensities in resonant absorption spectra --Sizes of moments/knight shifts --peak positions (``many-body ed-ep) (more on this later)

57 In practice: Available double counting corrections give Nd fairly close to Nd defined from DFT band theory calculations Example: rare earth nickelates ReNiO3 LuNiO3: insulator. 2 inequivalent Ni sites. Ni1 Ni2 GGA GGA+U Charge fixed by electrostatic effects. Very robust

58 *DFT+DMFT (Georges04,Kotliar06,Held06) Identify atomic-like d-orbital. Keep on-site d-d self energy arising from single-site DMFT approx using Slaterkanamori interactions DMFT self consistency eq: G ab QI =< ψ ia d (r) Ĝ(r, r,ω) ψ ib d (r ) > Same double counting issue as in DFT+U

59 Another example: High Tc cuprates Green dots: fully charge self-consistent DFT+DMFT calculations for U=7,10eV. Vertical lines: DFT band theory (Wein2K and GGA)

60 Nickelate superlattices Frontier orbital approach U=0 U=6eV Phys. Rev. Lett (2008) Phys. Rev. Lett. 103,

61 Atomic-like d orbitals M. J. Han, X. Wang, C. A. Marianetti, A. J. M, Phys. Rev. Lett (2011). *LaNiO3/LaXO3 (mainly X=Al) *VASP/Wannier DFT+DMFT (Non-self-consistent) *Computed orbital polarization *Several combinations of U and ed-ep U(eV) I M ! (ev)

62 Results 1: Define P by integrating spectral function for superlattice DOS (states/ev) DOS (states/ev) (a) P= (c) P= Energy (ev) Spectral functions (b) O p P= (d) Interactions decrease P P=0.09 x 2 -y 2 3z 2 -r Energy (ev) U(eV) Phase Diagram I M ! (ev)

63 Results 2: How many sheets to the Fermi surface (a) (b) Phase Diagram ky Π P= k x Π (c) ky Π P= k x Π (d) U(eV) I M ky Π P=0.05 ky Π k x Π P= k x Π ! (ev)

64 Results 2: Relation to Nd (a) Nd= (b) Nd=1.5 Phase Diagram ky Π P= k x Π (c) Nd=1.98 ky Π P= k x Π (d) Nd=1.43 U(eV) I M ky Π P=0.05 ky Π k x Π P= k x Π ! (ev)

65 Key point: Ni-O covalency Occupancy per orbital Band theory: electron moves from O to Ni, La2InNiO6 configuration is d 8 L According to band theory, LaNiO3 and derivatives are close to being negative charge transfer energy materials d 8 is high spin: both orbitals occupied

66 d 8 (high spin)=> no distortion Two electrons in high spin state=>no energy gain from distortion Negligible response to lattice distortion Over wide range of many-body phase space, Nd is close enough to d 8 that low-spin J-T state is disfavored 2-band Hubbard model with 1 el/ni unit cannot represent d 8 L physics

67 Phase diagram: bulk nickelates

68 Is the metal-insulator transition of the usual Mott/charge transfer kind ZSA phase diagram for cubic LaNiO3 (b) I U M [ev] ep-ed 4 VASP/MLWF DFT+DMFT non-self-consist

69 Is the metal-insulator transition of the usual Mott/charge transfer kind ZSA phase diagram for cubic LaNiO3 (b) I U?Where to place the materials? [ev] ep-ed M 8 6 4

70 Is the metal-insulator transition of the usual Mott/charge transfer kind ZSA phase diagram for cubic LaNiO3 (b) I U Replot in U-Nd space [ev] ep-ed M 8 6 4

71 Is the metal-insulator transition of the usual Mott/charge transfer kind ZSA phase diagram for cubic LaNiO3 (b) I U U [ev] Replot in U-Nd space I [ev] ep-ed M M N d

72 Is the metal-insulator transition of the usual Mott/charge transfer kind ZSA phase diagram for cubic LaNiO3 (b) I U U [ev] Replot in U-Nd space I GGA Nd nickelate [ev] ep-ed M M N d

73 Interpretation of nickelate MIT as standard ZSA transition is not plausible Metal-Insulator line occurs at Nd far from GGA Nd => either band theory is VERY wrong for Nd or standard ZSA picture not applicable: Nickelates not Mott materials U [ev] I GGA Nd nickelate M N d

74 What is really going on?

75 Evidence: Ni-O bond length Nickelates: ``charge order magnetic moment J A Alonso et al PRL (1999)

76 But `charge order on Ni implausible UNi~5-6eV implies energy COST to have different charges on different Ni sites Our GGA + U for LuNiO3 (GGA+U relaxed structure) thus--`charge order insignificant.

77 We find Hybridization wave =>site selective Mott transition Undistorted lattice Charge ordered lattice

78 Long-bond site: Weakly coupled to environment => local moment, Mott insulating Charge ordered lattice

79 Long-bond site: Weakly coupled to environment => local moment, Mott insulating Charge ordered lattice

80 Short-bond site: Strongly coupled to environment => d 8 L: spin holes: singlet ( band insulator ) Charge ordered lattice Energy gain from hybridization (singlet formation)

81 Key point: d-charge need not differ between sublattices Undistorted lattice Charge ordered lattice (of course, some small charge difference allowed by symmetry, but this is not the important physics)

82 Density Functional+ DMFT calculations substantiate this picture Band theory: crystal structures appropriate to LaNiO3 and LuNiO3 eg DOS extract Ni d and O p bands (max. loc. wannier).

83 DFT(+DMFT) in LuNiO3 structure:

84 DMFT, low T: magnetic order: very different ordered moments, almost same S 2, charge Red: <S(t) 2 ><=>Nd (Nd=1.24,1.20) Blue: Ordered moment (FM)

85 DMFT: Paramagnetic phase Long bond site: Curie Short bond site: constant (spin gap)

86 Self energy: pole on long-bond ( Mott ) site not on short bond ( singlet ) site

87 Energetics (preliminary): lattice distortion (hybridization wave) favored by DMFT ``LuNiO3

88 Energetics: LuNiO3 structure favors distortion much more than LaNiO3 structure LuNiO3 vs LaNiO3 structures (La pseudopotential)

89 Energetics: LuNiO3 structure favors distortion much more than LaNiO3 structure

90 Nickelate: summary Hybridization wave mechanism NOT charge order NOT standard charge transfer insulator Coupled to structural distortion Favored by particular bond-buckled crystal structures

91 Covalency in perovskites Generally important I M Band theory or standard double counting indicates that La2CuO4 is not a Mott insulator

92 Example: NiO Fit full p-d band complex: Add Slater-Kanamori U-J interactions on d-only. Solve. many-body calc DFT Karolak et al Journal of Electron Spectroscopy and Related Phenomena 181 (2010) choice of double counting (covalency) affects results (perhaps less strongly)

93 Summary Wide variety of behaviors Important role played by instability of d valence Energetics and state of the art of realistic theory of correlated electron materials Key (and still ill-understood) role of double counting correction Mott metal insulator transition only one part (maybe not the most important) aspect of transition metal oxide physics Important open question: is the ``standard model (atomic d, local correlation) the most correct and useful description?