1. University of California DavisKashiwa, July 27, 2007 From LDA+U to LDA+DMFT S. Y. Savrasov, Department of Physics, University of California, Davis Collaborators: Q. Yin, X. Wan, A. Gordienko (UC Davis) G. Kotliar, K Haule (Rutgers)

2. University of California DavisKashiwa, July 27, 2007 Content  From LDA+U to LDA+DMFT  Extension I: LDA+Hubbard 1 Approximation  Extension II: LDA+Cluster Exact Diagonalization  Applications to Magnons Spectra

3. University of California DavisKashiwa, July 27, 2007 U Idea of LDA+U is borrowed from the Hubbard Hamiltonian: LDA+U Spectrum of Antiferromagnet at half filling

4. University of California DavisKashiwa, July 27, 2007 LDA+U Orbital Dependent Potential The correction acts on the correlated orbitals only: The Schroedinger’s equation for the electron is solved with orbital dependent potential When forming Hamiltonian matrix EFEF U p d d

5. University of California DavisKashiwa, July 27, 2007 Main problem with LDA+U LDA+U is capable to recover insulating behavior in magnetically ordered state. However, systems like NiO are insulators both above and below the Neel temperature. Magnetically disordered state is not described by LDA+U Another example is 4f materials which show atomic multiplet structure reflecting atomic character of 4f states. Late actinides (5f’s) show similar behavior. Atomic limit cannot be recovered by LDA+U because LDA+U correction is the Hartree-Fock approximation for atomic self-energy, not the actual self-energy of the electron!

6. University of California DavisKashiwa, July 27, 2007 NiO: Comparison with Photoemission LSDA LDA+U Paramagnetic LDA Electronic Configuration: Ni 2+ O 2- d 8 p 6 (T 2g 6 E g 2 p 6 )

7. University of California DavisKashiwa, July 27, 2007 LDA+DMFT as natural extension of LDA+U In LDA+U correction to the potential is just the Hartree-Fock value of the exact atomic self energy. Why don’t use exact atomic self-energy itself instead of its Hartree-Fock value? This is so called Hubbard I approximation to the electronic self-energy. Next step: use self-energy from atom allowing to hybridize with conduction bath, i.e. finding it from the Anderson impurity problem. Impose self-consistency for the bath: full dynamical mean field theory is recovered. LDA+U LDA+ LDA+DMFT

8. University of California DavisKashiwa, July 27, 2007 Localized electrons: LDA+DMFT Electronic structure is composed from LDA Hamiltonian for sp(d) electrons and dynamical self-energy for (d)f-electrons extracted from solving Anderson impurity model Poles of the Green function have information about atomic multiplets, Kondo, Zhang-Rice singlets, etc. N(  ) d n ->d n+1   Better description compared to LDA or LDA+U is obtained d n ->d n-1

9. University of California DavisKashiwa, July 27, 2007 Exact Diagonalization Methods For capturing physics of localized electrons combination of LDA and exact diagonalization methods can be utilized: The cluster Hamiltonian is exact diagonalized The Green function is calculated: Self-energy is extracted:

10. University of California DavisKashiwa, July 27, 2007 Exact Diagonalization Methods In the limit of small hybridization V kd =0 this is reduced to calculating atomic d(f)-shell self-energy: Hubbard I approximation (Hubbard, 1961). If Hatree Fock estimate is used here, LDA+U method is recovered. Corrections due to finite hybridization can be alternatively evaluated using QMC, or NCA, OCA, SUNCA approximations (K. Haule, 2003)

11. University of California DavisKashiwa, July 27, 2007 Excitations in Atoms Electron Removal Spectrum Electron Addition Spectrum

12. University of California DavisKashiwa, July 27, 2007 Atomic Self-Energies have singularities Self-energy with a pole is required: Ground state energies for configurations d n, d n+1, d n-1 give rise to electron removal E n -E n-1 and electron addition E n -E n+1 spectra. Atoms are always insulators! or two poles in one-electron Green function Electron removal Electron addition Coulomb gap This is missing in DFT effective potential or LDA+U orbital dependent potential

13. University of California DavisKashiwa, July 27, 2007 Mott Insulators as Systems near Atomic Limit Classical systems: MnO ( d 5 ), FeO ( d 6 ), CoO ( d 7 ), NiO ( d 8 ). Neel temperatures 100-500K. Remain insulating both below and above T N Frequency dependence in self-energy is required: LDA/LDA+U, other static mean field theories, cannot access paramagnetic insulating state.

14. University of California DavisKashiwa, July 27, 2007 Electronic Structure calculation with LDA+Hub1 LDA+Hubbard 1 Hamiltonian is diagonalized Green Function is calculated Density of states can be visualized

15. University of California DavisKashiwa, July 27, 2007 LDA+Hub1 Densities of States for NiO and MnO Results of LDA+Hubbard 1 calculation: paramagnetic insulating state is recovered LHBUHB LHBUHB U U Dielectric Gap

16. University of California DavisKashiwa, July 27, 2007 NiO: Comparison with Photoemission Insulator is recovered, satellite is recovered as lower Hubbard band Low energy feature due to d electrons is not recovered! LHB UHB U Dielectric Gap

17. University of California DavisKashiwa, July 27, 2007 Americium Puzzle Density functional based electronic structure calculations:  Non magnetic LDA/GGA predicts volume 50% off.  Magnetic GGA corrects most of error in volume but gives m ~6  B (Soderlind et.al., PRB 2000).  Experimentally, Am has non magnetic f 6 ground state with J=0 ( 7 F 0 ) Experimental Equation of State (after Heathman et.al, PRL 2000) Mott Transition?“Soft” “Hard”

18. University of California DavisKashiwa, July 27, 2007 Photoemission in Am, Pu, Sm after J. R. Naegele, Phys. Rev. Lett. (1984). Atomic multiplet structure emerges from measured photoemission spectra in Am (5f 6 ), Sm(4f 6 ) - Signature for f electrons localization.

19. University of California DavisKashiwa, July 27, 2007 Am Equation of State: LDA+Hub1 Predictions LDA+Hub1 predictions:  Non magnetic f 6 ground state with J=0 ( 7 F 0 )  Equilibrium Volume: V theory /V exp =0.93  Bulk Modulus: B theory =47 GPa Experimentally B=40-45 GPa Theoretical P(V) using LDA+Hub1 Self-consistent evaluations of total energies with LDA+DMFT using matrix Hubbard I method. Accounting for full atomic multiplet structure using Slater integrals: F (0) =4.5 eV, F (2) =8 eV, F (4) =5.4 eV, F (6) =4 eV New algorithms allow studies of complex structures. Predictions for Am II Predictions for Am IV Predictions for Am III Predictions for Am I

20. University of California DavisKashiwa, July 27, 2007 Atomic Multiplets in Americium LDA+Hub1 Density of States Exact Diag. for atomic shell F (0) =4.5 eV F (2) =8.0 eV F (4) =5.4 eV F (6) =4.0 eV Matrix Hubbard I Method F (0) =4.5 eV F (2) =8.0 eV F (4) =5.4 eV F (6) =4.0 eV

21. University of California DavisKashiwa, July 27, 2007 Many Body Electronic Structure for 7 F 0 Americium Experimental Photoemission Spectrum after J. Naegele et.al, PRL 1984 Insights from LDA+DMFT: Under pressure energies of f 6 and f 7 states become degenerate which drives Americium into mixed valence regime. Explains anomalous growth in resistivity, confirms ideas pushed forward recently by Griveau, Rebizant, Lander, Kotliar, (2005)

22. University of California DavisKashiwa, July 27, 2007 Effective (DFT-like) single particle spectrum always consists of delta like peaks Real excitational spectrum can be quite different Bringing k-resolution to atomic multiplets

23. University of California DavisKashiwa, July 27, 2007 Many Body Calculations with speed of LDA (Savrasov, Haule, Kotliar, PRL2006) Non-linear over energy Dyson equation with pole representation for self energy is exactly reduced to linear set of equations in the extended space

24. University of California DavisKashiwa, July 27, 2007 Many Body Electronic Structure Method The proof lies in mathematical identity Green function G(r,r’,  ) Physical part of the electron is described by the first component of the vector  Electronic Green function is non interacting like but with more poles:

25. University of California DavisKashiwa, July 27, 2007 Many Body Electronic Structure for 7 F 0 Americium

26. University of California DavisKashiwa, July 27, 2007 Cluster Exact Diagonalization Cluster Hamiltonian The cluster Hamiltonian is exact diagonalized The Green function is calculated: Self-energy is extracted:

27. University of California DavisKashiwa, July 27, 2007 Electronic Structure calculation with LDA+CED LDA+  Hamiltonian is diagonalized Green Function is calculated Density of states can be visualized

28. University of California DavisKashiwa, July 27, 2007 Eg’s Illustration: Many Body Bands for NiO LDA+cluster exact diagonaization for NiO above T N : (atomic like LHB) (atomic like UHB) O-hole coupled to local d moment (Zhang-Rice like)

29. University of California DavisKashiwa, July 27, 2007 Generalized Zhang-Rice Physics NiO(d 8 ) CoO(d 7 ) FeO(d 6 ) MnO(d 5 ) S Ni =1 S O =1/2 S Cu =1/2 CuO 2 (d 9 ) Zhang-Rice Singlet (S tot =0) Doublet (S tot =1/2) Triplet (S tot =1) Quartet (S tot =3/2) Quintet (S tot =2) S O =1/2 S Mn =5/2 S O =1/2 S Fe =2 S O =1/2 S Co =3/2 J AF

30. University of California DavisKashiwa, July 27, 2007 NiO: LDA+CED compared with ARPES Dispersion of doublet

31. University of California DavisKashiwa, July 27, 2007 CoO: LDA+CED compared with ARPES Dispersion of triplet

32. University of California DavisKashiwa, July 27, 2007 LDA+CED for HTSCs: Dispersion of Zhang-Rice singlet

33. University of California DavisKashiwa, July 27, 2007 Magnons, Exchange Interactions, Tc’s Realistic treatment of magnetic exchange interactions in strongly-correlated systems: Spin waves, magnetic ordering temperatures Necessary input to Heisenberg, Kondo Hamiltonians Spin-phonon interactions, incommensurability, magnetoferroelectricity

34. University of California DavisKashiwa, July 27, 2007 Magnetic Force Theorem Exchange Constants via Linear Response (Lichtenstein et. al, 1987) Spin Wave spectra, Curie temperatures, Spin Dynamics (Antropov et.al, 1995) After Halilov, et. al, 1998 Fe

35. University of California DavisKashiwa, July 27, 2007 Exchange Constants, Spin Waves, Neel Tc’s Magnetic force theorem for DMFT has been recently discussed (Katsnelson, Lichtenstein, PRB 2000) Using rational representation for self-energy, magnetic force theorem can be simplified (X. Wan, Q. Yin, SS, PRL 2006) Expression for exchange constants looks similar to DFT However, eigenstates which describe Hubbard bands, quasiparticle bands, multiplet transitions, etc. appear here.

36. University of California DavisKashiwa, July 27, 2007 Spin Wave Spectrum in NiO

37. University of California DavisKashiwa, July 27, 2007 Calculated Neel Temperatures

38. University of California DavisKashiwa, July 27, 2007 Conclusion There are natural extensions of LDA+U method:  LDA+Hubbard 1 is a method where full frequency dependent atomic self- energy is used  LDA+ED is a method where self-energy is extracted from cluster calculations  LDA+DMFT is a general method where correlated orbitals are treated with full frequency resolution Many new phenomena (atomic multiplets, mixed valence, Kondo effect) can be studied with the electronic structure calculations