1. Correlation Effects in Itinerant Magnets : Towards a realistic Dynamical Mean Field Approach Gabriel Kotliar Physics Department Rutgers University In Electronic Structure and Computational Magnetism July 15-17 (2002)

2. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Outline  Dynamical Mean Field Theory: a tool for treating correlations in model Hamiltonians. Towards Realistic implementations of DMFT. Applications to Fe and Ni. Conclusions and outlook.

3. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Acknowledgements Collaborators and References: A. Lichtenstein M. Katsnelson and G. Kotliar Phys. Rev Lett. 87, 067205 (2001). I Yang S. Savrasov and G. Kotliar Phys. Rev. Lett. 87, 216405 (2001). Useful Discussions K. Hathaway and G. Lonzarich Support NSF and ONR

4. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Strong Correlation Problem Two limiting cases of the electronic structure problem are well understood. The high density limit ( spectrum of one particle excitations forms bands) and the low density limit (spectrum of atomic like excitations, Hubbard bands). Correlated compounds: electrons in partially filled shells. Not close to the well understood limits. Non perturbative regime. Standard approaches (LDA, HF ) do not work well.

5. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Motivations for going beyond density functional theory. DFT is a theory for ground state properties. Its Kohn Sham spectra can be taken a starting point for perturbative (eg. GW ) calculations of the excitation spectra and transport. This does not work for strongly correlated systems, eg oxides containing 3d, 4f, 5f elements. Character of the spectra (QP bands + Hubbard bands ) is not captured by LDA. LDA –GGA is less accurate in determining some ground state properties in correlated materials.

6. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS DMFT  DMFT simplest many body technique which describes correctly the open shell atomic limit and the band limit. Exact in the limit of large lattice coordination.  Band physics (i.e. kinetic energy) survive in the atomic limit (superexchange). Some aspects of atomic physics survive even in itinerant systems (J, U, Hubbard bands, satellites, L) Computations of one electron spectra, transport properties… Spectral density functional. Connects the one electron spectral function and the total energy.

7. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Mean-Field : Classical vs Quantum Classical case Quantum case Phys. Rev. B 45, 6497 A. Georges, G. Kotliar (1992)

8. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS DMFT Impurity cavity construction

9. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS C-DMFT C:DMFT The lattice self energy is inferred from the cluster self energy. Alternative approaches DCA (Jarrell et.al.) Periodic clusters (Lichtenstein and Katsnelson)

10. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS 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

11. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Solving the DMFT equations Wide variety of computational tools (QMC,ED….)Analytical Methods Reviews: A. Georges, G. Kotliar, W. Krauth and M. Rozenberg Rev. Mod. Phys. 68,13 (1996)]

12. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS From model hamiltonians to realistic calculations. DMFT as a method to be incorporated in electronic structure calculations. Important in regimes where local moments are present (e.g. NiO above its Neel temperature) Incorporate realistic structure and orbital degeneracy information in many body studies. Combination of electronic structure(LDA,GGA,GW) and many body methods (DMFT)

13. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Interface with electronic structure. Derive model hamiltonians, solve by DMFT (or cluster extensions). Total energy?  Full many body aproach, treat light electrons byt GW or screend HF, heavy electrons by DMFT [GK and Chitra, GK and S. Savrasov, P.Sun and GK cond-matt 0205522]  Treat correlated electrons with DMFT and light electrons with DFT (LDA, GGA +DMFT)

14. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Combining LDA and DMFT The light, SP electrons well described by LDA The heavier D electrons treat by model DMFT. LDA already contains an average interaction of the heavy electrons, subtract this out by shifting the heavy level (double counting term, Lichtenstein et.al.) Atomic physics parameters. U=F0 cost of double occupancy irrespectively of spin, J=F2+F4, Hunds energy favoring spin polarization.F2/F4=.6 Calculations of U, Edc, study as a function of these parameters.

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16. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Combine Dynamical Mean Field Theory with Electronic structure methods. Single site DMFT made correct qualitative predictions. Make realistic by: Incorporating all the electrons. Add realistic orbital structure. U, J….. Add realistic crystal structure. Allow the atoms to move.

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

18. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Realistic Calculations of the Electronic Structure of Correlated materials Combinining DMFT with state of the art electronic structure methods to construct a first principles framework to describe complex materials. Anisimov Poteryaev Korotin Anhokin and Kotliar J. Phys. Cond. Mat. 35, 7359 (1997). Lichtenstein and Katsenelson PRB (1998) Kotliar, Savrasov, in New Theoretical approaches to strongly correlated systems, Edited by A. Tsvelik, Kluwer Publishers, 2001) Savrasov Kotliar and Abrahams Nature 410, 793 (2001)) Kotliar, Savrasov, in New Theoretical approaches to strongly correlated systems, Edited by A. Tsvelik, Kluwer Publishers, 2001)

19. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Combining LDA and 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, subtract this out by shifting the heavy level (double counting term) The U matrix can be estimated from first principles (Gunnarson and Anisimov, Mc Mahan et. Al. Hybertsen et.al) or viewed as parameters

20. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Density functional theory and Dynamical Mean Field Theory DFT: Static mean field, electrons in an effective potential. Functional of the density. DMFT: Promote the local (or a few quasilocal Greens functions or observables) to the basic quantities of the theory. Express the free energy as a functional of those quasilocal quantities.

21. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Spectral Density Functional : effective action construction ( Fukuda, Valiev and Fernando, Chitra and Kotliar ). DFT, consider the exact free energy as a functional of an external potential. Express the free energy as a functional of the density by Legendre transformation.  DFT  (r)] Introduce local orbitals,   R (r-R)orbitals, and local GF G(R,R)(i  ) = The exact free energy can be expressed as a functional of the local Greens function and of the density by introducing  (r),G(R,R)(i  )] A useful approximation to the exact functional can be constructed, the DMFT +LDA functional. Savrasov Kotliar and Abrahams Nature 410, 793 (2001)) Full self consistent implementation.

22. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS LDA+DMFT-outer loop relax DMFT U E dc

23. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Realistic DMFT loop

24. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS LDA+DMFT functional  Sum of local 2PI graphs with local U matrix and local G

25. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Very Partial list of application of realistic DMFT to materials QP bands in ruthenides: A. Liebsch et al (PRL 2000) N phase of Pu: S. Savrasov et al (Nature 2001) MIT in V 2 O 3 : K. Held et al (PRL 2001) Magnetism of Fe, Ni: A. Lichtenstein et al PRL (2001) J-G transition in Ce: K. Held et al (PRL 2000); M. Zolfl et al PRL (2000). 3d doped Mott insulator La1-xSrxTiO3 (Anisimov et.al 1997, Nekrasov et.al. 1999, Udovenko et.al 2002) ………………..

26. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS DMFT Developed initially to treat correlation effects in model Hamiltonians. Review: A. Georges, G. Kotliar, W. Krauth and M. Rozenberg Rev. Mod. Phys. 68,13 (1996)] Extension to realistic setting [V. Anisimov, A. Poteryaev, M. Korotin, Anokhin and G. Kotliar, J. Phys. Cond. Mat 9, 7359 (1997). S. Savrasov, G. Kotliar and E. Abrahams, Nature 410, 793 (2001). ] Lichtenstein and Katsnelson [Phys.Rev. B 57, 6884(1998) ] Unlike DFT, DMFT computes both free energies and one electron (photoemission ) spectra and many other physical quantities at finite temperatures.

27. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS LDA+DMFT Spectral Density Functional ( Fukuda, Valiev and Fernando, Chitra and GK, Savrasov and GK ). DFT, consider the exact free energy as a functional of an external potential. Express the free energy as a functional of the local density by Legendre transformation. Introduce local orbitals,   R (r-R)orbitals, and local GF G(R,R)(i  ) = The exact free energy can be expressed as a functional of the local Greens function and of the density by introducing sources for  (r) and G and performing a double Legendre transformton

28. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Spectral Density Functional Formal construction of a functional of the d spectral density DFT is useful because good approximations to the exact density functional  DFT  (r)] exist, e.g. LDA, GGA A useful approximation to the exact functional can be constructed, the DMFT +LDA functional.

29. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS LDA+DMFT functional  Atom =Sum of all local 2PI graphs build with local Coulomb interaction matrix, parametrized by Slater integrals F0, F2 and F4 and local G.Express  in terms of AIM model.

30. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Outer loop relax U E dc Impurity Solver SCC G,  G0G0 DMFT LDA+U Hartree-Fock

31. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Outer loop relax U E dc Impurity Solver SCC G,  G0G0 DMFT LDA+U Imp. Solver: Hartree-Fock

32. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS LDA+DMFT Self-Consistency loop DMFT U E

33. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS LDA+DMFT and LDA+U Static limit of the LDA+DMFT functional, with  =  HF reduces to the LDA+U functional of Anisimov et.al. Crude approximation. Reasonable in ordered situations.

34. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS DMFT If the self energy matrix is weakly k dependent is the physical self energy. Since is a matrix, DMFT changes the shape of the Fermi surface DMFT is absolutely necessary in the high temperature “local moment”regime. LDA+U with an effective U is OK at low energy. DMFT is needed to describe spectra with QP and Hubbard bands or satellites.

35. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Applications of LDA+DMFT Organics Alpha-Gamma Cerium V2O3 Volume collapse in Pu Photoemission of ruthenates Doping driven Mott transition in LaSrTiO3 Itinerant Ferromagnetism Bucky Balls

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37. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Applications: Itinerant Ferromagnetism, Ni Fe Compromise in the resources used for the solution of the one electron problem, and the many body problem. Goal: obtain an overall approximate but consistent picture of how correlations affect physical properties. Estimate sensitivity on parameters. Tc, spectra, susceptibility, [QMC- impurity solver] [ASA, relatively small number of k points] Magnetic anisotropy [HF-impurity solver][full potential LMTO, large number of k points, non collinear magnetization]

38. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Case study Fe and Ni Archetypical itinerant ferromagnets LSDA predicts correct low T moment Band picture holds at low T 

39. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Iron and Nickel: crossover to a real space picture at high T

40. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Other aspects that require to treate correlations beyond LDA Magnetic anisotropy. L.S effect. LDA predicts the incorrect easy axis(100) for Nickel.(instead of the correct one (111) ) LDA Fermi surface in Nickel has features which are not seen in DeHaas Van Alphen ( G. Lonzarich) Photoemission spectra of Ni : 6 ev satellite 30% band narrowing, reduction of exchange splitting.

41. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS DMFT-QMC: Numerical Details 256 k points 10 5 - 10 6 QMC sweeps Analytic continuation via maximum entropy. Tight binding LMTO-ASA

42. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Photoemission Spectra and Spin Autocorrelation: Fe (U=2, J=.9ev,T/Tc=.8) (Lichtenstein, Katsenelson,GK prl 2001)

43. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Photoemission and T/Tc=.8 Spin Autocorrelation: Ni (U=3, J=.9 ev)

44. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Iron and Nickel:magnetic properties (Lichtenstein, Katsenelson,GK PRL 01)

45. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Ni and Fe: theory vs exp  eff    high T moment Fe 3.1 (theory) 3.12 (expt) Ni 1.5 (theory) 1.62 (expt) Curie Temperature T c Fe 1900 ( theory) 1043(expt) Ni 700 (theory) 631 (expt)

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47. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS  MAE is small (1  eV/Atom)  Ni: 2.8  eV/Atom easy axis 111 Fe: 1.4  eV/Atom easy axis 100 Long standing problem Early papers Van Vleck (PR 1937) Brooks (PR 1940) Magnetic anisotropy

48. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS  Trygg et.al (1995); SCF Total energy with large # of k-points; Wrong easy axis for Ni.  Other related works: Halilov et al. (1998) G. Schneider et al. (1997) Wang et al. (1993) Beiden et.al. (1998) LDA calculations

49. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Full-potential multiple kappa LMTO method. Pauli treatment of relativistic effects. Non-collinear intraatomic magnetism included. Explore different Edc. (Details I Yang Ph.D thesis) Generalized relativistic LDA+U with occupancies n  ’ Method

50. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Work of Trygg et.al. proves equivalence of special points and tetrahedra. Confirmed. (broadening 0.15  Ry.) Convergent E tot needs 15000 k’s. We use 28000k’s. Convergency checked to 100000 k’s. SUN E10K with 64 processors used. LDA results of Trygg et.al. reproduced: Ni 0.5  eV 001, exp. 2.8  eV 111, Fe 0.5  eV 001, exp. 1.4  eV 001. Numerical Considerations

51. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Studies of MAE as function of U and J. Both U and J influence magnetic moment which is OK in LDA: 0.6  B for Ni and 2.2  B for Fe. How to fix moment in LDA+U: Find M(U,J) and trace path for which moment does not change. LDA+U Results

52. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Magnetic moment as function of U and J for Ni Ni - M(U,J) N i - M(U,J)

53. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Magnetic moment as function of U and J for Fe Fe - M(U,J)

54. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS MAE as a function of U(J) U=1.9 eV, J=1.2 eV U=1.2 eV, J=0.8 eV Ni Fd

55. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS e g forming X 2 pocket egeg LDA vs LDA+U for Ni

56. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Ni U=2,J=.1 PT (Katsenelson and Lichtenstein)cond-matt 2002

57. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Conclusions Satellite in majority band at 6 ev, 30 % reduction of bandwidth, exchange splitting reduction from band theory value (.6ev) to.3 ev Spin wave stiffness controls the effects of spatial flucuations, it is about twice as large in Ni and in Fe. Single site should work for Ni, and overestimate Tc for Fe. Mean field calculations using measured exchange constants(Kudrnovski Drachl PRB 2001) right Tc for Ni but overestimates Fe, RPA corrections reduce Tc of Ni by 10% and Tc of Fe by 50%.

58. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Overall consistent picture of the effects of correlations on itinerant magnets using DMFT.  Can reproduce correct easy axis and MAE of Fe and Ni.  Can correct the Fermi surface of Ni. Conclusions

59. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Work in progress With existing techniques, derive practical formulae for the magnetic anisotropy of systems containing partially localized and itinerant electrons. Further tests of DMFT on interesting materials. Incorporate extensions of DMFT to incorporate frequency dependent interations (GW+DMFT) and to larger clusters.

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61. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS No changes of Fermi surface found LDA and LDA+U bands for Fe

62. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS LDA electronic structure for Ni E(k) for Ni

63. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Calculated Fermi surface for Ni using LDA+U. No artificial X 2 pocket Fermi Surface for Ni

64. Impurity Solver SCC G,  G0G0 DMFT LDA+U Hartree-Fock

65. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS GW+DMFT functional. S. Savrasov and GK. P. Sun and GK. (cond matt).

66. Realistic Theories of Correlated Materials ITP, Santa-Barbara July 20 – December 20 (2002) O.K. Andesen, A. Georges, G. Kotliar, and A. Lichtenstein http://www.itp.ucsb.edu/activities/future/

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

68. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS 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)]

69. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Minimize LDA functional Kohn Sham eigenvalues, auxiliary quantities.

70. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS LDA functional Conjugate field, V KS (r)

71. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Double counting term (Lichtenstein et.al) subtracts average correlation

72. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS However not everything in low T phase is OK as far as LDA goes.. Magnetic anisotropy puzzle. LDA predicts the incorrect easy axis(100) for Nickel.(instead of the correct one (111) LDA Fermi surface has features which are not seen in DeHaas Van Alphen ( Lonzarich) Use LDA+ U to tackle these refined issues, ( compare parameters with DMFT results )

73. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS DMFT Impurity cavity construction: A. Georges, G. Kotliar, PRB, (1992)] Weiss field

74. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Single site DMFT, functional of local Greens function G. Express in terms of Weiss field (semicircularDOS) Local self energy (Muller Hartman 89)

75. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS LDA+DMFT functional  Sum of local 2PI graphs with local Coulomb interaction matrix and local G

76. Calculated MAE for Ni and Fe using LDA+U method S. Y. Savrasov New Jersey Institute of Technology In collaboration with: Imseok Yang (Ph.D Thesis, RU) Gabriel Kotliar (RU) Sponsored by Office of Naval Research Grant No: ONR 4-2650 Phys. Rev. Lett. 87, 216405 (2001)

77. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Total Energy DFT job with huge k-point summation problem.  Eckard et.al (1987); Right order; Wrong easy axis for Fe.  Daalderlop et.al (1990); Force theorem; Wrong easy axis for Ni. Varying position of Fermi level, artificial X 2 pocket influences easy axis. Calculations

78. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Ni and Fe: theory vs exp  ( T=.9 Tc)/   ordered moment Fe 1.5 ( theory) 1.55 (expt) Ni.3 (theory).35 (expt)  eff    high T moment Fe 3.1 (theory) 3.12 (expt) Ni 1.5 (theory) 1.62 (expt) Curie Temperature T c Fe 1900 5 ( theory) 1043(expt) Ni 700 (theory) 631 (expt)

79. Correlations Many-body Hubbard interactions are important (not captured by LDA) DMFT: onsite correlations are treated exactly, both atomic and band limit are OK. Static limit of DMFT: LDA+U method: Self-energy  (  )->  (static) Solution of impurity model collapses to determination of n  Problem can be solved now.

80. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS 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. Many materials have electron states that are in between these two limiting situations and require the development of new electronic structure methods to predict some of its properties (spectra, energy, transport,….) DMFT simplest many body technique which treats simultaneously the open shell atomic limit and the band limit.

81. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Mean-Field : Classical vs Quantum Classical case Quantum case Phys. Rev. B 45, 6497 A. Georges, G. Kotliar (1992)