GW+DMFT approach for electronic structure calculations in real compounds

1. GW+DMFT approach for electronic structure calculations in real compounds N.Zeyn RRC “Kurchatov Institute”, Moscow S. Savrasov UCDavis, CA G. Kotliar Rutgers University, NJ Supported by NSF,DOE

2. Contents • Restrictions of DF method – semiconductors, f-metals, TMO- why we need something beyond LDA. • Green function method – GW approach – state of the art. • Green function method – DMFT approach-state of the art. • 4. Is effective interaction local in real crystals? GW+DMFT. • 5. Energy dependence of effective interaction - “U” • 6. Example – paramagnetic NiO.

3. Gap in semiconductor. Comparisonof LDA and GW with experiment

4. Comparison ofLDA andGWfor (0,9) nanotube. Gap changesfrom 0.08 eV to 0.17 eV

5. Intro. e-e interaction in homogeneous electron gas Perturbation series can be resummed in terms of screened interaction W(ε). (Hedin 1965) Series for thermodynamic energy (Luttiger-Ward functional ) can be sketched as ln(G)–ΣG+Φ; Σ=δ Φ /δG In perturbation theory first terms are GW diagram ~ Other Diagrams

8. LocalGWapproach in 3d and 4dmetals

9. Local GWapproach in 3d and 4dmetals

10. Variations of d-self-energy under iterations

12. Convergence of correlational part of the self-energy Σcorr in real space compared to bare Coulomb interaction in a.u. (1 a.u.= 27.2 eV) Test on transition metals. Matrix elements of bare Coulomb interaction Self-energy

13. Convergence of correlational part of the self-energy Σcorr compared to bare Coulomb interaction V (1 a.u.= 27.2 eV)Test in simple metals Self-energy Matrix elements of bare Coulomb interaction

14. Σexch, Σcorr for Si in a.u.(1 a.u.= 27.2 eV) Exchange part of Self-energy Correlational part of Self-energy

15. DMFT equations

16. DMFT equations Connection with Anderson model

17. Methods of DMFT equations solution Approximate methods – Hubbard I, Gutzwiller, etc Exact diagonalization method Second order perturbation theory in ∆/U Hirsh-Fye Monte-Carlo Continuous Time Monte-Carlo with expansion in U (Rubtsov) Continuous Time Monte-Carlo with expansion in ∆ (Millis, Haule)

18. General Framework Luttinger-Ward functional Generalized Luttinger-Ward functional* *Almbladh, von Barth and van Leeuwen, Int. J. Mod. Phys. B 13, 535 (1999) Chitra and Kotliar, Phys. Rev. B 63, 115110 (2001)

19. Approximation for Y wwww

20. Метод GW+DMFT R 0 On-site self-energySDMFT. Self-energy between sitesSGW . Biermann GeorgesLiechtenstein…. PRL (2002) Also can be reformulated using Luttinger-Word functional

21. GF+DMFT approach 7. We used LMTO-ASA method to build an appropriate basis Green function (ε-H0- Σ)-1 Π=-Σ(GG) W=V/(1+V Π); U=V(1+V Πrest) Σ= GW + Σsolver 8. Matsubara Green functions. To obtain DOS we make analytical continuation of ΣGW(iω) by Pade – approximant procedure.

22. Direct Coulomb interaction For homogeneous sphere with radius R

23. Pure Coulomb interaction F2=0.030 F4=0.009 F2=0.019 F4=0.003

24. Pure Coulomb interaction

25. Constrained LDA methodDederichs 1984, Gunnarson 1989, Anisimov 1991

26. GW approach to U Effective Hubbard model

27. Energy-dependent effective interaction between the 3d electrons

28. On-site U for the 3d series On-site U(0) is between 2-4 eV

29. Nearest-neighbour U for the 3d series Nearest-neighbour U(0) is between 0.2-1.0 eV

30. Static Hubbard U for the 3d series

31. Energy dependence of interaction in NiO

32. Antiferromagnetic NiO: LDA and GW calculations Faleev, Schilfgaarde, Kotani PRL 93 (2004)

33. Schematic levels in Mott insulator NiO

34. total total t2g t2g 250 eg eg 250 Op Op 200 200 150 ) ïëîòíîñòü ñîñòîÿíèé ) ïëîòíîñòü ñîñòîÿíèé 150 100 100 e e g( g( 50 50 0 0 -15 -10 -5 0 5 10 -15 -10 -5 0 5 10 e e in eV in eV Density of states(DOS)in paramagneticNiOLeft:DF approach-> металл(DOSat Fermi)Right :GW+DMFT->insulator (gap) Inexperiment NiO is an insulator with gap 4.5 eV

35. Hubbard and charge-transfer models

36. Experiment in NiO Hubbard model Charge-transfer model PRB 71 (2005)

37. LDA+EX-DIAG в NiO Savrasov et.al PRL (2008)

38. Conclusion • 1. Self-consistent GW method was implemented for 3D and 1D crystal structures. • 2. R-space convergence was investigated. • 3. GW+DMFT approach was applied for electronic structure calculations in Mott insulator – paramagnetic NiO. • 4. GW+DMFT with Моnte-Carlosolver - in progress.

39. 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

40. Self-Consistency Conditions The Hubbard U is screened within the impurity model such that the screened U is equal to the local W

41. Exact diagonalization method States

42. Second order perturbation theory PHYSICAL REVIEW B 72, 045111 2005 K. Haule, G. Kotliar

43. Effective interaction among electrons in a narrow band Suppose the bandstructure of a given solid can be well separated into a narrow band near the Fermi level and the rest, e.g., transition metals or 4f metals. We write the total polarisation as transitions only M. Springer and Aryasetiavan, PRB 57, 4364 (1998) T. Kotani, J. Phys. Cond. Matt. 12, 2413 (2000) Aryasetiavan, Imada, Georges, Kotliar, Biermann, Lichtenstein, PRB 70, 195104 (2004) I. Solovyev and M. Imada, cond-mat/0407786 v1 30 Jul 2004

44. Диаграммное разложение по гибридизацииD взвешивание диаграмм по Метрополису Millis et al PRB (2006) K. Haule PRB (2007)

45. One-site partition function Expansion in hybridization

47. Self-energy and effective interaction along imaginary axis