Merging First-Principles and Model Approaches

1. Merging First-Principles and Model Approaches ISSP 2007.07.25 Ferdi Aryasetiawan Research Institute for Computational Sciences, AIST, Tsukuba, Ibaraki 305-8568 – Japan Collaborators: Antoine Georges and Silke Biermann (Ecole Polytechnique, France) Takashi Miyake and Rei Sakuma (RICS-AIST)

2. Outline First Part • Motivations: • The need to go beyond one-particle picture in correlated materials • Previous works: LDA+U, LDA+DMFT • The GW approximation: success and difficulties • Combining GW and DMFT: • A first-principles scheme for correlated materials • A simplified implementation: • Application to ferromagnetic nickel Second Part (if time permits) Constrained RPA: Calculating the Hubbard U from first-principles

3. Spectral evolution as a function of U/bandwidth metal Can use LDA Difficult to treat within one-particle theory Can use LDA+U insulator Georges et al Rev. Mod. Phys.1996 (Dynamical Mean-Field Theory) Fujimori PRL 69, 1796 (1992)

4. orthorhombic cubic La/YTiO3

5. Experimentally SrVO3 and CaVO3 are metals LaTiO3 and YTiO3 are insulators These materials share similar electronic structure in LDA and they are all predicted to be metals [1] Solovyev, Hamada, and Terakura, PRB 53, 7158 (1996) (LDA+U: correct magnetic structure of La/YTiO3) [2] Pavarini, Biermann, Poteryaev, Lichtenstein, Georges, and Andersen, PRL 92, 176403 (2004) (LDA+DMFT: consistent description of metal-insulator transition)

6. Typical electronic structure of correlated materials: Partially filled narrow 3d or 4f band across the Fermi level The main action takes place here By slight distortion or pressure the ratio of U/bandwidth changes and the materials can undergo, e.g., phase transitions (metal-insulator). competition between kinetic energy (bandwidth) and U.

7. Competition between kinetic energy and U (itineracy and localisation) D Electrons prefer to “spread” themselves to lower their kinetic energy. U=0 When U is small it is preferable for the electrons to delocalise metal QP U/D>1 satellite For intermediate U it is a mixture of localised and delocalised electrons. U U/D>>1 When U is large it is costly for the electrons to hop  localised (Mott insulator) Lower Hubbard band Upper Hubbard band

8. Mapping to a Hubbard model D U=0 QP More realistic models take into account the underlying one-electron band structure LDA+U and LDA+DMFT U/D>1 satellite U U/D>>1 Lower Hubbard band Upper Hubbard band

9. To treat systems with strong on-site correlations:Previous works: LDA+U and LDA+DMFT Adjustable U Ad-hoc double-counting term Anisimov et al, J. Phys. Condens. Matter 9, 7359 (1997) (LDA+U, LDA+DMFT) Lichtenstein and Katsnelson, PRB 57, 6884 (1998) (LDA+DMFT)

10. One of the main features of correlated electrons is that they have both itinerant and localised characters. The Goal To construct a consistent theoretical scheme that can describe the electronic structure of correlated materials from first-principles. “Consistent” means that the scheme should be capable of describing the continuous transition from metal to insulator, i.e., it can describe both the itinerant and localised characters of the electrons.

11. Model Approaches: • Exact diagonalisation (Lanczos) • Quantum Monte Carlo (QMC) • Dynamical Mean-Field Theory (DMFT) • LDA+U and LDA+DMFT • First-Principles Methods: • Local Density Approximation (LDA) (ground states) • GW method (excited states) Insufficient to treat correlated materials Can treat strong correlations but need adjustable parameters Combine First-Principles and Model Approaches:GW+DMFT Electronic structure of correlated materials from first-principles Possible applications optical devices spintronics electronic transport nanotech.

12. The GW approximation Lars Hedin, Phys. Rev. 139, A796 (1965) Hartree-Fock approximation: G - - - - - - v G GW approximation: = = = = = W is a screened interaction

13. Screened exchange: from the poles of G Since W<v, it reduces the Hartree-Fock band gap. In addition we have another term arising from the poles of W Correlation hole: interaction between an electron and its screening hole

14. From Hybertsen and Louie, PRB34, 5390 (1986)

15. Assessing the GWA using the Polaron Hamiltonian: Exact solution: Plasmons satellites e+De

16. Quasi-particle and satellite energy Error in QP energy is smaller than error in satellite energy

18. Some difficulties with the GWA • There is no spin dependence in W, only in G: • Exchange is taken into account properly, but not correlation between the same spin. Collective excitations only arise from RPA: (1-Pv)= 0 (plasmons). Satellites arising from local correlations (atomic multiplets) are probably not fully captured. • The Hubbard Hamiltonian cannot be transformed into a polaron-type Hamiltonian for which GW is good. Usually start from a single Slater determinant: Difficult to treat systems that are inherently dominated by a few Slater determinants. Self-consistent GW appears to give poor spectra (?)

19. Band insulator Mott-Hubbard insulator Band insulator vs Mott-Hubbard insulator GW ImSfinite Shifts and broadens Quasi-Particle U DMFT ImSinfinite Removes QP and transfers weight to Hubbard bands (QP life-time ~ 1/ImS)

20. Strongly correlated systems are problematic for LDA: LDA often predicts (anti-ferromagnetic) insulators to be metals. Even the GW approximation may not be sufficient. DMFT: Map the lattice to an impurity embedded in a “bath”. “bath” U Dynamical A la Fukuyama A. Georges et al, Rev. Mod. Phys. 68, 13 (1996) G. Kotliar and D. Vollhardt, Physics Today (March 2004)

21. Effective dynamics for an impurity problem The Coulomb interaction is fully taken into account in one site (impurity), the rest of the sites (medium) is treated as an effective field with the dynamical mean-field bath

22. Dynamical Mean-Field Theory (DMFT) A. Georges et al, Rev. Mod. Phys. 68, 13 (1996) Green’s function Self-energy Self-consistency: Restore the lattice periodicity

23. Comparison between GW and DMFT

24. Basic physical idea of GW+DMFT R 0 The onsite self-energy is taken to be SDMFT. The off-site self-energy is approximated by SGW . P. Sun and G. Kotliar, PRB 66, 85120 (2002) S. Biermann, F.A. and A. Georges, PRL 90, 86402 (2003)

25. General Framework Luttinger-Ward functional Generalized Luttinger-Ward functional* *Almbladh, von Barth and van Leeuwen, Int. J. Mod. Phys. B13, 535 (1999)

26. Approximation for Y wwww Conserving approximation

27. Self-Consistency Conditions Additional condition: Determines U The Hubbard U is screened within the impurity model such that the screened U is equal to the local W

28. Self-consistency loop Impurity: given Weiss field G and U New Weiss field G and U Check self-consistency: Gloc=Gimp? Wloc=Wimp? Combine SGW and Simp

29. Simplified GW+DMFT scheme Non-self-consistent GW + DMFT (static impurity model) Offsite self-energy: corrected by GW Onsite self-energy: corrected by DMFT

30. Test: Application to ferromagnetic nickel(Simplified GW+DMFT scheme) Correlation problems in nickel within LDA:

31. ----- LDA majority GW+DMFT minority majority exp minority (Experimental data from Buenemann et al, cond-mat/0204142) Too large LDA bandwidth Nickel band structure S. Biermann et al, PRL 90, 86402 (2003)

32. GW+DMFT density of states of nickel Exchange splitting too large by 0.3 eV in LDA no 6 eV satellite in LDA

33. Next target: From V. Eyert

34. metal insulator

35. From Miyake

36. Koethe et al

37. Single impurity DMFT does not open up a gap in M1 Non-local self-energy is important  a good test for GW+DMFT From Biermann et al, PRL 94, 026404 (2005)

38. Preliminary GW result from Sakuma and Miyake a_1g O2p One-shot GW gives too small gap.  need DMFT?  self-consistency?

39. Weak satellites

40. The self-energy around the LDA energy is not linear.

41. Summary • GW+DMFT scheme: • -Potentially allows for ab-initio electronic structure calculations of • correlated systems with partially filled localized orbitals and • describes phase transitions. • -Unlike LDA+DMFT, • *The Hubbard U is determined self-consistently • *Avoids double counting. Challenges • Global self-consistency • Solving the impurity problem with an energy-dependent U • Treat all orbitals on equal footing