1. Electronic Structure Near the Mott transition Gabriel Kotliar Physics Department and Center for Materials Theory Rutgers University

2. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Outline Introduction to the strong correlation problem and to the Mott transition Some dynamical mean field ideas Applications to the Mott transition problem: some insights from studies of model Hamiltonians. Towards an electronic structure method: applications to materials.

3. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Momentum Space, bands, k in Brillouin zone is good quantum number. Landau Fermi liquid theory interactions renormalize away at low energy, simple band picture in effective field holds. The electron in a solid: wave picture Maximum metallic resistivity 200  ohm cm

4. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Standard Model of Solids Qualitative predictions: low temperature dependence of thermodynamics and transport Optical response, transitions between bands. Qualitative predictions. Filled bands-Insulators, Unfilled bands metals. Odd number of electrons metallicity. Quantitative tools: DFT, LDA, GGA, total energies,good starting point for spectra, GW,and transport

5. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS The electron in a solid: particle picture. NiO, MnO, …Array of atoms is insulating if a>>a B. Mott: correlations localize the electron e_ e_ e_ e_ Think in real space, solid collection of atoms High T : local moments, Low T spin-orbital order Superexchange

6. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Mott : Correlations localize the electron Low densities, electron behaves as a particle,use atomic physics, work in 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….) Quantitative calculations of Hubbard bands and exchange constants, LDA+ U, Hartree Fock. Atomic Physics. H H H+ H H H motion of H+ forms the lower Hubbard band H H H H- H H motion of H_ forms the upper Hubbard band

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

8. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Correlated Materials do big things Huge resistivity changes V 2 O 3. Copper Oxides..(La 2-x Ba x ) CuO 4 High Temperature Superconductivity. 150 K in the Ca 2 Ba 2 Cu 3 HgO 8. Uranium and Cerium Based Compounds. Heavy Fermion Systems,CeCu 6,m*/m=1000 (La 1-x Sr x )MnO 3 Colossal Magneto- resistance.

9. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Strongly Correlated Materials. Large thermoelectric response in CeFe 4 P 12 (H. Sato et al. cond-mat 0010017). Ando et.al. NaCo 2-x Cu x O 4 Phys. Rev. B 60, 10580 (1999). Huge volume collapses, Ce, Pu…… Large and ultrafast optical nonlinearities Sr 2 CuO 3 (T Ogasawara et.a Phys. Rev. Lett. 85, 2204 (2000) )

10. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS The Mott transition Electronically driven MIT. Forces to face directly the localization delocalization problem. Relevant to many systems, eg V2O3 Techniques applicable to a very broad range or problems.

11. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Mott transition in V 2 O 3 under pressure or chemical substitution on V-site

12. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Universal and non universal features. Top to bottom approach to correlated materials. Some aspects at high temperatures, depend weakly on the material (and on the model). Low temperature phase diagram, is very sensitive to details, in experiment (and in the theory).

13. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Mott transition in layered organic conductors S Lefebvre et al. cond-mat/0004455, Phys. Rev. Lett. 85, 5420 (2000)

14. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Failure of the Standard Model: NiSe 2-x S x Miyasaka and Takagi (2000)

15. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Phase Diagrams :V 2 O 3, Ni Se 2 -x S x Mc Whan et. Al 1971,. Czek et. al. J. Mag. Mag. Mat. 3, 58 (1976),

16. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Outline Introduction to the strong correlation problem and to the Mott transition. DMFT ideas Applications to the Mott transition problem: some insights from studies of models. Towards an electronic structure method: applications to materials: NiO, Pu, Fe, Ni, LaSrTiO3, ………. Outlook

17. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Hubbard model  U/t  Doping  or chemical potential  Frustration (t’/t)  T temperature Mott transition as a function of doping, pressure temperature etc.

18. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Limit of large lattice coordination Metzner Vollhardt, 89 Muller-Hartmann 89

19. 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)

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

21. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS DMFT: Effective Action point of view. R. Chitra and G. Kotliar Phys Rev. B. (2000). Identify observable, A. Construct an exact functional of =a,  [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.

22. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Example: DMFT for lattice model (e.g. single band Hubbard).Muller Hartman 89, Chitra and Kotliar 99. Observable: Local Greens function G ii (  ). Exact functional  [G ii (  )  DMFT Approximation to the functional.

23. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Extensions of DMFT. Renormalizing the quartic term in the local impurity action. EDMFT. Taking several sites (clusters) as local entity. CDMFT Combining DMFT with other methods. LDA+DMFT, GW+EDMFT or “GWU”…..

24. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Outline Introduction to the strong correlation problem. Essentials of DMFT Applications to the Mott transition problem: some insights from studies of models. Towards an electronic structure method: applications to materials Outlook

25. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Schematic DMFT phase diagram Hubbard model (partial frustration). Evidence for QP peak in V2O3 from optics. M. Rozenberg G. Kotliar H. Kajueter G Thomas D. Rapkine J Honig and P Metcalf Phys. Rev. Lett. 75, 105 (1995)

26. 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)

27. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS X.Zhang M. Rozenberg G. Kotliar (PRL 1993) Spectral Evolution at T=0 half filling full frustration. Three peak structure.

28. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Evolution of the Spectral Function with Temperature Anomalous transfer of spectral weight connected to the proximity to the Ising Mott endpoint (Kotliar Lange and Rozenberg Phys. Rev. Lett. 84, 5180 (2000)

29. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Insights from DMFT Three peak structure of the density of states. In the strongly correlated metallic regime the Hubbard bands are well formed.

30. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Insights from DMFT  The Mott transition is driven by transfer of spectral weight from low to high energy as we approach the localized phase  Control parameters: doping, temperature,pressure…

31. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS What about experiments?

32. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Parallel development: Fujimori et.al

33. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Mott transition in V 2 O 3 under pressure or chemical substitution on V-site

34. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Anomalous transfer of optical spectral weight V2O3 :M Rozenberg G. Kotliar and H. Kajuter Phys. Rev. B 54, 8452 (1996). M. Rozenberg G. Kotliar H. Kajueter G Tahomas D. Rapkikne J Honig and P Metcalf Phys. Rev. Lett. 75, 105 (1995)

35. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Anomalous transfer of spectral weight in v2O3

36. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Anomalous Spectral Weight Transfer: Optics Schlesinger et.al (FeSi) PRL 71,1748, (1993) B Bucher et.al. Ce 2 Bi 4 Pt 3 PRL 72, 522 (1994), Rozenberg et.al. PRB 54, 8452, (1996). ApreciableT dependence found. Below energy

37. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS. ARPES measurements on NiS 2-x Se x Matsuura et. al Phys. Rev B 58 (1998) 3690. Doniach and Watanabe Phys. Rev. B 57, 3829 (1998)

38. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Anomalous transfer of optical spectral weight, NiSeS. [Miyasaka and Takagi 2000]

39. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Anomalous Resistivity and Mott transition Ni Se 2-x S x Insights from DMFT: think in term of spectral functions (branch cuts) instead of well defined QP (poles )

40. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Recent exps. Moo et. al. (2003) Theory Held et. al.

41. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Transport in 2d organics. Limlet et. al.

42. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Strong correlation anomalies Metals with resistivities which exceed the Mott Ioffe Reggel limit. Transfer of spectral weight which is non local in frequency. Dramatic failure of DFT based approximations in predicting physical properties.

43. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Conclusions: generic aspects Three peak structure, quasiparticles and Hubbard bands. Non local transfer of spectral weight. Large resistivities.

44. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Insights from DMFT. Important role of the incoherent part of the spectral function at finite temperature Physics is governed by the transfer of spectral weight from the coherent to the incoherent part of the spectra. Real and momentum space pictures are needed as synthesized in DMFT.

45. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Outline Introduction to the strong correlation problem. Essentials of DMFT Applications to the Mott transition problem: some insights from studies of models. Towards an electronic structure method: applications to materials: Pu, Fe, Ni, Ce, LaSrTiO3, NiO,MnO,CrO2,K3C60,2d and quasi-1d organics, magnetic semiconductors,SrRuO4,V2O3…………. Outlook

46. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Interface DMFT with electronic structure. Derive model Hamiltonians, solve by DMFT (or cluster extensions).  Full many body aproach, treat light electrons by GW or screened HF, heavy electrons by DMFT.  Treat correlated electrons with DMFT and light electrons with DFT (LDA, GGA +DMFT)

47. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS DMFT: Effective Action point of view. R. Chitra and G. Kotliar Phys Rev. B. (2000). Identify observable, A. Construct an exact functional of =a,  [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.

48. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Spectral Density Functional : effective action construction ( Chitra and GK ). 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  )] Approximate functional using DMFT insights.

49. 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: Savrasov GK and Abrahams (Nature 2001) Dai Savrasov GK Migliori Letbetter and Abrahams (Science 2003) 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 2003) Paramagnetic Mott insulators. NiO MnO, Savrasov and GK( PRL 2002)……………………

50. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Case study in f electrons, Mott transition in the actinide series. B. Johanssen 1974 Smith and Kmetko Phase Diagram 1984.

51. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Physics of Pu

52. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Plutonium Puzzles 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.

53. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS DFT Studies LSDA predicts magnetic long range (Solovyev et.al.) Experimentally  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  phase of Pu, when full potential LMTO (Soderlind Eriksson and Wills) is used. This is usually taken as an indication that  Pu is a weakly correlated system Alternative approach Wills et. al. (5f)4 core+ 1f(5f)in conduction band.

54. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Shear anisotropy fcc Pu (GPa) C’=(C11-C12)/2 = 4.78 C44= 33.59 C44/C’ ~ 8 Largest shear anisotropy in any element! LDA Calculations (Bouchet et. al.) C’= -48

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

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

57. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Functional approach allows computation of linear response.(S. Savrasov and GK 2002) Apply to NiO, canonical Mott insulator. U=8 ev, J=.9ev Simple Impurity solver Hubbard 1.

58. Results for NiO: Phonons (Savrasov and Kotliar PRL 2002) Solid circles – theory, open circles – exp. ( Roy et.al, 1976 ) DMFT

59. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Phases of Pu

60. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Dai et. al.

61. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Epsilon Plutonium.

62. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Outline Introduction to the strong correlation problem. Essentials of DMFT Applications to the Mott transition problem: some insights from studies of models. Towards an electronic structure method: applications to materials: Outlook

63. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Outlook Local approach to strongly correlated electrons. Many extensions, make the approach suitable for getting insights and quantitative results in correlated materials.

64. 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 to the electronic structure of correlated materials.

65. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Outlook Systematic improvements, short range correlations, cluster methods, improved mean fields. Improved interfaces with electronic structure. Exploration of complex strongly correlated materials. Correlation effects on surfaces, large molecules, systems out of equilibrium, illumination, finite currents, aeging.

66. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Acknowledgements: Development of DMFT Collaborators: V. Anisimov,G. Biroli, R. Chitra, V. Dobrosavlevic, X. Dai, D. Fisher, A. Georges, H. Kajueter, K. Haujle, W.Krauth, E. Lange, A. Lichtenstein, G. Moeller, Y. Motome, O. Parcollet, G. Palsson, M. Rozenberg, S. Savrasov, Q. Si, V. Udovenko, I. Yang, X.Y. Zhang Support: NSF DMR 0096462 Support: Instrumentation. NSF DMR-0116068 Work on Fe and Ni: ONR4-2650 Work on Pu: DOE DE-FG02-99ER45761 and LANL subcontract No. 03737-001-02

67. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

68. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

69. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Expts’ Wong et. al.

70. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

71. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

72. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS E-DMFT references H. Kajueter and G. Kotliar (unpublished and Kajuter’s Ph.D thesis (1995)). Q. Si and J L Smith PRL 77 (1996)3391. R. Chitra and G.Kotliar Phys. Rev. Lett 84, 3678- 3681 (2000 )Phys. Rev. Lett 84, 3678- 3681 (2000 Y. Motome and G. Kotliar. PRB 62, 12800 (2000) R. Chitra and G. Kotliar Phys. Rev. B 63, 115110 (2001)Phys. Rev. B 63, 115110 (2001) S. Pankov and G. Kotliar PRB 66, 045117 (2002)

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

74. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Cluster extensions of DMFT Two impurity method. [A. Georges and G. Kotliar (1995 unpublished ) and RMP 68,13 (1996), A. Schiller PRL75, 113 (1995)] M. Jarrell et al Dynamical Cluster Approximation [Phys. Rev. B 7475 1998] Periodic cluster] M. Katsenelson and A. Lichtenstein PRB 62, 9283 (2000). G. Kotliar S. Savrasov G. Palsson and G. Biroli Cellular DMFT [PRL87, 186401 2001]

75. 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)

76. 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

77. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS DMFT plus other methods. DMFT+ LDA, 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. Savasov Kotliar and Abrahams. Application to delta Pu Nature (2001) Combining EDMFT with GW. Ping Sun and Phys. Rev. B 66, 085120 (2002). G. Kotliar and S. Savrasov in New Theoretical Approaches to Strongly Correlated Systems, A. M. Tsvelik Ed. 2001 Kluwer Academic Publishers. 259- 301. cond-mat/0208241

78. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS. ARPES measurements on NiS 2-x Se x Matsuura et. Al Phys. Rev B 58 (1998) 3690. Doniaach and Watanabe Phys. Rev. B 57, 3829 (1998)

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

80. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS QP in V2O3 was recently found Mo et.al

81. 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)

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

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

84. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Anomalous Spectral Weight Transfer: Optics Schlesinger et.al (FeSi) PRL 71,1748, (1993) B Bucher et.al. Ce 2 Bi 4 Pt 3 PRL 72, 522 (1994), Rozenberg et.al. PRB 54, 8452, (1996). AppreciableT dependence found. Below energy

85. 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.

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

87. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS DMFT: Effective Action point of view. R. Chitra and G. Kotliar Phys Rev. B. (2000). Identify observable, A. Construct an exact functional of =a,  [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. It is useful to introduce a Lagrange multiplier  conjugate to a,  [a,  It gives as a byproduct a additional lattice information.

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

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