The DFT+DMFT method and its implementation in Abinit

Transcription

1 The DFT+DMFT method and its implementation in Abinit Bernard Amadon CEA, DAM, DIF, F Arpajon, France New trends in computational approaches for many-body systems, June 2012, Sherbrooke p.1/70

2 Outline of the presentation DFT+DMFT: a short reminder/reformulation. Choice of the basis and local orbitals. Formalism. Physical considerations. Interaction and double counting and solver Description of the actual calculation Practical issues/details to carry out the calculation (PAW) Self-consistency issues: explanation, what has to be done. calculation of non diag occupancies to compute local densities Outcome of the calculation Spectral functions, self-energy. Energy. Some tests/examples. Conclusion. New trends in computational approaches for many-body systems, June 2012, Sherbrooke p.2/70

3 Localization of 3d, 4f and 5f electrons 3d and 4f are orthogonal to other orbitals only through the angular part φ(r) s p d V (3d) Nb (4d) Ta (5d) r (Å) φ(r) r(å) s p d f Ce (4f) Th (5f) New trends in computational approaches for many-body systems, June 2012, Sherbrooke p.3/70

4 Strong local correlations: Introduction (I) 3d,4f and 5f elements: localized atomic wavefunctions strong correlations. Small overlap: bands are narrow (width: W). Strong interactions "U" between electrons. Competition between band effect and correlation have to be described. New trends in computational approaches for many-body systems, June 2012, Sherbrooke p.4/70

5 Strong local correlations: Introduction (II) Hamiltonian to solve (i: électrons ): H = N [ r i +V ext (r i )]+ 1 2 i=1 i j 1 r i r j Strong correlation in Localized orbitals (f,d) Other orbitals: DFT(LDA/GGA) could be tried.. H LDA+Manybody = one electron term (DFT/LDA)+ U ˆn iˆn j 2 i j }{{} many body interaction N(N 1) E LDA+U = E LDA U + U n i n j 2 2 But in DFT+U: Electrons are frozen, only one Slater determinant, metastable states. i j New trends in computational approaches for many-body systems, June 2012, Sherbrooke p.5/70

6 DFT+U and the Dynamical Mean Field Theory (DMFT) DFT+U To describe interactions in solids with static mean field theory Main idea: An electron is described, in the effective field of all the other electrons. Self-consistent Hartree Fock problem. [Anisimov V I, Poteryaev A I, Korotin M A, Anokhin A O and Kotliar G J. Phys.: Condens. Matter (1997)] New trends in computational approaches for many-body systems, June 2012, Sherbrooke p.6/70

7 DFT+U and the Dynamical Mean Field Theory (DMFT) DFT+U To describe interactions in solids with static mean field theory Main idea: An electron is described, in the effective field of all the other electrons. Self-consistent Hartree Fock problem. DMFT To describe correlation in solids beyond static mean field theory. Main idea: An atom is isolated (in red), local correlations are described exactly, in the effective field of other atoms. Self-consistent Impurity problem: Anderson model [see review A. Georges, G. Kotliar, W. Krauth, and M. J. Rozenberg Rev. Mod. Phys. 68, 13 (1996) ] New trends in computational approaches for many-body systems, June 2012, Sherbrooke p.6/70

8 DFT+U and the Dynamical Mean Field Theory (DMFT) DFT+U Unrestricted Hartree Fock [Anisimov V I, Poteryaev A I, Korotin M A, Anokhin A O and Kotliar G J. Phys.: Condens. Matter (1997)] New trends in computational approaches for many-body systems, June 2012, Sherbrooke p.7/70

9 DFT+U and the Dynamical Mean Field Theory (DMFT) DFT+U Unrestricted Hartree Fock DMFT Configuration Interaction for correlated electron in a bath.. Only for correlated electrons on a single atom coupled to a bath! [see review A. Georges, G. Kotliar, W. Krauth, and M. J. Rozenberg Rev. Mod. Phys. 68, 13 (1996) ] New trends in computational approaches for many-body systems, June 2012, Sherbrooke p.7/70

10 The anderson model: general case ε f+u Γ T K W ε f t kf t kf t kf E <n n > [cf L. Kouwenhoven et L. Glazman, Physics World Jan 2001 p33] New trends in computational approaches for many-body systems, June 2012, Sherbrooke p.8/70

11 The anderson model: general case ε f+u Γ T K W ε f t kf t kf t kf E <n n > New trends in computational approaches for many-body systems, June 2012, Sherbrooke p.8/70

12 The DFT+DMFT scheme DFT+DMFT Loop over density H KS DFT χ R km Ψ kν ǫ KS DFT ν,k G 0 DMFTforfixed n(r) Impurity problem G n(r) f DFT+DMFT ν,ν,k G 1 0 = Σ + G 1 imp G imp Self-consistency condition G imp G latt Σ = G 1 0 G 1 imp Σ New trends in computational approaches for many-body systems, June 2012, Sherbrooke p.9/70

13 DMFT in an arbitrary basis (1) What is necessary to carry out effectively the DMFT calculation? The band structure and the one electron Hamiltonian H = ν,k Ψ kν ε kν Ψ kν A basis set for one electron quantities: B kα. The matrix element on the Hamiltonian can thus be computed as: H KS,αα (k) = B kα H B kα = ν B kα Ψ kν ε kν Ψ kν B kα. A definition for d/f local orbitals, let s call them χ R km. (m is an angular momentum index). DFT+DMFT is a local orbital dependent method. New trends in computational approaches for many-body systems, June 2012, Sherbrooke p.10/70

14 DMFT in an arbitrary basis (2) All one electron quantities can be expressed in the basis set B kα. Lets define projectors: Projectors: P R mα(k) χ R km B kα, P R mα(k) B kα χ R km. The Self energy is a local quantity in DMFT, thus: Σ = χ R Tm Σ Rimp mm χ R Tm mm TR New trends in computational approaches for many-body systems, June 2012, Sherbrooke p.11/70

15 DMFT in an arbitrary basis (3) Thus Σ αα (k,iω n ) = B kα Σ B kα = mm P R mα(k) Σ Rimp m,m P R m α (k). Note that the self-energy in this basis has a k-dependence. The lattice Green s function G α,α (k,iω n ) can be expressed in the complete Bloch Basis B kα. and projected to compute the local (impurity) Green s function. G imp mm (iω n ) = Pmα(k)P R m R α (k) k αα { [(iω n +µ)i H KS (k) Σ(k,iω n )] 1}, αα New trends in computational approaches for many-body systems, June 2012, Sherbrooke p.12/70

16 DMFT in an arbitrary basis (4): main idea A definition for d/f local orbitals χ R km. (m is an angular momentum index) A physically motivated choice. Non local correlations have to be minimized. A basis set for one electron quantities: B kα. Results should be independant of the basis. Numerical efficiency do limit the size of the basis. New trends in computational approaches for many-body systems, June 2012, Sherbrooke p.13/70

17 Formalism: DFT+DMFT implementation, LMTO LMTO ASA χ l,l,m,m (k) (Andersen 1975). LMTO s are an optimized atomic orbital Bloch basis. The local correlated subspace is thus a subset of the LMTO basis B kα. χ l=2,l =2,m,m (k) for d orbitals. First implementation in DMFT: Lichtenstein and Katsnelson 1998 Fast, efficient, but method dependent. New trends in computational approaches for many-body systems, June 2012, Sherbrooke p.14/70

18 Why DFT+DMFT in a plane wave code? Plane wave code do not make any approximation on the shape of the potential: the basis is complete. Supplemented by the PAW method, plane wave code are efficient even for localized systems (with d/f electrons) (see later) New trends in computational approaches for many-body systems, June 2012, Sherbrooke p.15/70

19 Formalism: DFT+DMFT implementation, MLWF Maximally Localized Wannier Functions. A Wannier function is by definition unitarily related to Bloch functions and form an orthonormal basis. w R km (r) = ν U m,ν (k) Ψ k,ν (r) (-6) From the Bloch Wannier function, one can construct the Wannier function on site T. ( ) V 3 wtm R = dke ikt wkm R 2π (-6) BZ (depends on r T only). Wanniers or NMTO (Pavarini et al 04, Anisimov et al 05) Maximally localized Wanniers and NMTO (Lechermann et al 06) New trends in computational approaches for many-body systems, June 2012, Sherbrooke p.16/70

20 Formalism: DFT+DMFT implementation, MLWF (2) Flexible: one can choose the extension of the Wannier functions. The correlated orbitals are a subset of the basis set. Might be problematic Disentanglement for large systems. f-electron systems. Other Wannier functions can however be easily built! New trends in computational approaches for many-body systems, June 2012, Sherbrooke p.17/70

21 DMFT in the Kohn Sham basis Let s choose the Bloch basis as the basis for one electron quantities. The band structure and the one electron Hamiltonian H KS = ν,k Ψ kν ε kν Ψ kν A basis set for one electron quantities: B kα. The matrix element on the Hamiltonian can thus be computed as: H KS,αα (k) = B kα H KS B kα = ν B kα Ψ kν ε kν Ψ kν B kα. A definition for d/f local orbitals, let s call them χ R km. (m is an angular momentum index) New trends in computational approaches for many-body systems, June 2012, Sherbrooke p.18/70

22 DMFT in the Kohn Sham basis Let s choose the Bloch basis as the basis for one electron quantities. The band structure and the one electron Hamiltonian H KS = ν,k Ψ kν ε kν Ψ kν A basis set for one electron quantities: Ψ kν. The matrix element on the Hamiltonian can thus be computed as: H KS,νν (k) = Ψ kν H Ψ kν = ε kν δ ν,ν A definition for d/f local orbitals, let s call them χ R km. (m is an angular momentum index) New trends in computational approaches for many-body systems, June 2012, Sherbrooke p.18/70

23 DMFT in the Kohn Sham basis (2) All one electron quantities can be expressed in the basis set Ψ kν. Lets define projectors: Projectors: P R mν(k) χ R km Ψ kν, P R mν(k) Ψ kν χ R km. P matrix is in general nonsquare. The Self energy is a local quantity in DMFT, thus: Σ = χ R Tm Σ Rimp mm χ R Tm mm T New trends in computational approaches for many-body systems, June 2012, Sherbrooke p.19/70

24 DMFT in the Kohn Sham Bloch basis (3) Thus Σ αα (k,iω n ) = Ψ kν Σ(k) Ψ kν = mm P R mν(k) Σ Rimp m,m P R mν(k). The lattice Green s function G α,α (k,iω n ) can be expressed in the complete Bloch Basis Ψ kν. and projected to compute the local (impurity) Green s function. G imp mm (iω n ) = Pmν(k)P R m R ν (k) k νν { [(iω n +µ ε kν )I Σ(k,iω n )] 1}, νν New trends in computational approaches for many-body systems, June 2012, Sherbrooke p.20/70

25 Choice of the basis set and local orbitals The local atomic-like orbitals χ R km will in general have a decomposition involving all Bloch bands (closure relation) χ R km = [ ] Ψ kν χ R km Ψ kν = Ψ kν Ψ kν χ R km ν If the basis set is restriced to a limited number of KS states in the window energyw: χ R km ν W Ψ kν χ R km Ψ kν = P R mν(k) Ψ kν ν χ R km need to be orthonormalized to give true Wannier functions wkm R. O R,R m,m (k) = χ R km χ R km = ν P R mν(k)p R m ν (k) New trends in computational approaches for many-body systems, June 2012, Sherbrooke p.21/70

26 Choice of the basis set and local orbitals wkm R is thus obtained though: wkm R = { [O(k)] 1/2} R,R R,m m,m χr km (-15) The localized basis depends on the energy extension of the window W. wkm R are less localized than initial χr km since the basis has a smaller energy range. The localized basis depends on the energy extension of the basis. If wkm R is chosen (through the window of energy), the whole framework is exact: There is no finite basis effect. wkm R Ψ kν = 0 if ν is outside the window of energy which defines wkm R. No convergence as a function of the size of the basis. New trends in computational approaches for many-body systems, June 2012, Sherbrooke p.22/70

27 Basis set and local orbitals: Physical considerations A reminder: hybridization in a diatomic molecule (oversimplified) H or V F or O New trends in computational approaches for many-body systems, June 2012, Sherbrooke p.23/70

28 An oversimplified derivation for the diatomic molecule VO φ V ε 2 Ψ 2 = βφ O αφ V β α ε 1 φ O Ψ 1 = αφ O +βφ V β α Two windows of energy are possible to compute χ = W Ψ i φ V Ψ i If only ε 2 is included, the correlated wavefunction is χ = Ψ 2 = β φ O α φ V and contains an Oxygen contribution If only ε 1 and ε 2 is included, the correlated wavefunction is χ = i Ψ i φ V Ψ i = φ V and is much more localized. New trends in computational approaches for many-body systems, June 2012, Sherbrooke p.24/70

29 Example of SrVO 3 (electrons/ev) total O-p V-t 2g V-e g ev metallic oxide with one d electron on the correlated atom (Vanadium). Hybridization between Oxygen and Vanadium. In a cubic environnement, d orbitals are splitted in two subgroups called t 2g and e g. New trends in computational approaches for many-body systems, June 2012, Sherbrooke p.25/70

30 SrVO3: band structure O-p V-t 2g V-e g (ev) R Γ X M Γ R Γ X M Γ R Γ X M Γ Fatbands Hybridization between O-p orbitals and V-t 2g orbitals New trends in computational approaches for many-body systems, June 2012, Sherbrooke p.26/70

31 Correlated orbitals for two windows of energy t2g bands only t2g bands and Oxygen p bands More extended Less extended From F. Lecherman, A. Georges, A. Poteryaev, S. Biermann, M. Posternak, A. Yamasaki and O.K. Andersen PRB 2006 New trends in computational approaches for many-body systems, June 2012, Sherbrooke p.27/70

32 Interaction parameters U and J t2g bands only t2g bands and Oxygen p bands More extended Less extended The interaction parameters between two electrons in orbitals m and m areu m,m = φ m φ m W(r,r ) φ m φ m. In the windows of energy increases, the value of U used should increase also, because the localization of correlated orbitals increases. New trends in computational approaches for many-body systems, June 2012, Sherbrooke p.28/70

33 Photoemission spectra of d elements. Quasiparticle peak Lower Hubbard band (~ 1.8eV) Upper Hubbard band (~2.5eV) From Morikawa et al (1995) Sekiyama 1992 YTiO 3 insulator: metal in LDA. SrVO 3 is a metal: metal en LDA, but without the peak at -1.8eV. New trends in computational approaches for many-body systems, June 2012, Sherbrooke p.29/70

34 (ev -1 ) Comparison of DFT+DMFT results MLWF Wannier from PLO G(τ ) τ (ev 5-1 ) (ev) Comparison of Spectral function for SrVO3 computed: Maximally Localized Wannier Functions (MLWF) Wannier functions obtained by Projection of Localized Orbitals (PLO). New trends in computational approaches for many-body systems, June 2012, Sherbrooke p.30/70

35 Interactions in DFT+U/DFT+DMFT E ee = 1 2 E ee = 1 2 σ 1,2,3,4 σ 1,3 [ ] 13 V ee 24 n σ 1,2n σ 3,4 +( 13 V ee V ee 42 )n σ 1,2n σ 3,4 [ 13 Vee 13 n σ 1n σ 3 +( 13 V ee V ee 31 )δ 13 n σ 1n σ ] 3 13 V ee 24 = 4π k=0,2,4,6 F k 2k +1 +k m= k m1 m m2 m3 m m4 Only density density terms are used in the interaction term: H = 1 3 1,3,σ U σ,σ 1,3 ˆnσ 1ˆn σ ,3,σ U σ, σ 1,3 ˆn σ 1ˆn σ 3 With U σ, σ 1,3 = 13 V ee 13 and U σ,σ 1,3 = 13 V ee V ee 31 New trends in computational approaches for many-body systems, June 2012, Sherbrooke p.31/70

36 Double counting corrections Double counting corrections: Atomic limit (or Full localized limit) [Lichtenstein(1995), Anisimov (1991)]: E FLL dc = t ( U 2 N(N 1) σ J 2 Nσ (N σ 1)) Around mean field version [Czyzyk(1994)] (delocalized limit): E AMF dc = t (UN N (N2 +N2 ) 2l (U J)) 2l+1 (Made to correct the delocalized limit.) New trends in computational approaches for many-body systems, June 2012, Sherbrooke p.32/70

37 Solvers available in the public version: Hubbard I Hubbard I solve the Anderson Impurity neglecting the hybridization between the impurity and the other atoms. H = m ǫ m c mc m We can compute the effective levels as m 1 m 2 U m1 m 2ˆn m1ˆn m2 ǫ m = χ m ˆ H lda χ m Σ DC µ = kν P mν (k)ǫ vk P νm(k) Σ DC µ. In the Hubbard I, Green s function is written as G atomique (iω n ) = 1 Z at AB A d B 2 iω n +E B E A (e βe A +e βe B ) E A = j n j ǫ j + 12 U 12 n 1 n 2 New trends in computational approaches for many-body systems, June 2012, Sherbrooke p.33/70

38 Example of input/output of HI for NiO == Print Diagonalized Energy levels -- polarization spin component polarization spin component Hubbard I: Energies as afunction of number of electrons Nelec Min. Ene. Max. Ener. HI HI HI HI HI HI HI HI HI HI HI New trends in computational approaches for many-body systems, June 2012, Sherbrooke p.34/70

39 Example of input/output of HI for NiO === Integrate green function > For Correlated Atom 1 Nb of Corr. elec. is: == The occupations (integral of the Green function) are == = In the atomic basis > For Correlated Atom 1 -- polarization spin component polarization spin component New trends in computational approaches for many-body systems, June 2012, Sherbrooke p.35/70

40 Solvers available in the public version: Hubbard I Drawback: number of electrons computed from the atom is an integer, no hybridization,... In progress (J. Bieder and BA): Implementation of CTQMC Strong coupling solver (see presentation of P. Werner).Done and under test. New trends in computational approaches for many-body systems, June 2012, Sherbrooke p.36/70

41 The DFT+DMFT scheme (1): a practical calculation H KS DFT DFT+DMFT Loop over density χ R km Ψ kν ǫ KS DFT ν,k G 0 DMFTforfixed n(r) Impurity problem G n(r) f DFT+DMFT ν,ν,k G 1 0 = Σ + G 1 imp G imp Self-consistency condition G imp G latt Σ = G 1 0 G 1 imp Σ First step: Make a converged DFT calculation. Wavefunctions Ψ kν and Eigenvalues ε kν are available. Unoccupied wavefunctions must be accurately computed in order to be used in the DMFT calculation. New trends in computational approaches for many-body systems, June 2012, Sherbrooke p.37/70

42 Diagonalise the Hamiltonian Solve DFT self-consistently Diagonalize the Hamiltonian with empty states Get Eigenvalues and eigenvectors. New trends in computational approaches for many-body systems, June 2012, Sherbrooke p.38/70

43 φi The Projector Augmented Wave Method ~ n = n + ( ~φi ) ~ p ~ ~ n =τ n i i PAW Efficiency with respect to usual plane wave calculations. Accuracy: nodal structure of valence wfc are reproduced. Flexibility: energy of several structures can be compared with the same basis. Frozen core approximation can be controlled. From M. Torrent and F. Jollet Torrent, Jollet, Bottin, Zerah, Gonze CMS 2008 New trends in computational approaches for many-body systems, June 2012, Sherbrooke p.39/70

44 The Projector Augmented Wave Method Ψ = Ψ + i p i Ψ φ i i p i Ψ ϕ i FIG. 3: Bonding p-σ orbital of the Cl 2 molecule and its decomposition of the wavefunction into auxiliary wavefunction and the two one-center expansions. Top-left: True and auxiliary wave function; top-right: auxiliary wavefunction and its partial wave expansion; bottom-left: the two partial wave expansions; bottom-right: true wavefunction and its partial wave expansion. Handbook of Materials Modeling; Bloechl, Kaestner, Foerst Implemented in Abinit. Torrent, Jollet, Bottin, Zerah, Gonze,CMS, 2008 New trends in computational approaches for many-body systems, June 2012, Sherbrooke p.40/70

45 Calculation of projections within PAW Projectors: P R mν(k) χ R km Ψ kν, P R mν(k) Ψ kν χ R km. P mν (k) = χ m Ψ kν + i p i Ψ kν ( χ m ϕ i χ m ϕ i ) However, atomic d or f wavefunctions are mainly localized inside spheres. P mν (k) = n i p ni Ψ kν χ m ϕ ni Check completness during contruction of atomic data. Check that PAW radius is not too small. Logarithmic derivatives Base completness relative to energy Data constructed with Atompaw (Holzwarth, Torrent and Jollet 2007.) New trends in computational approaches for many-body systems, June 2012, Sherbrooke p.41/70

46 The DFT+DMFT scheme (2) H KS DFT DFT+DMFT Loop over density χ R km Ψ kν ǫ KS DFT ν,k G 0 DMFTforfixed n(r) Impurity problem G n(r) f DFT+DMFT ν,ν,k G 1 0 = Σ + G 1 imp G imp Self-consistency condition G imp G latt Σ = G 1 0 G 1 imp Σ Fromε kν, compute DFT (LDA) Green s function: G Bloch ν,ν (iω n,k) = [(iω n +µ ε kν )I] 1 νν G imp mm (iω n ) = Pmν(k)P R m R ν (k) G Bloch ν,ν (iω n,k) k νν New trends in computational approaches for many-body systems, June 2012, Sherbrooke p.42/70

47 The DFT+DMFT scheme (3) H KS DFT DFT+DMFT Loop over density χ R km Ψ kν ǫ KS DFT ν,k G 0 DMFTforfixed n(r) Impurity problem G n(r) f DFT+DMFT ν,ν,k G 1 0 = Σ + G 1 imp G imp Self-consistency condition G imp G latt Σ = G 1 0 G 1 imp Σ 1 st iteration: DFT LDA Green s function used as Weiss field From Weiss Field G 0m,m, Anderson Impurity Model is solved to get impurity Green s function G m,m. New trends in computational approaches for many-body systems, June 2012, Sherbrooke p.43/70

48 The DFT+DMFT scheme (4) H KS DFT DFT+DMFT Loop over density χ R km Ψ kν ǫ KS DFT ν,k G 0 DMFTforfixed n(r) Impurity problem G n(r) f DFT+DMFT ν,ν,k G 1 0 = Σ + G 1 imp G imp Self-consistency condition G imp G latt Σ = G 1 0 G 1 imp Σ Self-energy is computed from Dyson Equation : Σ = G 1 0 G 1 The Lattice Green s function is computed with the self-energy. Fermi level ajusted to the number of electrons The Impurity Green s function is obtained by projection. Dyson equation is again used to recover new Weiss Field. New trends in computational approaches for many-body systems, June 2012, Sherbrooke p.44/70

49 Self energy and Lattice Green function Σ imp m,m is obtained from the Dyson equation, then: Σ αα (k,iω n ) = Ψ kν Σ(k) Ψ kν = mm P R mν(k) Σ imp m,m P R mν(k). The lattice Green s function G ν,ν (k,iω n ) can be expressed in the complete Bloch Basis Ψ kν and projected to compute the local (impurity) Green s function. { G ν,ν (k,iω n ) = [(iω n +µ ε kν )I Σ(k,iω n )] 1}, νν G imp mm (iω n ) = Pmν(k)P R m R ν (k) k νν { [(iω n +µ ε kν )I Σ(k,iω n )] 1}, νν New trends in computational approaches for many-body systems, June 2012, Sherbrooke p.45/70

50 The DFT+DMFT scheme (5) H KS DFT DFT+DMFT Loop over density χ R km Ψ kν ǫ KS DFT ν,k G 0 DMFTforfixed n(r) Impurity problem G n(r) f DFT+DMFT ν,ν,k G 1 0 = Σ + G 1 imp G imp Self-consistency condition G imp G latt Σ = G 1 0 G 1 imp Σ The DMFT Loop is done until convergency (of G, Σ). Then Occupations of electrons inside orbitals, have changed and thus, the local density must be updated. New trends in computational approaches for many-body systems, June 2012, Sherbrooke p.46/70

51 The self-consistency over electronic density The number of electron are recovered by a Fourier transform: f ν,ν (k) = iω n G ν,ν (k,iω n )e iω n0 + The Green function can be written in real space basis as: G(r,r ) = ] r [Ĝ(iωn ) r e iω n0 + iω n G(r,r ) = r Ψ kν G ν,ν (k,iω n ) Ψ kν r e iω n0 + iω n k,ν,ν The local density can be written as n(r) = G(r,r) n(r) = k,ν,ν Ψ kν (r)f ν,ν (k)ψ kν (r ) New trends in computational approaches for many-body systems, June 2012, Sherbrooke p.47/70

52 The DFT+DMFT scheme (6) H KS DFT DFT+DMFT Loop over density χ R km Ψ kν ǫ KS DFT ν,k G 0 DMFTforfixed n(r) Impurity problem G n(r) f DFT+DMFT ν,ν,k G 1 0 = Σ + G 1 imp G imp Self-consistency condition G imp G latt Σ = G 1 0 G 1 imp Σ From total density, the Hamiltonian H[n(r)] is diagonalized and new KS wavefunctions, projectors, and eigenvalues are computed for the next DMFT loop. At convergence of the electronic density n(r), the calculation is stopped. New trends in computational approaches for many-body systems, June 2012, Sherbrooke p.48/70

53 Analysis of DMFT calculations Spectral Function Impurity spectral function: From Green s function projected over local correlated orbitals Bloch states spectral function (k-resolved or sum): Green s function of Kohn Sham states. Self-energy Total energy New trends in computational approaches for many-body systems, June 2012, Sherbrooke p.49/70

54 SrVO3: band structure O-p V-t 2g V-e g (ev) R Γ X M Γ R Γ X M Γ R Γ X M Γ Fatbands Hybridization between O-p orbitals and V-t 2g orbitals New trends in computational approaches for many-body systems, June 2012, Sherbrooke p.50/70

55 (ev -1 ) Spectral functions MLWF Wannier from PLO G(τ ) τ (ev 5-1 ) (ev) Comparison of Spectral function for SrVO3 computed: Maximally Localized Wannier Functions (MLWF) Wannier functions obtained by Projection of Localized Orbitals (PLO). New trends in computational approaches for many-body systems, June 2012, Sherbrooke p.51/70

56 (ev -1 ) (ev -1 ) Formalism: Spectral function of SrVO O-p Bloch V-t2g projection (hybridization) V-t2g Bloch Dos partielle V-t 2g Bloch O 2p Bloch V t2g LDA+DMFT LDA (ev) Full orbitals. The position of O p bands % V d Redefinition of local orbitals More localized Amadon, Lechermann, Georges, Jollet, Wehling and Lichtenstein Phys. Rev. B 77, (2008) Anisimov, Kondakov, Kozhevnikov, Nekrasov et al PRB (2005) New trends in computational approaches for many-body systems, June 2012, Sherbrooke p.52/70

57 Functionals A functional is built from the partition function (Kotliar and Savrasov (2004)): Ω[ρ(r),G ab ;v KS (r), Σ ab ] LDA+DMFT = tr ln[iω n +µ v KS (r) χ. Σ.χ] dr(v KS v c )ρ(r) tr[g. Σ] + 1 drdr ρ(r)u(r r )ρ(r )+E xc [ρ(r)] 2 + ( ) Φ imp [G RR ab ] Φ DC[G RR ab ] R minimisation equations of LDA+DMFT. Total energy. E LDA+DMFT = E DFT λ ε LDA λ + H KS + H U E DC The one electron part of DFT is replaced by one electron part of DMFT, and DMFT interaction terms are added. New trends in computational approaches for many-body systems, June 2012, Sherbrooke p.53/70

58 Total energy: first, reminder of total energy in DFT E DFT = T DFT 0 +E xc+ha [n(r)]+ drv ext (r)n(r) Alternatively E DFT = ν,k f DFT ν,k ǫ DFT ν,k +E DFT DC [n(r)] with E DFT DC [n(r)] = E Ha [n(r)]+e xc [n(r)] v xc n(r)dr New trends in computational approaches for many-body systems, June 2012, Sherbrooke p.54/70

59 Total energy in the DFT+DMFT framework Thus, we have the following expression for the total energy in DFT+DMFT: EDFT+DMFT 2 = T0 DFT+DMFT +E xc+ha [n(r)]+ drv ext (r)n(r)+ H U E DC with T0 DFT+DMFT = ν,ν,k fdft+dmft ν,ν,k Alternatively, one can write: Ψ kν 2 2 Ψ kν E 1 DFT+DMFT = ν,k f DFT+DMFT ν,k ǫ KS DFT ν,k +E DFT DC [n(r)]+ H U E DC with E DFT DC [n(r)] = E Ha [n(r)]+e xc [n(r)] v xc n(r)dr Should give the same results at convergence. New trends in computational approaches for many-body systems, June 2012, Sherbrooke p.55/70

60 Input variables # == LDA+U usepawu 1 # Use U Hamiltonian dmatpuopt 3 # Choose density matrix (only for print out) lpawu 2-1 # Use Hubbard U for l=lpawu upawu ev # Value of U for the two species f4of2_sla # Value of F4/F2 to compute <m1m2 1/r m3m4> # == LDA+DMFT usedmft 1 # Enable DMFT dmftbandi 6 # First KS Band to use dmftbandf 12 # Last KS Band to use dmft_nwlo 100 # Nb of Freq in the Log Grid dmft_nwli # Nb of Freq in the Linear Matsubara dmft_iter 10 # Nb of Iter for the DMFT Loop dmftcheck 1 # Enable checks (Symmetry, Projection, Fourier) dmft_solv 2 # Choice of the solver dmft_rslf 1 # Read Self Energy from File (for Restart) dmft_mxsf 0.7 # Mixing Coefficient for Self Energy dmft_dc 1 # Double counting for DMFT #dmft_tollc # Tolerance over local charge # # for convergence of the DMFT Loop # (in the next version of ABINIT) occopt 3 # Fermi Dirac smearing tsmear 500 K # Temperature New trends in computational approaches for many-body systems, June 2012, Sherbrooke p.56/70

61 Automatic tests v6/t07 LDA calculation through the DMFT Loop (check) v6/t45 DMFT calculation with U=0, and U/=0 for several simple solvers v6/t46 DMFT calculation with two Ni atoms. v6/t47 DMFT calculation for f-orbitals (Gd) Several Internal checks: Projections Comparison of LDA occupations with wavefunctions or Green s function Nevertheless, always read the whole log file to check. New trends in computational approaches for many-body systems, June 2012, Sherbrooke p.57/70

62 Parallelization In the Hubbard one implementation, the most expensive task is the calculation and integration of Green s function. G ν,ν (k,iω n ) = { [(iω n +µ ε kν )I Σ(k,iω n )] 1} νν, This part is thus parallelized over logarithmic frequencies. Computed on a log frequency New trends in computational approaches for many-body systems, June 2012, Sherbrooke p.58/70

63 Comparison DFT+U/DFT+DMFT(HF) If three conditions are met, DFT+U and DFT+DMFT calculations can be compared for testing purposes. a large number of bands have to be used in order that Wannier function used in DMFT looks like atomic orbitals. a specific density matrix have to be used in DFT+U, namely: n R,σ m,m = kν f σ k,ν χ R m Ψ σ k,ν Ψσ k,ν χr m χ R χ R The allows for the renormalization of the projections as it is done in DFT+DMFT. a Hartree Fock self-energy need to be used to carry out the calculation. New trends in computational approaches for many-body systems, June 2012, Sherbrooke p.59/70

64 Isostructural transition in Cerium Isostructural transition V γ V α V γ = 15% α phase: Non-magnetic. γ phase: Paramagnetic β phase: Antiferromagnetic α phase: f 1 electron more delocalized. γ phase: e is localized Johansson 1974 New trends in computational approaches for many-body systems, June 2012, Sherbrooke p.60/70

65 Cerium: experimental spectra and LDA Experiment (Wuilloud et al 1983, Wieliczka et al 1984) Ce Ce (ev) Experimental photoemission spectra (ev) LDA density of states. Peak at the Fermi level only in the α phase. γ and α phase: high energy bands (-2 ev and 5 ev). bands at high energy not described in LDA. peak at the Fermi level not correct in LDA E dft lda (V): γ phase not stable New trends in computational approaches for many-body systems, June 2012, Sherbrooke p.61/70

66 Comparison DFT+U/DFT+DMFT(HF) 10 5 LDA+DMFT(HF) LDA+U (ev) Spectral function of γ-cerium, LDA+U and LDA+DMFT (Static) With the LDA+U Self-energy, the DMFT recover the LDA+U results at convergence of the KS basis. New trends in computational approaches for many-body systems, June 2012, Sherbrooke p.62/70

67 Comparison DFT+U/DFT+DMFT(HF) 10 5 LDA+DMFT(HF) LDA+U (ev) a (a.u.) B 0 (GPa) PAW/LDA+U PAW/LDA+DMFT(HF) New trends in computational approaches for many-body systems, June 2012, Sherbrooke p.63/70

68 Results: Spectral function of Ce 2 O NSCF SCF (ev) LMTO-ASA +DMFT PAW + DMFT L. Pourovskii, B. A., S. Biermann and A. Georges PRB (2007) B. Amadon JPCM 2012 New trends in computational approaches for many-body systems, June 2012, Sherbrooke p.64/70

69 Results: Spectral function of Ce 2 O XPS+BIS SCF (ev) Spectral function of Ce 2 O 3, in LDA+DMFT (Hubbard I) with and without self-consistency New trends in computational approaches for many-body systems, June 2012, Sherbrooke p.65/70

70 Results: Structural parameters of Ce 2 O 3 a (A) B 0 (Mbar) Exp(Barnighausen 1985) PAW/LDA+U(AFM)(Da Silva 07) PAW/LDA+U(AFM) PAW/LDA+DMFT (H-I) NSCF PAW/LDA+DMFT (H-I) SCF ASA/LDA+DMFT(H-I) NSCF (Pourovskii 2007) ASA/LDA+DMFT(H-I) SCF (Pourovskii 2007) Lattice parameteraand Bulk modulus B 0 of Ce 2 O 3. New trends in computational approaches for many-body systems, June 2012, Sherbrooke p.66/70

71 Results: Converged Electronic Density. Difference between electronic densities computed in the LDA+U (left)/lda+dmft (right) and in LDA for Ce 2 O 3. Blue (resp. green-red) area corresponds to positive (resp negative) value of the difference. New trends in computational approaches for many-body systems, June 2012, Sherbrooke p.67/70

72 Implementation of the LDA+DMFT (self-consistency) a (a.u.) B 0 (GPa) Exp[Jeong 2004] /21 PAW/LDA+U PAW/LDA+DMFT NSCF (H-I) PAW/LDA+DMFT SCF (H-I) ASA/LDA+DMFT NSCF (H-I) ASA/LDA+DMFT SCF (H-I) Lattice parameter a and Bulk modulus B 0 of γ Cerium according to experimental data and calculations New trends in computational approaches for many-body systems, June 2012, Sherbrooke p.68/70

73 Conclusion DFT+DMFT physical choices Interactions U, J Definition of local correlated orbitals: Wannier functions Technical choices Basis for the one electron Green function: KS in Abinit Solver: Hubbard one In progress and/or not yet distributed: CT-Quantum Monte Carlo (see talk of P. Werner) Spin-orbit Improve parallelism Details of the Abinit implementation can be found in Amadon B, Lechermann F, Georges A, Jollet F, Wehling T O and Lichtenstein A I 2008 Phys. Rev. B Amadon B, 2012 J. Phys.: Condens. Matter New trends in computational approaches for many-body systems, June 2012, Sherbrooke p.69/70

74 Some Review, lectures notes, and introductary articles A. Georges, G. Kotliar, W. Krauth, and M. J. Rozenberg Rev. Mod. Phys. 68, 13 (1996) A. Georges cond-mat/ Lectures on the Physics of Highly Correlated Electron Systems VIII (2004) 3, AIP Conf. Proc G. Kotliar and D. Vollhardt Physics Today Mars 2004 G. Kotliar, S. Y. Savrasov, K. Haule, V. S. Oudovenko, O. Parcollet, C. A. Marianetti, Rev. Mod. Phys. 78, 865 (2006) K. Held, Advances in Physics (2007) New trends in computational approaches for many-body systems, June 2012, Sherbrooke p.70/70