1. Electronic Structure of Correlated Materials : a DMFT Perspective Gabriel Kotliar Physics Department and Center for Materials Theory Rutgers University In Realistic Theories, GRC on Correlated Electrons. June 29-July 3 rd 2002 Supported by the NSF DMR 0096462

2. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Outline Introduction Basic ideas of Dynamical Mean Field Theory and its extensions. Qualitative successes of DMFT. Realistic implementation of DMFT. Illustrations: NiO (with S. Savrasov) Fe and Ni (with Lichtenstein and Katsnelson) Outlook.

3. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Weakly correlated electrons:band theory. Simple conceptual picture of the ground state, excitation spectra, transport properties of many systems (simple metals, semiconductors,….). A methods for performing quantitative calculations. (Density functional theory, in various approximations).

4. Success story : Density Functional Linear Response Tremendous progress in ab initio modelling of lattice dynamics & electron-phonon interactions has been achieved ( Review: Baroni et.al, Rev. Mod. Phys, 73, 515, 2001 ) (Savrasov, PRB 1996)

5. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Limitations of LDA LDA spectra can be taken a starting point for perturbative (eg. GW ) calculations of excitation spectra and transport. THIS DOES NOT WORK for strongly correlated systems, eg oxides containing 3d, 4f, 5f elements. CHARACTER OF THE EXCITATION SPECTRA is not captured by LDA. LDA does not have good predicted power for ground state properties in this system either.

6. 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). Standard approaches (LDA, HF ) do not work well. Dynamical Mean Field Theory. Treats atoms and bands. Treats QP bands and Hubbard bands. Exact in large dimensionality.

7. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Strongly correlated electrons Large degeneracy. Low energy scales. Many Competing forms of long range order (various forms of charge-spin-orbital and even currents) Quasidegenerate ground states, with different forms of magnetic order. Competition between different possible states, frustration, phase separation. Tunability. Intricate microsctrucure. Mesoscale.

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

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

10. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS C-DMFT C:DMFT The lattice self energy is inferred from the cluster self energy.

11. 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

12. 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)]

13. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS DMFT: Methods of Solution

14. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS A (non comprehensive )list of extensions of DMFT Two impurity method. [A. Georges and G. Kotliar, A. Schiller K Ingersent ] Bethe Peirels [ A. Georges and G. Kotliar] Dynamical Cluster Approximation [M. Jarrell et. Al. Phys. Rev. B 7475 1998] Periodic cluster [Lichtenstein and Katsnelson]. Cellular Dynamical Mean Field Theory [G. Kotliar et.al] Extended DMFT [Sachdev and Ye, Parcollet and Georges, H. Kajueter and G. Kotliar, Q. Si and J L Smith, R. Chitra and G. Kotliar] Combination with lowest order Perturbation theory for the light orbitals [Savrasov Kotliar, Ping Sun]

15. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Insights from model DMFT studies  Canonical phase diagram at integer occupation. Low temperature Ordered phases. Stability depends on details (chemistry and crystal structure)  High temperature behavior around Mott endpoint, more universal regime, captured by simple models treated within DMFT. Role of magnetic frustration.

16. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Schematic DMFT phase diagram Hubbard model (partial frustration ) M. Rozenberg G. Kotliar H. Kajueter G Thomas D. Rapkikne J Honig and P Metcalf Phys. Rev. Lett. 75, 105 (1995)

17. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Kuwamoto Honig and Appell PRB (1980)

18. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Mott transition in pyrites: NiSe 2-x S x Miyasaka and Takagi (2000)

19. 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)

20. 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…

21. 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)

22. 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)

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

24. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS General Lessons Coherence –incoherence crossover. Mott transition in one band model. Transfer of spectral weight. Coexistence of atomic like and band like excitations at finite temperatures. Anomalous transport. Simple laws for transfer of spectral weight around special points.

25. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Realistic DMFT methods. Spectral functions. Finite temperatue. Excitations. Ground state properties are a byproduct of spectra. Can be computed more reliably being less sensitive on long distance details. High temperature. NON PERTURBATIVE, using the Weiss field as a variable one can cross the barrier where skeleton PT theory breaks down.

26. 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.

27. 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

28. 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)

29. 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 Anisimov Hybertsen et.al) or viewed as parameters

30. 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.

31. 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.

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

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

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

35. 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) ………………..

36. Failures of lda NiO dielectric constant. LSDA:35.7 Exp:5.7 Lattice dynamics cannot be predicted: Optical G-phonon in MnO within LSDA: 3.04 THz, Experimentally: 7.86 THz ( Massidda, et.al, PRL 1999 ) Bulk modulus for metallic Plutonium is one order of magnitude too large within LDA (214 GPa vs. 30 GPa) Also elastic constants are off. ( Bouchet, et.al, J.Phys.C, 2001 )

37. 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= J=.9 Simple Impurity solver Hubbard 1.

38. Results for NiO: Phonons Solid circles – theory, open circles – exp. ( Roy et.al, 1976 )

39. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS NiO U=8ev, J=1ev, Savrasov and GK (2002)

40. 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 Curie behavior at high temperatures. Crossover between a real space and a momentum space prediction. 

41. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Iron and Nickel: crossover to a real space picture at high T (Lichtenstein, Katsnelson and Kotliar Phys Rev. Lett 87, 67205, 2001 )

42. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Photoemission Spectra and Spin Autocorrelation: Fe (U=2, J=.9ev,T/Tc=.8) (Lichtenstein, Katsenelson,Kotliar Phys Rev. Lett 87, 67205, 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  /   ordered moment Fe 2.5 ( theory) 2.2(expt) Ni.6 (theory).6(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)

46. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Role of theory Orient the thinking about materials. Visualization. Generate Refine questions, ask about deviation from DMFT.

47. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Outlook. Many qualitative aspects of the Mott transition in clusters need to be understood. The notions and the calculations of U’s need to be refined a revisited. (E-DMFT). Replacing LDA part by simple low order diagrammatic scheme (local GW) RG techniques and cluster impurity solvers. Small clusters may be needed for accurate computations of critical temperatures. Role of long wavelength fluctuations? Many materials to be studied, and insights to be gained.

48. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Acknowledgements: Development of DMFT Collaborators: V. Anisimov, C. Bolech, G. Biroli, R. Chitra, V. Dobrosavlevic, D. Fisher, A. Georges, H. Kajueter, V. Kancharla, W.Krauth, E. Lange, A. Lichtenstein, G. Moeller, Y. Motome, G. Palsson, O. Parcollet, M. Rozenberg, S. Savrasov, Q. Si, V. Udovenko,X.Y. Zhang Support: National Science Foundation..

49. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

50. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Future Challenges Develop user friendly interfaces, for first principles calculations of realistic DMFT, for visualization of spectra, resolved in real space, momentum space and orbital space. FAT DMFT. [Done for LDA, S. Savrasov, Material Information and DESIGN Laboratory] and for further code development.

51. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Outlook  The Strong Correlation Problem:How to deal with a multiplicity of competing low temperature phases and infrared trajectories which diverge in the IR  Strategy: advancing our understanding scale by scale  Generalized cluster methods to capture longer range magnetic correlations  New structures in k space. Cellular DMFT

52. 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)

53. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Photoemission V2O3 Held et.al. PRL 2001

54. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS V2O3 LDA+DMFT Held et.al. PRL 2001

55. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Outlook Extensions of DMFT implemented on model systems, 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, ………

56. 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.

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

58. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Dynamical Mean Field Theory, cavity construction A. Georges G. Kotliar 92

59. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Extended DMFT electron phonon

60. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Extended DMFT e.ph. Problem

61. 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)

62. 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

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

64. 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

65. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Comments on LDA+DMFT Static limit of the LDA+DMFT functional, with  =  HF reduces to LDA+U DMFT retain correlations effects in the absence of orbital ordering. (for example treating the impurity model in the Hubbard 1 approximation). Functional formulation allows calculation of total energies and linear response. Gives the local spectra and the total energy simultaneously, treating QP and H bands on the same footing.