Constrained RPA: Calculating the Hubbard U from First-Principles

1. Bonn 2008.01.10-12 Constrained RPA:Calculating the Hubbard U from First-Principles Ferdi Aryasetiawan Research Institute for Computational Sciences Tsukuba, Ibaraki 305-8568 – Japan Collaborators: Takashi Miyake (Tsukuba) Masa Imada (ISSP, Tokyo) Antoine Georges, Silke Biermann (Ecole Polytechnique, Palaiseau) Krister Karlsson (Skoevde)

2. Motivationsand aims: Many-electron Hamiltonian is too complicated to be solved directly. Isolate correlated bands and downfold weakly correlated bands Systematic way of performing the downfolding First-principles (parameter-free) method. Adjustable parameters may give nice results but not necessarily for good reasons.

3. Related works on the Hubbard U Seminal work on U (constrained LDA): O Gunnarsson, OK Andersen, O Jepsen, J Zaanen, PRB 39, 1708 (1989) VI Anisimov and O Gunnarsson, PRB 43, 7570 (1991) Improvement on constrained LDA M Cococcioni and S de Gironcoli, PRB 71, 035105 (2005) Nakamura et al (PRB 2005) Random-Phase Approximation (RPA): M Springer and FA, PRB 57, 4364 (1998) T Kotani, J. Phys.: Condens. Matter 12, 2413 (2000) FA, M Imada, A Georges, G Kotliar, S Biermann, AI Lichtenstein, PRB 70, 195104 (2004) IV Solovyev and M Imada, PRB 71, 045103 (2005) IV Solovyev, cond-mat/05066632

4. Instead of downfolding the Hamiltonian directly we downfold the response function. Divide the total polarization into: The rest of the polarization Typical electronic structure of strongly correlated materials t_2g e_g Aim: To find the effective interaction among electrons living in the t_2g band

5. Polarization function

6. Effective interaction among electrons in a narrow band: Constrained RPA (cRPA) Identity: can be viewed as an energy-dependent effective interaction between the 3d electrons • Advantages: • Energy-dependent U • Full matrix U • U(r,r’) is basis-independent Asymptotically Long range ! FA, M Imada, A Georges, G Kotliar, S Biermann, AI Lichtenstein, PRB 70, 195104 (2004)

8. Static Hubbard U for the 3d transition metal series For a more accurate model it may be necessary to include nearest-neighbour U

9. Sensitivity to the choice of energy window Green 3d Red 4s 3d 3d-4s hybridised E window (eV) U (-2.0, 4.0) 3.7 (-3.0, 4.0) 6.3 (-4.0, 4.0) 7.0 4s

10. Energy window (eV) U (-2.0, 1.5) 7.9 (-1.5, 1.5) 7.6 (-1.0, 1.5) 5.7 (-0.5, 1.5) 3.3 Energy window (eV) U (-2.0, 1.7) 6.6 (-1.5, 1.7) 5.4 (-1.0, 1.7) 4.3 (-0.7, 1.7) 3.2 cLDA U=6 eV

11. U as a function of eliminated bands Ni Band 2 to 6 are eliminated Ce compare with cLDA=5.4 eV V Band 2 to 8 are eliminated The 4f bands of Ce correspond to band 2 to 8. The 3d bands of Ni and V correspond to band 2 to 6

12. For the 3d transition metals and Cerium the following hybrid criterion has been used: Lower bound: eliminate the lowest band (4s) Upper bound: use energy cut off

13. Roles of screening channels: Vanadium vs Nickel (early vs late transition metals)

14. Vanadium: • Eliminating all transitions from the 3d bands has little effect on U(0)(green) •  In early transition elements the screening for U (0) is provided by the 4s electrons. • Nickel: • In contrast to vanadium, eliminating all transitons from the 3d bands • has a large effect on U(0) (green) •  In late transition elements screening from 4s electrons alone are not sufficient • to obtain U(0). The 3d electrons contribute significantly to screening. Eliminating transitions from the 4s band has no effects on W(0) for both V and Ni(red) 3d screening is metallic, very efficient in screening a point charge without help from the 4s electrons. W(0) is rather constant across the 3d series.

15. Connection between constrained LDA and constrained RPA Janak’s theorem From the Kohn-Sham equation: Constrained RPA: Constrain transitions in Constrained LDA: Constrain hoppings dielectric function U von Barth, The Electronic Structure of Complex Systems,Vol 113 NATO series B: Physics p67. M Springer and FA, PRB 57, 4364 (1998)

16. Constrained LDA Super Cell Transition metal or rare earth atom Hopping from and to 3d orbitals is cut off “impurity” Change the 3d charge on the impurity, keeping the system neutral, do a self-consistent calculation and calculate the change in the 3d energy level  U(3d).

17. SrVO3 t_2g Only O2p screening e_g “self- screening” compare with cLDA=9 eV U as a function of eliminated transitions (c.f. similar result, Solovyev, cond-mat/0506632)

18. SrVO3: 1 d system Eliminating all transitions from the 3d bands (red curve) has almost no influence on U(0)

19. Comparison between cRPA and cLDA for 3d transition metals U (cLDA) U (cRPA) W (RPA) The comparison is not clear cut because the 3d band is not completely isolated.

20. Breathing or Orbital relaxation In constrained LDA calculations, the 3d/4f orbitals are allowed to relax. Relaxing the 3d orbitals are equivalent to polarising them. In the language of RPA: 3d3d, 3d4d, 3d5d, etc. transitions Not allowed Allowed In constrained LDA, the 3d/4f orbitals should be fixed. In constrained LDA calculations orbital relaxation compensates for the lack of self-screening.

21. Energy dependence of U Gd Ce

22. Influence of energy dependence of U Spectral function of Ni from the Hubbard model with a static U, compared with the “true” one

23. The real and imaginary part of the self-energy from the Hubbard model with a static U compared with the “true” self-energy. “true” “true” Hubbard model Hubbard model The Hubbard model should work if the high energy part of is well separated from the low energy part

24. Self-energy of Ni from the Hubbard model with an energy-dependent Hubbard U

25. Maximally localized generalized Wannier function Marzari and Vanderbilt, PRB56, 12847 (1997) Souza, Marzari and Vanderbilt, PRB65, 035109 (2001) Wannier function Spread of Wannier function Use Wannier functions as basis for a model Hamiltonian

26. On-site interaction at w=0 LMTO-ASA (the head – partial wave) Maxloc Wannier Hubbard U Ni Fully screened W Full-Potential LMTO-GW (Takashi Miyake)

27. The screened exchange interaction J of some 3d metals Fe Ni Filled black triangle: fully screened J Cu Empty blue triangle: J calculated according to cRPA A non-negligible reduction of about 20 % from the bare atomic value is found.

28. Wannier orbitals obtained by maximising U Form a linear combination of maxloc Wannier orbitals in real space: Max. loc. Wannier Edmiston and Ruedenberg, Rev. Mod. Phys. 35, 457 (1963)

29. We have defined an anti-Hermitian matrix F Steepest ascent: which ensures that Construct

30. The Hubbard U calculated in maxU Wannier orbitals are surprisingly close to the values calculated in the maximally localised Wannier orbitals.

31. Check the procedure Have not found the global maximum of U?

32. Summary • cRPA allows for a systematic calculation of U: Full U matrix, energy-dependent U • In early transition metals, the 4s electrons do most of the screening for U • but in late transition metals, the 3d-electron screening contributes significantly to U. • In transition metals, the 3d electrons are very efficient in screening • a point charge (metallic screening)  W is almost constant across the series. • One source of discrepancy between cLDA and cRPA may be attributed to self-screening • * 3dnon-3d transitions in transition metals, • * O2p3d in SrVO3, • Orbital relaxation? • Self-screening and orbital relaxation tend to cancel each other. • Energy dependence of U can be large, even at low energy. • How to find a static U that takes into account the variation in energy. • How to solve an impurity model with an energy-dependent U • Maximally localised Wannier orbitals together with constrained RPA • provide an unambiguous way of constructing low-energy model Hamiltonians. •  Applications to GW+DMFT