From LDA+U to LDA+DMFT

1. 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. Content • From LDA+U to LDA+DMFT • Extension I: LDA+Hubbard 1 Approximation • Extension II: LDA+Cluster Exact Diagonalization • Applications to Magnons Spectra

3. U LDA+U Idea of LDA+U is borrowed from the Hubbard Hamiltonian: Spectrum of Antiferromagnet at half filling

4. d EF U p d LDA+U Orbital Dependent Potential The Schroedinger’s equation for the electron is solved with orbital dependent potential When forming Hamiltonian matrix The correction acts on the correlated orbitals only:

5. 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. NiO: Comparison with Photoemission Electronic Configuration: Ni 2+O2- d8p6 (T2g6Eg2p6) LDA+U Paramagnetic LDA LSDA

7. 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. LDA+ LDA+ LDA+U Next step: use self-energy from atom allowing to hybridize with conduction bath, i.e. finding it from the Anderson impurity problem. LDA+ LDA+ Impose self-consistency for the bath: full dynamical mean field theory is recovered. LDA+DMFT LDA+

8. 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(w) dn->dn+1 dn->dn-1 w 0 Better description compared to LDA or LDA+U is obtained

9. 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. Exact Diagonalization Methods In the limit of small hybridization Vkd=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 hybridizationcan be alternatively evaluated using QMC, or NCA, OCA, SUNCA approximations (K. Haule, 2003)

11. Excitations in Atoms Electron Removal Spectrum Electron Addition Spectrum

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

13. Mott Insulators as Systems near Atomic Limit Classical systems: MnO (d5), FeO (d6), CoO (d7), NiO (d8). Neel temperatures 100-500K. Remain insulating both below and above TN LDA/LDA+U, other static mean field theories, cannot access paramagnetic insulating state. Frequency dependence in self-energy is required:

14. Electronic Structure calculation with LDA+Hub1 LDA+Hubbard 1 Hamiltonian is diagonalized Green Function is calculated Density of states can be visualized

15. LDA+Hub1 Densities of States for NiO and MnO Results of LDA+Hubbard 1 calculation: paramagnetic insulating state is recovered LHB UHB U Dielectric Gap LHB UHB U Dielectric Gap

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

17. Americium Puzzle Experimental Equation of State (after Heathman et.al, PRL 2000) “Soft” Mott Transition? “Hard” 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~6mB (Soderlind et.al., PRB 2000). • Experimentally, Am hasnon magnetic f6ground state with J=0(7F0)

18. Photoemission in Am, Pu, Sm Atomic multiplet structure emerges from measured photoemission spectra in Am (5f6), Sm(4f6) - Signature for f electrons localization. after J. R. Naegele, Phys. Rev. Lett. (1984).

19. Am Equation of State: LDA+Hub1 Predictions 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. Theoretical P(V) using LDA+Hub1 Predictions for Am I LDA+Hub1 predictions: • Non magnetic f6ground state with J=0(7F0) • Equilibrium Volume: Vtheory/Vexp=0.93 • Bulk Modulus: Btheory=47 GPa Experimentally B=40-45 GPa Predictions for Am II Predictions for Am III Predictions for Am IV

20. Atomic Multiplets in Americium LDA+Hub1 Density of States Matrix Hubbard I Method F(0)=4.5 eV F(2)=8.0 eV F(4)=5.4 eV F(6)=4.0 eV 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

21. Many Body Electronic Structure for 7F0 Americium Insights from LDA+DMFT: Under pressure energies of f6 and f7 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) Experimental Photoemission Spectrum after J. Naegele et.al, PRL 1984

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

23. 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. Many Body Electronic Structure Method Green function G(r,r’,w) The proof lies in mathematical identity S(w) 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. Many Body Electronic Structure for 7F0 Americium

26. Cluster Exact Diagonalization Cluster Hamiltonian The cluster Hamiltonian is exact diagonalized The Green function is calculated: Self-energy is extracted:

27. Electronic Structure calculation with LDA+CED LDA+S Hamiltonian is diagonalized Green Function is calculated Density of states can be visualized

28. Illustration: Many Body Bands for NiO LDA+cluster exact diagonaization for NiO above TN: (atomic like UHB) O-hole coupled to local d moment (Zhang-Rice like) Eg’s (atomic like LHB)

29. SO=1/2 SCu=1/2 SO=1/2 SFe=2 SO=1/2 SMn=5/2 SO=1/2 SNi=1 SO=1/2 SCo=3/2 Generalized Zhang-Rice Physics JAF CuO2(d9) Zhang-Rice Singlet (Stot=0) NiO(d8) CoO(d7) MnO(d5) FeO(d6) Doublet (Stot=1/2) Triplet (Stot=1) Quintet (Stot=2) Quartet (Stot=3/2)

30. NiO: LDA+CED compared with ARPES Dispersion of doublet

31. CoO: LDA+CED compared with ARPES Dispersion of triplet

32. LDA+CED for HTSCs: Dispersion of Zhang-Rice singlet

33. 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. Magnetic Force Theorem Exchange Constants via Linear Response (Lichtenstein et. al, 1987) Spin Wave spectra, Curie temperatures, Spin Dynamics (Antropov et.al, 1995) Fe After Halilov, et. al, 1998

35. 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. Spin Wave Spectrum in NiO

37. Calculated Neel Temperatures

38. 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