1. Optical Properties of Strongly Correlated Electrons: A Dynamical Mean Field Approach G. Kotliar Physics Department and Center for Materials Theory Rutgers University

2. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Outline Correlated Electrons and the Dynamical Mean Field Theory (DMFT) framework. Restricted Sum Rules and Transfer of Optical Spectral Weight. Optics near the temperature driven Mott transition. The Cerium alpha-gamma transition, Mott transition or Kondo collapse ? A perspective from optics.

3. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS References, Collaborators. DMFT: Reviews: A. Georges G. Kotliar W. Krauth and M. Rozenberg RMP68, 13, 1996 Gabriel Kotliar and Dieter Vollhardt Physics Today 57,(2004). Optical transfer or spectral weight near the Mott transition. M. Rozenberg G. Kotliar and H. Kajueter PRB 54, 8452, (1996).  DMFT Optics V. Udovenko S. Savrasov K. Haule and G. Kotliar Cond-matt 0209336.  Alpha-Gamma Cerium. K. Haule V. Udovenko S. Savrasov and G. Kotliar. Cond-matt 0403086.

4. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS MAIN MESSAGE DMFT is a working tool (under constant development). Theory (DMFT) and experiments (optical conductivity) complement each other extraordinary well. Interpretation. Predictions. Access to regimes that cannot be easily reached in real materials.

5. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS “Standard Model “. Kohn Sham reference system Excellent starting point for computation of spectra in perturbation theory in screened Coulomb interaction GW. Bethe Salpeter equation for optics.

6. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS “Standard Model” fails when Correlations localize the electron Hubbard bands. One particle excitations: corresponding to adding or removing electrons. In solids they broaden by their incoherent motion (eg. Mott insulators NiO, CoO MnO….) 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 Optical conductivity, start from atomic physics and broaden the atomic transitions (on site processes). Transitions to neighboring atomic states (transitions between the Hubbard bands ). One needs a tool that treats quasiparticle bands and Hubbard bands on the same footing to contain the band and atomic limit. DMFT!

7. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Strong correlation anomalies Metals with resistivities which exceed the Mott Ioffe Reggel limit. Gigantic linear and non linear responses. Dramatic failure of DFT based approximations (say DFT-GW) in predicting physical properties. Breakdown of the rigid band picture.

8. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Transfer of optical spectral weight non local in frequency Schlesinger et.al (FeSi) PRL 71,1748, (1993) B Bucher et.al. Ce 2 Bi 4 Pt 3 PRL 72, 522 (1994), Neff depends on T

9. 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

10. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS RESTRICTED SUM RULES M. Rozenberg G. Kotliar and H. Kajueter PRB 54, 8452, (1996). ApreciableT dependence found. Below energy

11. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS RESTRICTED SUM RULES M. Rozenberg G. Kotliar and H. Kajueter PRB 54, 8452, (1996). ApreciableT dependence found. Below energy

12. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS DMFT Cavity Construction. A. Georges and G. Kotliar PRB 45, 6479 (1992). First happy marriage of a technique from atomic physics and a technique band theory. Reviews: A. Georges G. Kotliar W. Krauth and M. Rozenberg RMP68, 13, 1996 Gabriel Kotliar and Dieter Vollhardt Physics Today 57,(2004)

13. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Self-Consistency loop. S. Savrasov and G. Kotliar (2001) and cond-matt 0308053 DMFT U E

14. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Spectral Density Functional : effective action construction G. Kotliar, and S. Savrasov, in New Theoretical approaches to strongly correlated systems, edited by A.M. Tsvelik, Kluwer Academic Publishers, 259 (2001); S. Y. Savrasov and G. Kotliar, Phys. Rev. B 69, 245101 (2004).) 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  )]

15. 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). G. Kotliar, and S. Savrasov, in New Theoretical ap- proaches to strongly correlated systems, edited by A. M. Tsvelik, Kluwer 259 (2001); S. Y. Savrasov and G. Kotliar, Phys. Rev. B 69, 245101 (2004).

16. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS LDA+DMFT Formalism.

17. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Optics formula double pole single pole One divergence integrated out!

18. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Comments on LDA+DMFT Gives the local spectra and the total energy simultaneously, treating QP and H bands on the same footing. Gives an approximate starting point, for perturbation theory in the non local part of the Coulomb interactions. [See for example, P. Sun and G. Kotliar PRL ]. Good approximate starting point for optics.

19. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Outline Correlated Electrons and the Dynamical Mean Field Theory (DMFT) framework. Restricted Sum Rules and Transfer of Optical Spectral Weight. Optics near the temperature driven Mott transition. The Cerium alpha-gamma transition, Mott transition or Kondo collapse ? A perspective from optics. Doping driven Mott transition in La 1-x Sr x TiO 3. A perspective from the optical conductivity.

20. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Pressure Driven Mott transition

21. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Insights from DMFT  Low temperature Ordered phases. Stability depends on chemistry and crystal structure  High temperature behavior around Mott endpoint, more universal regime, captured by simple models treated within DMFT. Role of magnetic frustration.

22. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Schematic DMFT phase diagram of a partially frustrated integered filled Hubbard model. M. J. Rozenberg, G. Kotliar, H. Kajueter, G. A. Thomas, D. H. Rapkine, J. M. Honig, and P. Metcalf, Phys. Rev. Lett. 75, 105, 1995

23. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Spectral Evolution at T=0 half filling full frustration X.Zhang M. Rozenberg G. Kotliar (PRL 70,16661993) Spectra of the strongly correlated metallic regime contains both quasiparticle-like and Hubbard band-like features. Mott transition is driven by transfer of spectral weight.

24. 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 nd Rozenberg Phys. Rev. Lett. 84, 5180 (2000)

25. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Consequences for the optical conductivity 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 Anomalous transfer of spectral weight

27. 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)

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

29. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Optical transfer of spectral weight, kappa organics. Eldridge, J., Kornelsen, K.,Wang, H.,Williams, J., Crouch, A., and Watkins, D., Sol. State. Comm., 79, 583 (1991).

30. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Epilogue, the search for a quasiparticle peak and its demise, photoemission, transport. Confirmation of the DMFT predictions  ARPES measurements on NiS2-xSex Matsuura et. Al Phys. Rev B 58 (1998) 3690. Doniaach and Watanabe Phys. Rev. B 57, 3829 (1998)  S.-K. Mo et al., Phys Rev. Lett. 90, 186403 (2003).  Limelette et. al. [Science] G. Kotliar [Science].

31. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Case study Cerium. Study the alpha to gamma transition. Test the approach, in a well studied setting. Differentiate between the Kondo volume collapse picture and the Mott transition picture.

32. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Overview  Various phases : isostructural phase transition (T=298K, P=0.7GPa)  (fcc) phase [ magnetic moment (Curie-Wiess law) ]   (fcc) phase [ loss of magnetic moment (Pauli-para) ] with large volume collapse  v/v  15  (  -phase a  5.16 Å  -phase a  4.8 Å) volumesexp.LDALDA+U  28Å 3 24.7Å 3  34.4Å 3 35.2Å 3   -phase (localized) : High T phase  Curie-Weiss law (localized magnetic moment),  Large lattice constant  Tk around 60-80K   -phase (localized) : High T phase  Curie-Weiss law (localized magnetic moment),  Large lattice constant  Tk around 60-80K   -phase (delocalized:Kondo-physics) : Low T phase  Loss of Magnetism (Fermi liquid Pauli susceptibility) - completely screened magnetic moment  smaller lattice constant  Tk around 1000-2000K   -phase (delocalized:Kondo-physics) : Low T phase  Loss of Magnetism (Fermi liquid Pauli susceptibility) - completely screened magnetic moment  smaller lattice constant  Tk around 1000-2000K

33. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Qualitative Ideas. Johanssen, Mott transition of the f electrons as a function of pressure. Ce alpha gamma transition. spd electrons are spectators. Mathematical implementation, “metallic phase” treat spdf electrons by LDA, “insulating phase” put f electron in the core. Allen and Martin. Kondo volume collapse picture. The dominant effect is the spd-f hybridization.

34. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Qualitative Ideas “screened moment alpha phase” Kondo effect between spd and f takes place. “unscreend moment gamma phase” no Kondo effect (low Kondo temperature). Mathematical implementation, Anderson impurity model in the Kondo limit suplemented with elastic terms. (precursor of DMFT ideas, but without self consistency condition).

35. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Photoemission&experiment A. Mc Mahan K Held and R. Scalettar (2002) K. Haule V. Udovenko and GK. (2003)

36. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS X.Zhang M. Rozenberg G. Kotliar (PRL 1993) A. Georges, G. Kotliar (1992) Phys. Rev. B 45, 6497 Unfortunately photoemission cannot decide between the Kondo collapse picture and the Mott transition picture. Evolution of the spectra as a function of U, half filling full frustration, Hubbard model!!!!

37. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Resolution: Turn to Optics! Qualitative idea. The spd electrons have much larger velocities, so optics will be much more senstive to their behavior. See if they are simple spectators (Mott transition picture ) or wether a Kondo binding unbinding takes pace (Kondo collapse picture).

38. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

39. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS LDA and LDA+DMFT studies.K.Haule et. al.

40. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Optical Conductivity Temperature dependence.

41. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Origin of the features.

42. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Conclusion The anomalous temperature dependence and the formation of a pseudogap, suggests that the Kondo collapse picture is closer to the truth for Cerium.  Possible experimental verification in Ce(ThLa) alloys. Qualitative agreement with experiments, quantitative discrepancies. (see however J.Y. Rhee, X. Wang, B.N. Harmon, and D.W. Lynch, Phys. Rev. B 51, 17390 (1995) ). 

43. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Conclusion Dynamical mean field theory, a first principles approach to the computation of physical properties of correlated materials. Tool under construction! Many improvements are possible. Already giving interesting results. Violations of the restricted sum rule near the temperature driven Mott transition of the order or 5 -10 %. Prediction of DMFT. Verified in experiments.

44. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Conclusion Complementary tool to photoemission/inverse photoemission. Experimental advantages. Ex. V2O3, Cerium. Future work, investigate vertex corrections. Future work Where does the spectral weight go ? Future work, study more materials.

45. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

46. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS La 1-x Sr x O 3 Adding holes to a Mott insulator in three dimensions. For very small doping,(x<.07) interesting spin and orbital order takes place, non universal physics and lattice distortions are important. Small energy scales, larger dopings more robust universal behavior. Magnetic frustration. Good system to applyDMFT.

47. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Optical Conductivity

48. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Optical conductivity

49. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Realistic Computation of Optical Properties : La 1-x Sr x TiO 3

50. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Conclusion Reasonable agreement, between theory and experiments at both low and high energy. The dependence of Neff on doping is due to the changes in the effective mass.

51. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS (Tokura et. Al. 1993)A doped Mott insulator:La x Sr 1-x O 3

52. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS DMFT calculation U near the Mott transition, Rozenberg et.al 94

53. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Hall Coefficient, electron like.

54. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS La 1-x Sr x TiO 3 photoemission

55. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Evolution of spectra with doping U=4

56. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Haule et. al.

57. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS