Gabriel Kotliar Physics Department and Center for Materials Theory Rutgers University

1. Towards First-Principles Electronic Structure Calculations of Correlated Materials Using Dynamical Mean Field Theory (DMFT). Gabriel Kotliar Physics Department and Center for Materials Theory Rutgers University CMSN-Workshop on Predictive Capabilities for Strongly Correlated Systems UT November 7-9 2003

2. Outline , Collaborators, References A. Poteryaev, A. Lichtenstein and G. Kotliar (preprint) (2003) S.Savrasov G. Kotliar and E. Abrahams, Nature 410,793 (2001). X. Dai,S. Savrasov, G. Kotliar,A. Migliori, H. Ledbetter, E. Abrahams  Science,  Vol300, 954 (2003) Funding: Basic Energy Sciences DOE.. DMFT and electronic structure calculations Case study 1: Ti2O3 Case study 2: Elemental Pu Conclusions: Future developments.

3. Two limits of the electronic structure problem are well under control. Band limit, (LDA or GGA)+ GW, gives good spectra and total energy. Physical properties are accessible in perturbation theory in the screened Coulomb interactions Well separated atoms, in the presence of spin orbital long range order, expansion around the atomic limit, unrestricted HF, and LDA+U work well for ordered Mott insulators. Challenge ahead: materials that are not in either one of these regimes. Requires combination of many body theory and one electron theory. Strongly Correlated Electrons THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

4. Strongly correlated systems are usually treated with model Hamiltonians • Conceptually one wants to restrict the number of degrees of freedom by eliminating high energy degrees of freedom. • In practice other methods (eg constrained LDA are used) THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

5. Reduce the quantum many body problem to a one site or a cluster of sites, in a medium of non interacting electrons obeying a self consistency condition. Instead of using functionals of the density, use more sensitive functionals of the one electron spectral function. [density of states for adding or removing particles in a solid, measured in photoemission] Dynamical Mean Field Theory THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

6. DMFT A. Georges, G. Kotliar (1992) Phys. Rev. B 45, 6497 THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

7. DMFT action and self consistency condition In general tk is large matrix H[k] , U is a matrix In the case of cluster is a matrix and is not the self energy, (but can be used to estimate the lattice self energy by projection) THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

8. Solving the DMFT equations • Review: A. Georges, G. Kotliar, W. Krauth and M. Rozenberg Rev. Mod. Phys. 68,13 (1996)] THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

9. DMFT: Effective Action point of view.R. Chitra and G. K Phys Rev. B.62 12715(2000), 63 115110(2001) S Savrasov and G. K. cond-matt 0308053 • Identify observable, A. Construct an exact functional of <A>=a, G [a] which is stationary at the physical value of a. • Example, density in DFT theory. (Fukuda et. al.) • When a is local, it gives an exact mapping onto a local problem, defines a Weiss field. • The method is useful when practical and accurate approximations to the exact functional exist. Example: LDA, GGA, in DFT. • DMFT, build functionals of the LOCAL spectral function. • Exact functionals exist. We also have good approximations! • Extension to an ab initio method. Functional of greens function of electric field and electronic field, functional of the density and the local greens function. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

10. Observable: Local Greens function Gii (w). Exact functional G [Gii (w) ]. DMFT Approximation to the functional.(Muller Hartman 89) DMFT as an approximation to the exact functional of the Greens function, DMFT as a truncation of the BK functional of the full Greens function. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

11. LDA+DMFT (II) Edc U DMFT THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

12. C-DMFT: test in one dimension. (Bolech, Kancharla GK cond-mat 2002) Gap vs U, Exact solution Lieb and Wu, Ovshinikov Nc=2 CDMFT vs Nc=1 THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

13. 1d Hubbard U/t=4 exact diag 2+6.Capone Civelli and GK 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. ROAD 1: Derive model Hamiltonians, solve by DMFT (or cluster extensions). V.Anisimov A Poteryaev V.Korotin V.Anokin andG Kotliar J. Phys. Cond. Mat. 35, 7359 (1997).A.Lichtenstein and M Katsnelson PRB (1998). ROAD 2: Define a functional of the density and of the local Greens function and extremize the functional to get coupled equations for the density and the spectral function and compute total energies. G. Kotliar, S.Savrasov, in New Theoretical approaches to strongly correlated systems, Edited by A. Tsvelik, Kluwer Publishers, (2001). Interfacing DMFT with band theory THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

16. 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) . This defines H. The U matrix can be estimated from first principles (Gunnarson and Anisimov, McMahan et.al. Hybertsen et.al) of viewed as parameters. Anisimov Poteryaev Korotin Anhokin and Kotliar J. Phys. Cond. Mat. 35, 7359 (1997). LDA+DMFT (I) THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

17. Application to Ti2O3 THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

18. Isostructural to V2O3. All the qualitative physics of the high temperature part of the phase diagram of V2O3 can be understood within single site DMFT. Computations with a realistic density of states, and multiorbital impurity model (K. Held and D. Vollhardt ) substantial quantiative improvement. Is the same thing true in Ti2O3? Metal to insulator transition in Ti2O3 THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

19. Ti2O3 V2O3 : Resistivities THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

20. Ti2O3 Structure THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

21. Relevant Orbitals: Goodenough picture THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

22. As a function of temperature, there is no magnetic transition in Ti2O3, unlike V2O3 As a function of temperature, there is no structural change, unlike V2O3 which becomes monoclinic at low temperatures. In V2O3 the distance between the Vanadium pairs incrases as the temperature decreases. In Ti2O3 the distance between the Vanadium pairs decreases as one lowers the temperature. LTS 250 K, HTS 750 K. Ti2O3 vs V2O3 THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

23. Band Structure Calculations always produce a good metal. L.F. Mattheiss, J. Phys.: Condens. Matter 8, 5987 (1996) Unrestricted Hartree Fock calculations produce large antiferromagnetic gap. M. Cati, G. Sandrone, and R. Dovesi, Phys. Rev. B. f55 , 16122 (1997). Earlier work. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

24. Ti2O3 LDA-DOS HTS LTS THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

25. Impurity solver. Multiband QMC. Derivation of the effective Hamiltonian. Massive downfolding with O Andersen’s new Nth order LMTOS. Coulomb interactions estimated using dielectric constant W=.5 ev. U on titanium 2 ev. J= .2 ev. Methodology:1 and 2 site CDMFT THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

26. Single site DMFT fails. LTS THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

27. Two-site CDMFT for beta=20, and beta=10 (T=500,1000) Poteryaev Lichtenstein and GK THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

28. Important role played by the Coulomb nn repulsion. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

29. Application to Plutonium THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

30. Small amounts of Ga stabilize the d phase (A. Lawson LANL) THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

31. Elastic Deformations Uniform compression:Dp=-B DV/V Volume conserving deformations: F/A=c44Dx/L F/A=c’ Dx/L In most cubic materials the shear does not depend strongly on crystal orientation,fcc Al, c44/c’=1.2, in Pu C44/C’ ~ 7largest shear anisotropy of any element. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

32. 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 30-35% lower than experiment This is the largest discrepancy ever known in DFT based calculations. DFT studies THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

33. LSDA predicts magnetic long range (Solovyev et.al.) Experimentally d Pu is not magnetic. If one treats the f electrons as part of the core LDA overestimates the volume by 30% DFT in GGA predicts correctly the volume of the a phase of Pu, when full potential LMTO (Soderlind Eriksson and Wills) is used. This is usually taken as an indication that a Pu is a weakly correlated system DFT Studies THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

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

35. Energy vs Volume [GGA+U=4 ev] EXPT: Bcc 14.7 Fcc 15.01 THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

36. GGA+U spectra THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

37. Predicts plutonium to be magnetic. Different theories of alpha and delta. Other problems with LDA+U THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

38. Atomic sphere approximation. Ignore crystal field splittings in the self energies. Fully relativistic non perturbative treatment of the spin orbit interactions. Impurity solver: interpolative scheme using slave bosons (low frequency ) and eqn of motion (high frequency). DMFT - Technical details [spectra and energy] THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

39. Full potential LMTO with two kappas. Linear response method in LMTO’s (S. Savrasov) Impurity solver: lowest order projection (Roth method) in the equations of motion. DMFT- Phonon Spectra THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

40. Pu: DMFT total energy vs Volume (Savrasov Kotliar and Abrahams 2001) THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

41. Pu Spectra DMFT(Savrasov) EXP (Arko Joyce Morales Wills Jashley PRB 62, 1773 (2000) THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

42. Alpha and delta Pu THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

43. Alpha phase is also a correlated metal. It differs from delta in the relative weight of the resonance and the Hubbard band. Consistent with resistivity and specific heat measurements. Similar conclusions A. Mc Mahan K. Held and R. Scalettar, for the alpha to gamma transition in Cerium. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

44. Spectra Method E vs V Summary LDA LDA+U DMFT

45. Electrons are the glue that hold the atoms together. Vibration spectra (phonons) probe the electronic structure. Phonon spectra reveals instablities, via soft modes. Phonon spectrum of Pu had not been measured. Short distance behavior of the elastic moduli. Phonon Spectra THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

46. Phonon freq (THz) vs q in delta Pu X. Dai et. al. Science vol 300, 953, 2003 THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

47. Inelastic X Ray. Phonon energy 10 mev, photon energy 10 Kev. E = Ei - Ef Q =ki - kf THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

48. Expt. Wong et. al. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

49. Expts’ Wong et. al. Science 301. 1078 (2003) Theory Dai et. al. Science 300, 953, (2003) THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

50. C’=(C11-C12)/2 = 4.78 GPa C’=3.9 GPa C44= 33.59 GPa C44=33.0 GPa C44/C’ ~ 7 Largest shear anisotropy in any element! C44/C’ ~ 8.4 Shear anisotropy. Expt. vs Theory THE STATE UNIVERSITY OF NEW JERSEY RUTGERS