Wannier Functions in the context of the Dynamical Mean-Field Approach to strongly correlated materials

Transcription

1 Wannier Functions in the context of the Dynamical Mean-Field Approach to strongly correlated materials Frank Lechermann I. Institute for Theoretical Physics, University of Hamburg, Germany Ab-initio Many-Body Theory San Sebastian p./6

2 Outline Survey: Basis sets in electronic-structure theory Wannier functions: definition and general properties Constructing Wannier(-like) functions Dynamical Mean-Field Theory using Wannier functions Example: The puzzling physics of BaVS 3 Conclusions and outlook. p./6

3 Basis sets in electronic-structure theory Ground state energy of a many-electron system: E v [ρ] = T s [ρ] + d 3 rρ(r)v(r) + d 3 rd 3 r ρ(r)ρ(r ) r r + E xc [ρ] ρ : electronic charge density v : external potential (ions) T s : kinetic energy of noninteracting electrons Kohn-Sham equations in solids: ] [ + v KS(r) ψ kν (r) = ε kν ψ kν (r) ψ kν (r) : effective single-particle Bloch function, ρ(r) P kν f kν ψ kν (r) Representation: eigenvalue equation: ψ kν (r) = α ckν α B k α(r) α [ H k αα ε kν S k αα ] c kν α =. p.3/6

4 Basis sets in electronic-structure theory Representation: ψ kν (r) = α c kν α Bk α (r) tight binding (LCAO) plane waves ψ kν (r) = X Rnlm c kν Rnlm Bk Rnlm (r) ψ kν (r) = X G c kν G Bk G (r) B k Rnlm (r) = X T e ik T φ Rnlm (r T) B k G (r) = ΩC e i(k+g) r no orbital-dependent interaction in standard DFT freedom of choice chemical picture itinerancy programming computational cost. p.4/6

5 Wannier functions: definition crystal wave function ψ kν (r) is periodic in reciprocal lattice for fixed r: ψ kν (r) = w Tν (r) e ik T w Tν (r) = dkψ kν (r) e ik T Ω B T Ω B Bloch theorem: ψ(r + T) = e ik T ψ(r) w Tν (r) = w ν (r T) Wannier function: w ν (r) = Ω B Ω B dkψ kν (r) single function for νth band not unique since ψ kν ψ kν e iϕ k form complete orthonormal set: Z dr wν (r T)w ν (r T ) = δ νν δ TT Ω C X wν (r T)w ν (r T) = δ(r r) Tν can be chosen to be purely real or purely imaginary. p.5/6

6 Wannier functions: localization properties Wannier functions atomic orbitals? one-dimensional case: [Kohn, PR 5 (959)] isolated band: WF can be chosen exponentially localized, i.e., w(r) e ar two limits: w ν (r) 8 < : e ε ν r two- and three-dimensional case: : tightly bound r : nearly free quantum chemistry limit: Boys orbitals [Boys, RMP 3 (96)] localization via composite bands and band-projection techniques [Blount, Solid State Phys. 3 (96)] [des Cloizeaux, PR 35 (964)] maximally-localized Wannier functions [Marzari and Vanderbilt, PRB 56 (997)] Nth order muffin tin orbitals (NMTO) (intuitive localization) [Andersen, Saha-Dasgupta, PRB 6 ()] exact proof: exponential localization for insulators that display time-reversal symmetry [Brouder et al. PRL 98 (7)]. p.6/6

7 Wannier functions: maximally-localized [Marzari and Vanderbilt, PRB 56 (997)] w ν (r) = Ω B Ω B dk ν U (k) νν ψ kν (r) unitary matrix U (k) may be chosen in order to enforce certain properties on the Wannier functions Define F = ( r ν r ν), O ν = dro w ν (r) ν Recipe: Determine U (k) as the transformation that minimizes F isolated set of bands (n =, N): from ψ nk (r)=e ik R u nk (r) calculate N N matrix M (k,q) nm = u nk u mk+q and express Ω via M -5 ε-ε F (ev) silicon L Γ X U entangled set of bands (n =, N): determine most smooth Wannier-like bands and perform localization [Souza et al., PRB 65 ()]. p.7/6

8 Wannier functions: NMTO and projection From downfolding to NMTO: [Andersen, Saha-Dasgupta, PRB 6 ()] standard Löwdin downfolding of basis set into active (φ A ) and passive (φ P ) orbitals φ A (ε) H φ A (ε) = φ A H φ A + φ A H φ P φ P ε H φ P φ P H φ A linearize φ P ε H φ P NMTO: construct φ A via intuitive localization for N reference energies For N large and upon orthonormalization NMTOs converge to Wannier functions for the active bands Band-projection techniques: [des Cloizeaux, PR 35 (964)] Generate v T from local trial function t T via band-projection v Tν = ˆP t T = X k ψ k ψ k t T v T are nonorthonormal Wannier-like functions (NWFs) with overlap S T = v v T = t ˆP t T additional localization by going to dual functions y T [He and Vanderbilt, PRL 86 ()] y = X T `S T v T y v T = y t T = δ T. p.8/6

9 Dynamical Mean-Field Theory (DMFT) Hubbard model at half filling occupation empty single double U t U t U t U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U ideal metal correlated metal Mott insulator Introduce time-dependent Weiss field to map lattice problem onto impurity problem by integrating out the effect of all lattice sites but one: Σ imp G G imp DMFT: G loc = G imp G loc (iω n ) = X k iω n + µ ε k Σ imp (iω n ) U The dynamical mean-field G (τ τ ) allows to take care of all local quantum fluctuations within DMFT. The theory is designed to treat both quasiparticles and states originating from atomic-like excitations on equal footing. [Georges and Kotliar, PRB 45 (99)] [Metzner and Vollhardt, PRL 6 (989)] G (τ τ ). p.9/6

10 LDA+DMFT: Implementation schemes Various ways of combining DFT(LDA) with DMFT have been introduced: Earlier realizations: Correlated orbitals within the LMTO basis set {B k Rlm } [V.I. Anisimov, A.I. Poteryaev, M.A. Korotin, A.O. Anokhin, and G. Kotliar, JPCM 9, 7359 (997)] [A.I. Lichtenstein and M.I. Katsnelson, PRB 57, 6884 (998)] Wannier functions for the correlated subspace: NMTO [E. Pavarini, S. Biermann, A. Poteryaev, A.I. Lichtenstein, A. Georges, and O.K. Andersen, PRL (4)] Projection method [V. I. Anisimov, D. E. Kondakov, A. V. Kozhevnikov, I. A. Nekrasov, Z. V. Pchelkina, J. W. Allen, S.-K. Mo, H.-D. Kim, P. Metcalf, S. Suga, A. Sekiyama, G. Keller, I. Leonov, X. Ren and D. Vollhardt, PRB 7 59 (5)] Maximally-localized [FL, A. Georges, A. Poteryaev, S. Biermann, M. Posternak, A. Yamasaki, and O.K. Andersen, PRB 74, 5 (6)]. p./6

11 LDA+DMFT: correlated subspace so far only single-band, model-type hopping: G loc (iω n ) = X k iω n + µ ε k Σ(iω n ) Start from full lattice Green s function of the correlated solid: ] G(r,r ;iω n ) = r [iω n + µ + ˆv KS ˆΣ r Self-energy Σ lives in correlated subspace C defined by set of localized orbitals {χ Rm (r)} centered at site R: Σ(r,r ;iω n ) Tmm χ m(r R T) Σ mm (iω n )χ m (r R T) Define projection operator onto correlated subspace C: ˆP (C) R m C χ Rm χ Rm Ĝ loc = ˆP (C) R (C) Ĝ ˆP R. p./6

12 LDA+DMFT: representation! DMFT self-consistency condition: Ĝ imp = Ĝ loc = ˆP (C) R Ĝ ˆP (C) R Representation in arbitrary basis set {B k α} G imp mm (iω n ) = X k X αα χ k m B k α Σ αα (k, iω n ) = X mm B k α χ k m Σ mm (iω n ) χ k m B k α n [(iω n + µ) H KS (k) Σ(k, iω n )] o αα Bk α χ k m Choice: Wannier functions for {B k α W} and {χ k m C} (W C) w kα X T e ik T w Tα = X ν W U (k) αν ψ kν Ĥ (W) KS (k) = X ˆΣ (C) (iω n ) = αα W X mm C,k w kα H αα (k) w kα w km Σ(iω n ) w km G imp mm (iω n ) = P j h i ff k (iω n + µ) H (W) KS (k) Σ(C) (iω n ) mm example: H KS (k) H sp(k) H sp,d (k) H d,sp (k) H dd (k) A, Σ(iω n ) Σ dd (iω n ) A. p./6

13 Strongly correlated materials: principles FIG. 59. Variation of d p for the LaMO 3 series with the metal M element. From Mahadevan et al., 996. [Mattheis PRB 5 (97)] [Zaanen, Sawatzky, Allen, PRL 55 (985)] [Imada et al. RMP 7 (998)] Example: SrVO 3 is a 3d transition-metal oxide with full cubic symmetry: ε ε F (ev) DOS (/ev) total V(t g ) V(e g ) O(p) R Γ X M Γ E-E F (ev). p.3/6

14 Strongly correlated materials: cuprates ε d ε p splitting is rather small for copper-oxides... Single-band NMTO Wannier Function (from O.K. Andersen and coworkers). p.4/6

15 LDA+DMFT: flowchart 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 ] construct initial Ĝ DMFT loop ρ update compute new chemical potential µ ρ(r) = ρ KS (r) + ρ(r) (Appendix A) Ĝ = Ĝ loc + ˆΣ imp impurity solver G imp mm (τ τ ) = ˆT ˆd mσ (τ) ˆd m σ (τ ) Simp ˆΣ imp = Ĝ Ĝ imp self-consistency condition: construct Ĝloc Ĝ loc = ˆP (C) R [Ĝ KS (ˆΣimp ˆΣ )] (C) dc ˆP R double-counting term Σ dc is a tricky issue ( GW+DMFT?) most applications so far: W=C Full self-consistency over the charge density only realized in MTO(-like) schemes. p.5/6

16 Application: BaVS 3 the vanadium sulfide shows three continuous phase transitions: T 4 K : hexagonal to orthorhombic structural transition T 7 K : metal-to-insulator transition (MIT) from Curie-Weiss metal to paramagnetic insulator, structural transition to monoclinic phase T 3 K : incommensurate antiferromagnetic transition [Sayetat et al. J. Phys. C 5 67] orthorhombic (Cmc ) structure at T = K: [Ghedira et al., J. Phys. C ] - zigzag VS 3 chains - two formula units in primitive cell - d inter VV dintra VV [Fagot et al., PRL 9 964]: large one-dimensional structural fluctuations along c axis above MIT. p.6/6

17 The intriguing physics of BaVS 3 Hall coefficient: resistivity/mag. susceptibility: charge density wave below MIT: [Booth et al., PRB 6 485] [Graf et al., PRB 5 37] [Inami et al., PRB ] V(3d ) system e g e g mutually hybridizing A g orbitals along c axis narrow E g bands at the Fermi level MIT vanishes at critical pressure [Forró et al., PRL ] 3d A g E g e g A g E g E g atomic hexagonal orthorhombic [Massenet et al., J. Phys. Chem. Solids ]. p.7/6

18 LDA results for metallic (Cmc ) BaVS 3 narrow (.7 ev) E g bands at the Fermi level broader (.7 ev) folded A g band k F along Γ-Z:.94c experimental k F : qc CDW =.5c ε ε F (ev) 7K<T<4K: orthorhombic Cmc Γ C Y Γ Z E T Z q c CDW 6 A g 6 Fermi surface not flattened orbital populations? nature of E g states (Curie-Weiss behavior)? DOS (/ev) E g E g e g e g S(3p) E-E F (ev). p.8/6

19 Wannier functions for low-energy states ε-ε F (ev) LDA band structure Γ C Y Γ Z E T Z Wannier functions in crystal-field basis derived from maximally-localized construction [Marzari and Vanderbilt, PRB ] [Souza, Marzari, and Vanderbilt, PRB ] A g E g E g.5. E g E g A g DOS (/ev) E-E F (ev) Hoppings in mev A g E g E g A g -E g p.9/6

20 LDA+DMFT: integrated properties impurity on-site interaction Hamiltonian (U, U = U J, U = U 3J): Ĥ U = U X ˆn m ˆn m + U X ˆn mσˆn m σ + U X ˆn mσˆn m σ m mm σ m m mm σ m m integrated spectral function: temperature dependence: DOS (/ev) LDA LDA+DMFT A g E g E g ρ (/ev) ρ (/ev) β= ev - β=3 ev - A g E g E g ω (ev) orbital fillings: (n tot = ) U,J (ev) A g E g E g., , ω (ev) magnetic susceptibility: χ (loc) 4 3 U/J=7 U/J=4 A g E g E g total T/ (K). p./6

21 LDA+DMFT Quasiparticle states Self-energy: Σ(iω n ) = RΣ(iω n ) + IΣ(iω n ) analytical continuation and expansion: RΣ mm (ω + i + ) RΣ mm () + ` [Z ] mm ω O(ω ) IΣ mm (ω + i + ) Γ mm ω + O(ω 3 ) det[(ω k Z (H LDA (k) + RΣ() µ)] =. LDA LDA+DMFT ε-ε F (ev) -. Γ C Y Γ Z E T Z. ε-ε F (ev). -. Γ M/ A/ Z Γ. p./6

22 ( Reflectivity E c Reflectivity E c Recent measurements Angle-resolved photoemission (ARPES) [Mitrovic et al., PRB 75 (7)] [Mo, Allen et al., APS March Meeting 5] Optics [Kézsmárki et al., PRL ] - cm - ) A g E g E c K 45K 6K 73K 85K K 5K 3K.. 3 E g E A * g S( z ) E g c S(3p) V(3d) Energy (ev) E c E c E c E c p./6

23 BaVS 3 below MIT: insulating CDW state Cmc structure: orthorhombic two equivalent V atoms in unit cell d (chain) VV = 5.37 a.u. T < T MIT : Im structure [Fagot et al., Solid State Sci. 7 78] monoclinic doubling of unit cell four inequivalent V atoms tetramerization (trimerization) dominant k F distortion d (chain) VV =.7 a.u. d (chain) VV =.7 a.u. d (chain) VV =+.9 a.u. d (chain) VV =+. a.u. V(4) ( d VS =. a.u.) V(3) ( d VS =+. a.u.) V() ( d VS =+.34 a.u.) V() ( d VS =+.6 a.u.). p.3/6

24 LDA+DMFT for insulating BaVS3 Wannier functions from LDA LDA + cluster-dmft with 4-site impurity Eg majority occupation on V()/V() mixed Ag /Eg occupation on V(3)/V(4) substantial intersite Σ(ω) between V(3)/V(4) - Γ Z A M L U = 3.5 ev, J =.7 ev Im (4 K) Wannier-DOS Ag Eg Eg V(4) V(4) V(3) V() ρ (/ev) - V DOS (/ev) ε-εf (ev) V(3) V() V() V() - E-EF (ev) ω (ev) 3 4. p.4/6

25 LDA+DMFT for insulating BaVS 3. A g - A g A g - E g E g - A g E g - E g. Σ (ev) - V V Σ (ev) -.. V-V V-V3. - Re Σ Im Σ V3 ω n (ev) V4 -. V3-V4..5. ω n (ev) Re Σ Im Σ V4-V.5. onsite correlations intersite correlations. p.5/6

26 Conclusions LDA+DMFT may be implemented in any basis set of choice Definition of correlated subspace not unique Low-energy hamiltonian depends on given problem Wannier functions can provide important insight GENERALLY: More implementation and testing on the basis set issue of realistic strongly correlated systems has to be done... BaVS 3 poses interesting test case in strongly correlated physics Exhibits competing itinerant and localized states DFT-LDA not sufficient to treat the compound adequately LDA+DMFT capable of revealing basic mechanisms. p.6/6