1. Dynamical Mean Field Theory for Electronic Structure Calculations Gabriel Kotliar Physics Department and Center for Materials Theory Rutgers University CECAM workshop on realistic studies of correlations Lyon July 25-29 th 2001

2. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Outline Choice of Basis. Realistic self consistency condition Integration with LDA. Effective action formulation. Comparison with LDA and LDA+U Some examples in real materials, transition metals and actinides.

3. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Acknowledgements: Collaborators, Colleagues, Support ….. S. Lichtenstein (Nijmeigen), E Abrahams (Rutgers) G. Biroli (Rutgers), R. Chitra (Rutgers- Jussieux), V. Udovenko (Rutgers), S. Savrasov (Rutgers-NJIT) G. Palsson, I. Yang (Rutgers) NSF, DOE and ONR

4. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS DMFT Impurity construction: A. Georges, G. Kotliar, PRB, (1991)] Weiss field

5. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Elements of the Dynamical Mean Field Construction and C-DMFT. Definition of the local degrees of freedom Expression of the Weiss field in terms of the local variables (I.e. the self consistency condition) Expression of the lattice self energy in terms of the cluster self energy.

6. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Cellular DMFT : Basis selection

7. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Lattice action

8. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Elimination of the medium variables

9. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Determination of the effective medium.

10. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Connection between cluster and lattice self energy. The estimation of the lattice self energy in terms of the cluster energy has to be done using additional information Ex. Translation invariance C-DMFT is manifestly causal: causal impurity solvers result in causal self energies and Green functions (GK S. Savrasov and G. Palsson) Improved estimators for the lattice self energy are available (Biroli and Kotliar) In simple cases C-DMFT converges faster than other causal cluster schemes.

11. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Realistic DMFT self consistency loop

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

13. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS DMFT+QMC (A. Lichtenstein, M. Rozenberg)

14. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Schematic DMFT phase diagram one band Hubbard model (half filling, semicircular DOS, partial frustration) Rozenberg et.al PRL (1995)

15. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Recent phase diagram of the frustrated Half filled Hubbard model with semicircular DOS (QMC Joo and Udovenko).

16. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Case study: IPT half filled Hubbard one band (Uc1) exact = 2.1 (Exact diag, Rozenberg, Kajueter, Kotliar 1995), (Uc1) IPT =2.4 (Uc2) exact =2.95 (Projective self consistent method, Moeller Si Rozenberg Kotliar PRL 1995 ) (Uc 2 ) IPT =3.3 (T MIT ) exact =.026+_.004 (QMC Rozenberg Chitra and Kotliar PRL 1999), (T MIT ) IPT =.5 (U MIT ) exact =2.38 +-.03 (QMC Rozenberg Chitra and Kotliar PRL 1991), (U MIT ) IPT =2.5 For realistic studies errors due to other sources (for example the value of U, are at least of the same order of magnitude).

17. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Realistic implementation of the self consistency condition H and S, do not commute Need to do k sum for each frequency DMFT implementation of Lambin Vigneron tetrahedron integration (Poteryaev et.al 1987)

18. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Solving the impurity Multiorbital situation and several atoms per unit cell considerably increase the size of the space H (of heavy electrons). QMC scales as [N(N-1)/2]^3 N dimension of H Fast interpolation schemes (Slave Boson at low frequency, Roth method at high frequency, + 1 st mode coupling correction), match at intermediate frequencies. (Savrasov et.al 2001)

19. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Good method to study the Mott phenomena Evolution of the electronic structure between the atomic limit and the band limit. Basic solid state problem. Solved by band theory when the atoms have a closed shell. Mott’s problem: Open shell situation. The “”in between regime” is ubiquitous central them in strongly correlated systems. Some unorthodox examples Fe, Ni, Pu …………….

20. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Two Roads for calculations of the electronic structure of correlated materials Crystal Structure +atomic positions Correlation functions Total energies etc. Model Hamiltonian

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

22. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Minimize LDA functional

23. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS LDA+U functional

24. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS 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 of viewed as parameters

25. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Spectral Density Functional : effective action construction ( Fukuda, Valiev and Fernando, Chitra and GK ). 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 sources for  (r) and G and performing a Legendre transformation,  (r),G(R,R)(i  )]

26. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Spectral Density Functional The exact functional can be built in perturbation theory in the interaction (well defined diagrammatic rules )The functional can also be constructed from the atomic limit, but no explicit expression exists. 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.

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

28. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Comments on LDA+DMFT Static limit of the LDA+DMFT functional, with  =  HF reduces to LDA+U Removes inconsistencies of this approach, Only in the orbitally ordered Hartree Fock limit, the Greens function of the heavy electrons is fully coherent Gives the local spectra and the total energy simultaneously, treating QP and H bands on the same footing.

29. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS LDA+DMFT Connection with atomic limit Weiss field

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

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

32. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS 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 G.Kotliar, funcional formulation for full self consistent implementation (2001)

33. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Functional Approach The functional approach offers a direct connection to the atomic energies. One is free to add terms which vanish quadratically at the saddle point. Allows us to study states away from the saddle points, All the qualitative features of the phase diagram, are simple consequences of the non analytic nature of the functional. Mott transitions and bifurcations of the functional.

34. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Functional Approach G. Kotliar EPJB (1999)

35. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Case study in f electrons, Mott transition in the actinide series

36. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Pu: Anomalous thermal expansion (J. Smith LANL)

37. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Small amounts of Ga stabilize the  phase

38. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Delocalization-Localization across the actinide series o f electrons in Th Pr U Np are itinerant. From Am on they are localized. Pu is at the boundary. o Pu has a simple cubic fcc structure,the  phase which is easily stabilized over a wide region in the T,p phase diagram. o The  phase is non magnetic. an equilibrium volume of the  phase  Is 35% lower than experiment o Many LDA, GGA studies ( Soderlind et. Al 1990, Kollar et.al 1997, Boettger et.al 1998, Wills et.al. 1999) give an equilibrium volume of the  phase  Is 35% lower than experiment o This is one of the largest discrepancy ever known in DFT based calculations.

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

40. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS 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  phase of Pu, when full potential LMTO (Soderlind and Wills). This is usually taken as an indication that  Pu is a weakly correlated system

41. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Conventional viewpoint Alpha Pu is a simple metal, it can be described with LDA + correction. In contrast delta Pu is strongly correlated. Constrained LDA approach (Erickson, Wills, Balatzki, Becker). In Alpha Pu, all the 5f electrons are treated as band like, while in Delta Pu, 4 5f electrons are band-like while one 5f electron is deloclized. Same situation in LDA + U (Savrasov and Kotliar, Bouchet et. Al. ) Delta Pu has U=4, Alpha Pu has U =0.

42. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Problems with the conventional viewpoint of Pu The specific heat of delta Pu, is only twice as big as that of alpha Pu. The susceptibility of alpha Pu is in fact larger than that of delta Pu. The resistivity of alpha Pu is comparable to that of delta Pu. Only the structural and elastic properties are completely different.

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

44. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Anomalous Resistivity J. Smith LANL

45. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS MAGNETIC SUSCEPTIBILITY

46. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Dynamical Mean Field View of Pu ( Savrasov Kotliar and Abrahams, Nature 2001) Delta and Alpha Pu are both strongly correlated, the DMFT mean field free energy has a double well structure, for the same value of U. One where the f electron is a bit more localized (delta) than in the other (alpha). Is the natural consequence of the model hamiltonian phase diagram once electronic structure is about to vary. This result resolves one of the basic paradoxes in the physics of Pu.

47. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Pu: DMFT total energy vs Volume

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

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

50. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Earlier Studies of Magnetic Anisotropy Erickson Daalderop

51. 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 Main challenge, finite T properties (Lichtenstein’s talk). Magnetic anisotropy puzzle. LDA predicts the incorrect easy axis for Nickel. LDA Fermi surface has features which are not seen in DeHaas Van Alphen ( Lonzarich)

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

53. 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)

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

55. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Iron and Nickel:magnetic properties (Lichtenstein, Katsenelson,GK cond-mat 0102297)

56. 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 ( theory) 1043(expt) Ni 700 (theory) 631 (expt)

57. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Fe and Ni Satellite in minority band at 6 ev, 30 % reduction of bandwidth, exchange splitting reduction.3 ev Spin wave stiffness controls the effects of spatial flucuations, it is about twice as large in Ni and in 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 Ni moment

59. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Fe moment\

60. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Magnetic anisotropy Ni

61. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Magnetic anisotropy Fe

62. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Magnetic anisotropy

63. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Conclusion The character of the localization delocalization in simple( Hubbard) models within DMFT is now fully understood, nice qualitative insights.  This has lead to extensions to more realistic models, and a beginning of a first principles approach interpolating between atoms and band, encouraging results for simple elements

64. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS DMFT Review: A. Georges, G. Kotliar, W. Krauth and M. Rozenberg Rev. Mod. Phys. 68,13 (1996)] Weiss field

65. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Outlook Systematic improvements, short range correlations. Take a cluster of sites, include the effect of the rest in a G0 (renormalization of the quadratic part of the effective action). What to take for G0: DCA (M. Jarrell et.al), CDMFT ( Savrasov and GK ) include the effects of the electrons to renormalize the quartic part of the action (spin spin, charge charge correlations) E. DMFT (Kajueter and GK, Si et.al)

66. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Outlook Extensions of DMFT implemented on model systems, (e.g. Motome and GK ) carry over to more realistic framework. Better determination of Tcs. First principles approach: determination of the Hubbard parameters, and the double counting corrections long range coulomb interactions E-DMFT Improvement in the treatement of multiplet effects in the impurity solvers, phonon entropies, ………