Spectral Density Functional Theory Sergej Savrasov Financial support NSF US DOE LANL
Collaborators and Content Constructing New Functionals to Access Energetics and Spectra of Correlated Solids Phonons in Paramagnetic Mott Insulators with G. Kotliar Nature 410, 793 (2001) Phys. Rev. Lett. 90, 056401 (2003) arxiv:cond-mat/0308053 www.aps.org
Collaborators and Content Phonons & Phases in Plutonium with X. Dai, G. Kotliar, E. Abrahams H. Ledbetter, A. Migliori Nature 410, 793 (2001) SCIENCE VOL 300 9 MAY 2003 www.sciencemag.org
Density Functional Linear Response Tremendous progress in ab initio modelling lattice dynamics & electron-phonon interactions using LDA has been achieved
Superconductivity & Transport in Metals (after Savrasov et.al, PRB 1996)
Novel superconductors: MgB 2, LiBC,, MgCNi 3, etc Superconductivity in MgB 2 was recently studied using density functional linear response (after Y. Kong et.al. PRB 64, 020501 (R) 2002) Doped LiBC is predicted to be a superconductor with T c ~20 K (in collaboration with An, Rosner, Pickett, PRB 66, 220602(R) 2002) MgCNi 3 is recently predcted to be a strongly anharmonic superconductor (in collaboration with Ignatov, Tyson, PRB Rapids in press) Phosphorus under pressure and its implications for spintronics. In collaboration with Ostanin, Trubitsin, Staunton, PRL 91, 087002 (2003)
Problems with LDA theory LDA fails to reproduce spectra in strongly correlated systems, such as heavy fermions, high-tc s, Mott insulators, etc. Whether total energies & lattice dynamics as ground state property can be predicted accurately? Optical Γ-phonon in MnO within LSDA: 3.04 THz, Experimentally: 7.86 THz (Massidda, et.al, PRL 1999) Ground state volume of δ-pu is 30% too small within LDA. Bulk modulus is one order of magnitude too large (214 GPa vs. 30 GPa) Also elastic constants are off. (Bouchet, et.al, J.Phys.C, 2001)
Physics of Local Coulomb Interaction U: Hubbard 1964, Gutzwiller,, 1963, Anderson, 1961 Importance of on-site Coulomb correlations: Hubbard model at half-filling N(E) N(E) N(E) W Lower & upper Hubbard bands U Renormalization of quasiparticle band WZ E E E
Dynamical Mean-Field Theory Realistic Spectrum for strongly-correlated system using DMFT N(E) WZ E U N(E) LDA prediction W E
DMFT & Quantum Monte Carlo Simulations Development of semicircular density of states as U increases (Georges, Kotliar, Rosenberg, Krauth, RMP 1995)
Electronic Structure + Many-Body Effects Integration of advances: density functional electronic structure and many-body DMFT. Anisimov, Poteryaev, Korotin, Anokhin, Kotliar, J. Phys. Cond. Mat. 35, 7359 (1997) A Lichtenstein, M. Katsenlson, Phys. Rev. B 57 6884 (1998) Significant progress due to recent series of publications by the groups from IMF Ekaterinburg, University of Augsburg, LLNL Livermore, etc. SS,Kotliar, Abrahams, full self-consistent implementation and calculations of Pu: Nature 410, 793 (2001). Bierman+Ferdi+George, PRL 2003 with full GW+DMFT method
A functional theory alternative to DFT? New energy functional Γ has to be designed to access energetics and spectra in strongly correlated systems. Dynamical matrix Λ = µν RR ' 2 δ Γ δ R δ R µ ν as a derivative of the functional can be studied. Which variable X in Γ(X) is most appropriate when searching for extremum δγ / δ X = 0?
Green Function Theory Effective action formulation (Chitra, Kotliar, PRB 2001) S = dxψ x + V x x + + 2 ( )[ τ ext( )] ψ( ) + + dxdx ' ψ ( x) ψ ( x ') v ( x x ') ψ( x) ψ( x ') C x = (, r τ ) Adding an auxiliary source field to the system to probe Green function + SJ [ ] = S+ dxdxj ' ( xx, ') ψ( x') ψ ( x) + W[ J] = ln D[ ψψ]exp( S'[ J]) J ( xx, ') Eliminate source in favor of conjugate field using Legendre transform, obtain Free Energy Baym-Kadanoff Functional Γ BK[ G] = W[ J] Tr[ JG]
Freedom to construct functionals Choosing various source fields, various functionals can be obtained: Example 1. DFT (Fukuda et.al., 1994). Local source J(x) SJ [ ] S dxj( x) ψ( x) ψ + ( x) = + probes the density => Density Functional Γ DFT [ρ] is obtained. Example 2,3,4... Choose appropriate source obtain TD-DFT, Spin Polarized DFT, LDA+U, In all cases free energy of the system is accessed in extremum. (Kotliar, Savrasov, in New Theoretical Approaches to Strongly Correlated Systems, 2001)
Spectral Density Functional Theory Spectral Density Functional Theory is obtained using local source J ( xx, ') = J( xx, ') θ ( rr, ') which probes local Green function loc loc G (, r r', iω ) = G(, r r', iωθ ) (, r r') Total Free Energy is accessed Γ δγ δ G SDF SDF [ G ] loc loc Local Excitational Spectrum is accessed via G loc = 0
Kinetic energy & generalization of Kohn-Sham Idea To obtain kinetic functional: Γ [ G ] = K [ G ] +Φ [ G ] SDF loc SDF loc SDF loc introduce fictious particles which describe local Green function: G(, rr', ω) = G KS kj (, r r', ω) = ψ kj kjω() r ψkj ω(') r ω ψ E kjω K [ G ] K [ G] SDF loc SDF Exactly as in DFT: * kj() r ψkj(') r ω E K [ ρ] K [ G ] DFT DFT KS kj
Local self-energy energy of spectral density functional Spectral Density Functional looks similar to DFT Γ [ ψ ] = f E M ( r, r ', iω) G( r, r ', iω) drdr ' kjω kjω kjω eff kj iω iω + ρ() rv () rdr+ E [ ρ] +Φ [ G ] ext H xc loc Effective mass operator is local by construction and plays auxiliary role exactly like Kohn-Sham potential in DFT M (, rr', ω) = [ V () r + V ()] r δ( r r') + eff ext H δ G δφ xc (, r r', ω) Energy dependent Kohn-Sham (Dyson) equations give rise to energy-dependent band structure Ekj ω 2 ψkj ω + eff ω ψkj ω = kjωψkj ω Ekj ω Ekj ω ( r) M ( r, r', ) ( r') dr' E ( r) has physical meaning in contrast to Kohn-Sham spectra. are designed to reproduce local spectral density loc f kjω 1 = ( iω + µ E ) kjω
Local Dynamical Mean Field Approximation Exchange-correlation functional Φ [ G ] is unknown Local dynamical mean field approximation for Φ xc[ Gloc] Sum of two-particle diagrams constructed with local Green function G loc and bare Coulomb interaction v C Remarkably, that sum can be performed by mapping onto auxiliary quantum impurity model subjected to self-consistency condition (Georges, Kotliar, 1992) S = dxψ ( x) G ( x, x') ψ( x) + dxdx' ψ ( x) ψ ( x') v ( x x') ψ( x) ψ( x') imp Ω loc + -1 + + 0 Ω G xc 1 1 0 loc int loc loc ( xx, ') = G ( xx, ') + M ( xx, ') C
Elimination of U : Dynamically Screened Interaction Dynamically screened interaction W(r,r,ω) can be viewed as another variable in Baym-Kadanoff functional Γ BK [G,W]. G = G0 G0ΣintG W = v v ΠW C C The same works in spectral density functional theory: Γ SDF [G loc, W loc ] A local source is introduced and probes W in part of the space: W (, rr', ω) = W(, rr', ωθ ) (, rr') loc To write functional explicitly, introduce Kohn-Sham interaction: W(, rr', ω) = Wrr (, ', ω) = W (, rr', ω), r Ω, r' Ω G = G0 G0MintG W = v v PW C loc c loc Functional Γ BK [G, W] is extremized both over G and over W C
Self-Consistency
Further Approximations LDA+DMFT method and its static limit: LDA+U (Anisimov et.al, 1990, Anisimov+Kotliar team, 1997, Held+Nekrasov+Vollhard, 2001, McMahan+Held+Scalettar, 2001) Divide electrons onto light and heavy. Solve local impurity problem for heavy electrons only. Assume that LDA works for light electrons. Local GW approximation (Kotliar+SS, 2001, Zein+Antropov 2002_ Solves impurity model using GW diagram: M xc =-G loc W loc Eliminates problems of input U and double counting. Bierman+Ferdi+George, PRL 2003 with full GW+DMFT method beyond SDFT
Linear Response with Spectral Density Functional New spectral density functional provides foundation for studying lattice dynamics of strongly correlated systems (Savrasov, Kotliar, PRL 2003) Λ = µν RR ' 2 δ ΓSDF δ R δ R µ ν This requires evaluations of changes in local Green function ( Kohn-Sham Green function) and ( Kohn-Sham ) self-energy with respect to the displacements: δg( ω) δ M ( ), eff ω δr δr
Test Studies of NiO and MnO NiO, MnO are classical Mott-Hubbard insulators. LDA (LSDA, LSDA+U) works for magnetically ordered phases only. Disordered moment regime cannot be accessed by LDA which would give a metal. Inclusion of dynamical self-energy effects is needed to access the phonon spectra of paramagnetic regime.
DOSes for paramagnetic NiO and MnO Results of LDA+DMFT calculation with Hubbard 1 solver
NiO: : Phonons in LDA, LDA+U, LDA+Hubbard1 Solid circles theory, open circles exp. (Roy et.al, 1976) LDA, magnetic LDA+U, magnetic LDA+Hub1, non-mag. (after Savrasov, Kotliar, PRL 90, 056401, 2003)
NiO and MnO: : comparisons of various theories
Phonons & Phases in Plutonium Variety of crystal structures (6) Huge volume expansion between δ and α: V δ = 1.25 V Anomalous thermal expansion of δ phase. Anomalious volume collapse between δ and ε. Strong quasiparticle peak at Fermi energy. Anomalous resistivity behavior. Temperature independent susceptibility and absence of long-range magnetic order. Unknown phonon dispersions. α
Total Energy as a function of volume for Pu (after Savrasov, Kotliar, Abrahams, Nature 2001)
Photoemission in α- and δ-phases (Arko et.al., PRB 2000)
DMFT Phonons in fcc δ-pu C 11 (GPa) C 44 (GPa) C 12 (GPa) C'(GPa) Theory 34.56 33.03 26.81 3.88 Experiment 36.28 33.59 26.73 4.78 (after Dai, Savrasov, Kotliar,Ledbetter, Migliori, Abrahams, Science, 9 May 2003) (experiments from Wong et.al, Science, 22 August 2003)
DMFT Phonons in bcc ε-pu
Instable transverse mode at q=(110) in ε-pu
Importance of Phonon Entropy in δ ε Transition Via instability, there is a lot of phonon entropy in ε-phase Free energy difference between δ and ε-pu Phonon entropy is important in phase transitions of Pu (in contrast to α γ transition in Ce) Accounting for phonon contribution to free energy, correct order of T c can be obtained.
Conclusion Spectral density functional theory is manifestly basis set dependent theory, yet it delivers exact total energies and local spectra as long as exact functional is used. Self-energy of SDFT is local by construction and appears similar to Kohn-Sham potential of DFT Local Green function can be defined in general Hilbert space of orbitals χ α (r-r) Grr (, ', ω) = χα( r RG ) αβ( ω, R R') χβ(' r R') αβ as for example LMTO representation for diagonal elements Local dynamical mean-field approximation is a practical device (exactly as LDA within DFT) to perform total energy and linear response calculations in strongly correlated systems such as Pu