Strongly Correlated Electron Systems: a DMFT Perspective

1. Strongly Correlated Electron Systems: a DMFT Perspective Gabriel Kotliar Physics Department and Center for Materials Theory Rutgers University • Brookhaven National Lab February 17th 2005

2. Introduction to Dynamical Mean Field Theory (DMFT) ideas. • Application 1 : Optical Conductivity in Cerium. Mott transition or volume collapse. [K. Haule et. al. PRL 2005] • Application 2 :The approach to the Mott transition in Kappa Organics and Cuprates. [O. Parcollet G. Biroli and GK PRL ]Work with M. Civelli M. Capone O. Parcollet V. Sarma B. Kyung A.M. Tremblay and D. Senechal]

3. 1 2 4 3 A. Georges and G. Kotliar PRB 45, 6479 (1992). G. Kotliar,S. Savrasov, G. Palsson and G. Biroli, PRL 87, 186401 (2001) .

4. Site Cell. Cellular DMFT. C-DMFT. G. Kotliar,S.. Savrasov, G. Palsson and G. Biroli, Phys. Rev. Lett. 87, 186401 (2001) tˆ(K) hopping expressed in the superlattice notations. • Other cluster extensions (DCA Jarrell Krishnamurthy, Katsnelson and Lichtenstein periodized scheme, Nested Cluster Schemes Schiller Ingersent ), causality issues, O. Parcollet, G. Biroli and GK cond-matt 0307587 (2003)

5. Dynamical Mean Field Theory • Captures the non local Gaussian physics of model and the local non Gaussian effects. • Assigns to each lattice model a quantum impurity problem, (local degrees of freedom in a Gaussian bath) which describes its local physics. • Spectral density functional. Impurity model generates the “exact spectra” of a system. Good starting point for doing perturbation theory in correlated materials. [ Analogy with Kohn Sham System].

6. Testing CDMFT (G.. Kotliar,S. Savrasov, G. Palsson and G. Biroli, Phys. Rev. Lett. 87, 186401 (2001) ) with two sites in the Hubbard model in one dimension. [V. Kancharla C. Bolech and GK PRB 67, 075110 (2003)][[M.CaponeM.Civelli V Kancharla C.Castellani and GK P. R B 69,195105 (2004) U/t=4. ]

7. M. Rozenberg et. al. Phys. Rev. Lett. 75, 105 (1995) T/W Phase diagram of a Hubbard model with partial frustration at integer filling. Thinking about the Mott transition in single site DMFT.

8. Mott transition in layered organic conductors S Lefebvre et al.

9. Single site DMFT and kappa organics

10. ARPES measurements on NiS2-xSexMatsuura et. Al Phys. Rev B 58 (1998) 3690. Doniaach and Watanabe Phys. Rev. B 57, 3829 (1998)Mo et al., Phys. Rev.Lett. 90, 186403 (2003). .

11. Two paths for calculation of electronic structure of strongly correlated materials Crystal structure +Atomic positions Model Hamiltonian Correlation Functions Total Energies etc. DMFT ideas can be used in both cases.

12. Electronic Structure and EDMFT Phys. Rev. B 62, 12715 (2000) Introduce Notion of Local Greens functions, Wloc, Gloc G=Gloc+Gnonloc . Ex. Ir>=|R, r> Gloc=G(R r, R r’) dR,R’’ Sum of 2PI graphs One can also view as an approximation to an exact Spectral Density Functional of Gloc and Wloc) Approximations……………...

13. Convergence in R space of first self energy correction for Si in a.u. (1 a.u.= 27.2 eV) N. Zein GW self energy Self energy correction beyond GW Coordination Sphere Coordination Sphere Lowest order graph in the screened coulomb interaction (GW approximation) treated self consistently reproduces the gap of silicon. [Exp : 1.17 ev, GW 1.24 ] W. Ku, A. Eguiluz, PRL 89,126401 (2002)

14. C11 (GPa) C44 (GPa) C12 (GPa) C'(GPa) Theory 34.56 33.03 26.81 3.88 Experiment 36.28 33.59 26.73 4.78 DMFT Phonons in fcc d-Pu ( Dai, Savrasov, Kotliar,Ledbetter, Migliori, Abrahams, Science, 9 May 2003) (experiments from Wong et.al, Science, 22 August 2003)

15. Case study Elemental 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.

16.  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) Overview • -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

17. Qualitative Ideas. • Johanssen, Mott transition of the f electrons as a function of pressure. Ce alpha gamma transition. spd electrons are spectators. • Mathematical implementation, “itinerant phase” treat spdf electrons by LDA, “localized phase” put f electron in the core. • Allen and Martin. Kondo volume collapse picture. The dominant effect is the spd-f hybridization.

18. 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).

19. Photoemission&experiment • A. Mc Mahan K Held and R. Scalettar (2002) • K. Haule V. Udovenko S. savrasov and GK. (2003)

20. 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!!!! X.Zhang M. Rozenberg G. Kotliar PRL 70,1666(1993).A. Georges, G. Kotliar (1992) Phys. Rev. B 45, 6497

21. 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 whether a Kondo binding unbinding takes pace (Kondo collapse picture).

23. Optical Conductivity Temperature dependence.

24. Origin of the features.

25. 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)) .

27. modeled to triangular lattice t t’ k-(ET)2X are across Mott transition ET = Insulating anion layer X-1 conducting ET layer [(ET)2]+1

28. Evolution of the spectral function at low frequency. If the k dependence of the self energy is weak, we expect to see contour lines corresponding to Ek = const and a height increasing as we approach the Fermi surface.

29. Evolution of the k resolved Spectral Function at zero frequency. (Kappa organics Parcollet Biroli and GK PRL, 92, 226402. (2004)) ) U/D=2.25 U/D=2 Uc=2.35+-.05, Tc/D=1/44

30. Mechanism for hot spot formation

31. Mott transition in cluster

32. Phenomena Resembles what is seen in cuprate superconductors. • “Fermi Arcs” M. Norman et. al. Nature Vol 392 (1998). • Theoretical approaches. M Rice Umklapp Scattering C.Honerkamp, M.Salmhofer, N.Furukawa, T.M.Rice,% • cond-mat/9912358. PRB 63, (2001) 035109. • F. H. L. Essler, A. M. Tsvelik, cond-mat/0409491.

33. High Temperature Superconductors

34. RVB phase diagram of the Cuprate Superconductors • P.W. Anderson. Connection between high Tc and Mott physics. Science 235, 1196 (1987) • Connection between the anomalous normal state of a doped Mott insulator and high Tc. • Baskaran Zhou and Anderson Slave boson approach. <b> coherence order parameter. • k, D singlet formation order parameters.

35. RVB phase diagram of the Cuprate Superconductors. Superexchange. G. Kotliar and J. Liu Phys.Rev. B 38,5412 (1988) • The approach to the Mott insulator renormalizes the kinetic energy Trvb increases. • The proximity to the Mott insulator reduce the charge stiffness , TBE goes to zero. • Superconducting dome. Pseudogap evolves continously into the superconducting state.

36. . Cellular DMFT and Dynamical RVB • Retains ideas of the slave boson mean field and make many of the results more solid but also removes many difficulties. • Can treat coherent and incoherent spectra on the same footing. • Can treat dynamical fluctuations between different singlet order parameters. • Allows the investigation of broken symmetries. Superconductivity, Antiferromagnetism, etc.

37. Superconductivity in the Hubbard model role of the Mott transition and influence of the super-exchange. (M. Capone V. Kancharla. CDMFT+ED, 4+ 8 sites t’=0) . RVB

38. Superconductivity and Antiferromagnetism t’=0 M. Capone V. Kancharla (see also CPT Senechal and Tremblay ).

39. CDMFT and Dynamical RVB . • Allows the investigation of the normal state underlying the superconducting state, by forcing a symmetric Weiss function, follow the normal state near the Mott transition. • Earlier studies (Katsnelson and Lichtenstein, M. Jarrell, M Hettler et. al. Phys. Rev. B 58, 7475 (1998). T. Maier et. al. Phys. Rev. Lett 85, 1524 (2000) ) used QMC as an impurity solver and DCA as cluster scheme. • We use exact diag ( Krauth Caffarel 1995 with effective temperature 32/t=124/D ) as a solver and Cellular DMFT as the mean field scheme.

40. Follow the “normal state” with doping. Evolution of the spectral function at low frequency. If the k dependence of the self energy is weak, we expect to see contour lines corresponding to Ek = const and a height increasing as we approach the Fermi surface.

41. Hole doped case t’=-.3t, U=16 t n=.71 .93 .97

42. K.M . Shen et. al (2004). For a review Damascelli et. al. RMP (2003)

43. Approaching the Mott transition: CDMFT Picture • Qualitative effect, momentum space differentiation. Formation of hot –cold regions is an unavoidable consequence of the approach to the Mott insulating state! • D wave gapping of the single particle spectra as the Mott transition is approached. • Similar scenario was encountered in previous study of the kappa organics. O Parcollet Biroli and Kotliar PRL, 92, 226402. (2004) .

44. Electron Doped Case t’=.3 t, U=16 tn=.7 ,.93, .96

45. Approaching the Mott transition: CDMFT picture. • Qualitative effect, momentum space differentiation. Formation of hot –cold regions is an unavoidable consequence of the approach to the Mott insulating state! • General phenomena, BUT the location of the cold regions depends on parameters. • Quasiparticles are now generated from the Mott insulator at (p, 0). • Consistenty with many numerical studies of electron hole asymmetry in t-t’ Hubbard models Tohyama Maewawa Phys. Rev. B 67, 092509 (2003) Senechal and Tremblay. PRL 92 126401 (2004) Kusko et. al. Phys. Rev 66, 140513 (2002)

46. Consistent with experiments. Armitage et. al. PRL (2001).Momentum dependence of the low-energy Photoemission spectra of NCCO

47. To test if the formation of the hot and cold regions is the result of the proximity to Antiferromagnetism, we studied various values of t’/t, U=16.

48. Electron doped case t’=.9t U=16tn=.69 .92 .96

49. Approaching the Mott transition: • Qualitative effect, momentum space differentiation. Formation of hot –cold regions is an unavoidable consequence of the approach to the Mott insulating state! • General phenomena, but the location of the cold regions depends on parameters. • With the present resolution, t’ =.9 and .3 are similar. However it is perfectly possible that at lower energies further refinements and differentiation will result from the proximity to different ordered states.