Gabriel Kotliar Physics Department and Center for Materials Theory Rutgers University

1. Dynamical Mean Field Theory in Electronic Structure Calculations:Applications to solids with f and d electrons Gabriel Kotliar Physics Department and Center for Materials Theory Rutgers University Joint work with Sergej Savrasov X International Workshop on computational Material Science Total energy and Force Methods 11-13 January 2001 Trieste

2. Outline • Problems posed by the electronic structure of strongly correlated electrons • Dynamical Mean Field Theory Concepts, DMFT and DFT • A case study of system specific properties: f electrons in d Pu (S. Savrasov , GK) • A case study involving d electrons La1-xSrxTiO3 THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

3. Mott : Correlations localize the electron Low densities, electron behaves as a particle,use atomic physics, real space One particle excitations: Hubbard Atoms: sharp excitation lines corresponding to adding or removing electrons. In solids they broaden by their incoherent motion, Hubbard bands (eg. bandsNiO, CoO MnO….) High T, local moments Magnetic and Orbital Ordering at low T Quantitative calculations of Hubbard bands and exchange constants, LDA+ U, Hartree Fock. Atomic Physics. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

4. The Strong Correlation Problem Two limiting cases of the electronic structure of solids are understood:the high density limit and the limit of well separated atoms. High densities, the is electron be a wave, use band theory, k-space One particle excitations: quasi-particle,quasi-hole bands, collective modes. Density Functional Theory with approximations suggested by the Kohn Sham formulation, (LDA GGA) is a successful computational tool for the total energy, and a good starting point for perturbative calculation of spectra, such as GW.…………………… THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

5. Localization vs Delocalization Strong Correlation Problem • A large number of compounds with electrons which are not close to the well understood limits (localized or itinerant). • These systems display anomalous behavior (departure from the standard model of solids). • Neither LDA or LDA+U or Hartree Fock works well • Dynamical Mean Field Theory: Simplest approach to the electronic structure, which interpolates correctly between atoms and bands THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

6. La1-xSrxTiO3 photoemission THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

7. DMFT for lattice hamiltonians k independent S k dependent G, Local Approximation Treglia et. al 1980 THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

8. How to compute S ? View locally the lattice problem as a (multiorbital) Anderson impurity model The local site is now embedded in a medium characterized by THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

9. How to determine the medium • Use the impurity model to compute S and the impurity local Greens function. Require that impurity local Greens function equal to the lattice local Greens function. Weiss field THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

10. Solving the DMFT equations • Wide variety of computational tools (QMC, NRG,ED….) • Analytical Methods THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

11. DMFT • Construction is easily extended to states with broken translational spin and orbital order. • Large number of techniques for solving DMFT equations for a review see • A. Georges, G. Kotliar, W. Krauth and M. Rozenberg Rev. Mod. Phys. 68,13 (1996)] THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

12. DMFT : effective action construction (Fukuda, Valiev Fernando , Chitra GK). • Select a set of local orbitals. • Define a frequency dependent, local Greens function (spectral function) • The energy can be expressed as a functional of the local Greens function (R. Chitra and G. Kotliar PRB 2000) • The functional can be built in perturbation theory in the interaction (well defined diagrammatic rules ) • The functional can also be constructed from the atomic limit. • A useful approximation to the exact functional can be constructed, this Is the DMFT functional. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

13. Effective action Fukuda, Valiev and Ferndando, Chitra GK c dynamical analog of kohn Sham potential Reduces to self energy when self energy is local, approximation. Sum of all local graphs which are 2PI THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

14. Two roads for ab-initio calculation of electronic structure of strongly correlated materials Crystal structure + Atomic positions Model Hamiltonian Correlation Functions Total Energies etc. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

15. LDA+DMFT • The light, SP (or SPD) electrons are extended, well described by LDA • The heavy, D (or F) electrons are localized,treat by DMFT. • LDA already contains an average interaction of the heavy electrons, substract this out by shifting the heavy level (double counting term) The U matrix can be estimated from first principles THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

16. LDA+DMFT loop (in a tight binding basis, say LMTO’s cka )given interaction matrix U • 0) Guess r (r), G0ab(i w) • 1) Form Vxc , Solve AIM to get S, and local Greens function of heavy orbitals. • 2) Form LMTO Matrix , overlap matrix and heavy level shift E , form G(k, i w) • 3) Recompute the density and Weiss function G0ab(i w) to go back to 1. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

17. LDA+DMFT • To implement step 3 we use Notice the Weiss field,E and self energies use only heavy block, while H is full. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

18. LDA+DMFT • An exact functional by expresses the total energy as a functional of the local spectral function (or local Greens function) of the heavy electrons, and of the total density. • The construction proceeds by Legendre transformation (G. Kotliar and S. Savrasov 2001). • LDA+DMFT useful approximation for this functional. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

19. LDA+DMFT References • V. Anisimov, A. Poteryaev, M. Korotin, A. Anokhin and G. Kotliar, J. Phys. Cond. Mat. 35, 7359-7367 (1997). • A Lichtenstein and M. Katsenelson Phys. Rev. B 57, 6884 (1988). • S. Savrasov and GK full self consistent implementation (2001) THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

20. Delocalization Localization across the actinide series THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

21. Pu: Complex Phase Diagram (J. Smith LANL) THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

22. Small amounts of Ga stabilize the d phase THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

23. Problems with LDA • DFT in the LDA or GGA is a well established tool for the calculation of ground state properties. • Many studies (Freeman, Koelling 1972)APW methods • ASA and FP-LMTO Soderlind et. Al 1990, Kollar et.al 1997, Boettger et.al 1998, Wills et.al. 1999) give • an equilibrium volume of the d phaseIs 35% lower than experiment • This is the largest discrepancy ever known in DFT based calculations. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

24. Problems with LDA • LSDA predicts magnetic long range order which is not observed experimentally (Solovyev et.al.) • If one treats the f electrons as part of the core LDA overestimates the volume by 30% • LDA predicts correctly the volume of the a phase of Pu, when full potential LMTO (Soderlind and Wills). This is usually taken as an indication that a Pu is a weakly correlated system. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

25. Other Methods • LDA+ U (Savrasov and Kotliar 2000, Bouchet et. Al 2000) predicts correct volume of Pu with the constrained LDA estimate of U=4 ev. However, it predicts spurious magnetic long range order and a spectra which is very different from experiments. • Ideal system to try realistic DMFT THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

26. Pu: DMFT total energy vs Volume (Savrasov 00) THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

27. Lda vs Exp Spectra THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

28. Pu Spectra DMFT(Savrasov) EXP (Arko et. Al) THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

29. Pu Specific Heat THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

30. Anomalous ResistivityJ. Smith LANL THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

31. (Tokura et. Al. 1993)A doped Mott insulator:LaxSr1-xO3 THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

32. DMFT calculation, U near the Mott transition, Rozenberg et.al 94 THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

33. Hall Coefficient, electron like. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

34. Conclusions • DMFT is a many body approach which is able to describe on the same footing the atomic limit (open shell atoms) and bands. Works even in the localization Delocalization crossover. [Improvement in Eq.Volume and spectra and other quantities (relative to LDA GGA) in Pu and titanites,prototype systems] • Progress in many body theory results in better understanding of materials: Pu, LaSrTiO3………. • Future work: improve treatment of local Hamiltonian multiplets. • Future work: improve treatment of spd electrons (E-DMFT) Coulomb screening, connection with GW • Extension to multiple site clusters (DCA M. Jarrell et. Al., C-DMFT G.Kotliar et.al, Two impurity DMFT Schiller, Ingersent Georges Kotliar, …..) THE STATE UNIVERSITY OF NEW JERSEY RUTGERS