The Mott Transition and the Challenge of Strongly Correlated Electron Systems.

1. The Mott Transition and the Challenge of Strongly Correlated Electron Systems. G. Kotliar Physics Department and Center for Materials Theory Rutgers PIPT Showcase Conference UBC Vancouver May 12th 2005

2. Outline • Correlated Electron Materials. • Dynamical Mean Field Theory. • The Mott transition problem: qualitative insights from DMFT. • Towards first principles calculations of the electronic structure of correlated materials. Pu Am and the Mott transition across the actinide series.

3. The Standard Model of Solids • Itinerant limit. Band Theory. Wave picture of the electron in momentum space. . Pauli susceptibility. • Localized model. Real space picture of electrons bound to atoms. Curie susceptibility at high temperatures, spin-orbital ordering at low temperatures.

4. Correlated Electron Materials • Are not well described by either the itinerant or the localized framework . • Compounds with partially filled f and d shells. Need new starting point for their description. Non perturbative problem. New reference frame for computing their physical properties. • Have consistently produce spectacular “big” effects thru the years. High temperature superconductivity, colossal magneto-resistance, huge volume collapses……………..

5. Large Metallic Resistivities

6. Transfer of optical spectral weight non local in frequency Schlesinger et. al. (1994), Vander Marel (2005) Takagi (2003 ) Neff depends on T

7. Breakdown of the standard model of solids. • Large metallic resistivities exceeding the Mott limit. Maximum metallic resistivity 200 mohm cm • Breakdown of the rigid band picture. Anomalous transfer of spectral weight in photoemission and optics. • The quantitative tools of the standard model fail.

8. MODEL HAMILTONIAN AND OBSERVABLES Parameters: U/t , T, carrier concentration, frustration : Local Spectral Function Limiting case itinerant electrons Limiting case localized electrons Hubbard bands

9. Limit of large lattice coordination Metzner Vollhardt, 89 Muller-Hartmann 89

10. Mean-Field Classical vs Quantum Classical case Quantum case A. Georges, G. Kotliar Phys. Rev. B 45, 6497(1992) Review: G. Kotliar and D. Vollhardt Physics Today 57,(2004)

11. Realistic Descriptions of Materials and a First Principles Approach to Strongly Correlated Electron Systems. • Incorporate realistic band structure and orbital degeneracy. • Incorporate the coupling of the lattice degrees of freedom to the electronic degrees of freedom. • Predict properties of matter without empirical information.

12. LDA+DMFT V. Anisimov, A. Poteryaev, M. Korotin, A. Anokhin and G. Kotliar, J. Phys. Cond. Mat. 35, 7359 (1997). • The light, sp (or spd) electrons are extended, well described by LDA .The heavy, d (or f) electrons are localized treat by DMFT. Use Khon Sham Hamiltonian after substracting the average energy already contained in LDA. • Add to the substracted Kohn Sham Hamiltonian a frequency dependent self energy, treat with DMFT. In this method U is either a parameter or is estimated from constrained LDA • Describes the excitation spectra of many strongly correalted solids. .

13. Spectral Density Functional • Determine the self energy , the density and the structure of the solid self consistently. By extremizing a functional of these quantities. (Chitra, Kotliar, PRB 2001, Savrasov, Kotliar, PRB 2005). Coupling of electronic degrees of freedom to structural degrees of freedom. Full implementation for Pu. Savrasov and Kotliar Nature 2001. • Under development. Functional of G and W, self consistent determination of the Coulomb interaction and the Greens functions.

14. Mott transition in V2O3 under pressure or chemical substitution on V-site. How does the electron go from localized to itinerant.

15. The Mott transition and Universality Same behavior at high tempeartures, completely different at low T

16. COHERENCE INCOHERENCE CROSSOVER T/W Phase diagram of a Hubbard model with partial frustration at integer filling.  M. Rozenberg et.al., Phys. Rev. Lett. 75, 105-108 (1995). .

17. V2O3:Anomalous transfer of spectral weight Th. Pruschke and D. L. Cox and M. Jarrell, Europhysics Lett. , 21 (1993), 593 M. Rozenberg G. Kotliar H. Kajueter G Tahomas D. Rapkikne J Honig and P Metcalf Phys. Rev. Lett. 75, 105 (1995)

18. Anomalous transfer of optical spectral weight, NiSeS. [Miyasaka and Takagi 2000]

19. Anomalous Resistivity and Mott transition Ni Se2-x Sx Crossover from Fermi liquid to bad metal to semiconductor to paramagnetic insulator.

20. Single-site DMFT and expts

21. Conclusions. • Three peak structure, quasiparticles and Hubbard bands. • Non local transfer of spectral weight. • Large metallic resistivities. • The Mott transition is driven by transfer of spectral weight from low to high energy as we approach the localized phase. • Coherent and incoherence crossover. Real and momentum space. • Theory and experiments begin to agree on a broad picture.

22. Mott Transition in the Actinide Series

23. Pu phases: A. Lawson Los Alamos Science 26, (2000) LDA underestimates the volume of fcc Pu by 30%. Within LDA fcc Pu has a negative shear modulus. LSDA predicts d Pu to be magnetic with a 5 ub moment. Experimentally it is not. Treating f electrons as core overestimates the volume by 30 %

24. Total Energy as a function of volume for PU (Savrasov, Kotliar, Abrahams, Nature ( 2001) Non magnetic correlated state of fcc Pu.

25. Double well structure and d Pu Qualitative explanation of negative thermal expansion[ G. Kotliar J.Low Temp. Physvol.126, 1009 27. (2002)]See also A . Lawson et.al.Phil. Mag. B 82, 1837 ] Natural consequence of the conclusions on the model Hamiltonian level. We had two solutions at the same U, one metallic and one insulating. Relaxing the volume expands the insulator and contract the metal.

26. Phonon Spectra • Electrons are the glue that hold the atoms together. Vibration spectra (phonons) probe the electronic structure. • Phonon spectra reveals instablities, via soft modes. • Phonon spectrum of Pu had not been measured.

27. Phonon freq (THz) vs q in delta Pu X. Dai et. al. Science vol 300, 953, 2003

28. Inelastic X Ray. Phonon energy 10 mev, photon energy 10 Kev. E = Ei - Ef Q =ki - kf

29. 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)

30. J. Tobin et. al. PHYSICAL REVIEW B 68, 155109 ,2003

31. First Principles DMFT Studies of Pu • Pu strongly correlated element, at the brink of a Mott instability. • Realistic implementations of DMFT : total energy, photoemission spectra and phonon dispersions of delta Pu. • Clues to understanding other Pu anomalies. Qualitative Insights and quantitative studies. Double well. Alpha and Delta Pu.

32. Approach the Mott point from the right Am under pressure Experimental Equation of State (after Heathman et.al, PRL 2000) Mott Transition? “Soft” “Hard” Density functional based electronic structure calculations: • Non magnetic LDA/GGA predicts volume 50% off. • Magnetic GGA corrects most of error in volume but gives m~6mB (Soderlind et.al., PRB 2000). • Experimentally, Am hasnon magnetic f6ground state with J=0(7F0)

33. Mott transition in open (right) and closed (left) shell systems. Realization in Am ?? S S g T Tc Log[2J+1] ??? Uc J=0 U U g ~1/(Uc-U)

35. Cluster Extensions of Single Site DMFT

36. Conclusions Future Directions • DMFT: Method under development, but it already gives new insights into materials……. • Exciting development: cluster extensions. Allows us to see to check the accuracy of the single site DMFT corrections, and obtain new physics at lower temperatures and closer to the Mott transition where the single site DMFT breaks down. • Captures new physics beyond single site DMFT , i.e. d wave superconductivity, and other novel aspects of the Mott transition in two dimensional systems. • Allow us to focus on deviations of experiments from DMFT. • DMFT and RG developments

37. Some References • Reviews: A. Georges G. Kotliar W. Krauth and M. Rozenberg RMP68 , 13, (1996). • Reviews: G. Kotliar S. Savrasov K. Haule V. Oudovenko O. Parcollet and C. Marianetti. Submitted to RMP (2005). • Gabriel Kotliar and Dieter Vollhardt Physics Today 57,(2004)

38. Am Equation of State: LDA+DMFT Predictions (Savrasov Kotliar Haule Murthy 2005) Self-consistent evaluations of total energies with LDA+DMFT . Accounting for full atomic multiplet structure using Slater integrals: F(0)=4.5 eV, F(2)=8 eV, F(4)=5.4 eV, F(6)=4 eV New algorithms allow studies of complex structures. Theoretical P(V) using LDA+DMFT Predictions for Am I LDA+DMFT predictions: • Non magnetic f6ground state with J=0(7F0) • Equilibrium Volume: Vtheory/Vexp=0.93 • Bulk Modulus: Btheory=47 GPa Experimentally B=40-45 GPa Predictions for Am II Predictions for Am III Predictions for Am IV

39. Photoemission Spectrum from 7F0 Americium LDA+DMFT Density of States Matrix Hubbard I Method F(0)=4.5 eV F(2)=8.0 eV F(4)=5.4 eV F(6)=4.0 eV Experimental Photoemission Spectrum (after J. Naegele et.al, PRL 1984)

40. J. C. Griveau et. al. (2004)

41. K. Haule , Pu- photoemission with DMFT using vertex corrected NCA.

42. Cluster Extensions of DMFT

43. Pu is not MAGNETIC, alpha and delta have comparable susceptibility and specifi heat.

44. More important, one would like to be able to evaluate from the theory itself when the approximation is reliable!! And captures new fascinating aspects of the immediate vecinity of the Mott transition in two dimensional systems…..

45. Cluster Extensions of Single Site DMFT

46. Some References • Reviews: A. Georges G. Kotliar W. Krauth and M. Rozenberg RMP68 , 13, (1996). • Reviews: G. Kotliar S. Savrasov K. Haule V. Oudovenko O. Parcollet and C. Marianetti. Submitted to RMP (2005). • Gabriel Kotliar and Dieter Vollhardt Physics Today 57,(2004)

47. 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)

48. Total Energy as a function of volume for Pu W (ev) vs (a.u. 27.2 ev) (Savrasov, Kotliar, Abrahams, Nature ( 2001) Non magnetic correlated state of fcc Pu. Zein Savrasov and Kotliar (2004)

49. DMFT : What is the dominant atomic configuration ,what is the fate of the atomic moment ? • Snapshots of the f electron :Dominant configuration:(5f)5 • Naïve view Lz=-3,-2,-1,0,1, ML=-5 mB, ,S=5/2 Ms=5 mB . Mtot=0 • More realistic calculations, (GGA+U),itineracy, crystal fields G7 +G8, ML=-3.9 Mtot=1.1. S. Y. Savrasov and G. Kotliar, Phys. Rev. Lett., 84, 3670 (2000) • This moment is quenched or screened by spd electrons, and other f electrons. (e.g. alpha Ce). • Contrast Am:(5f)6

50. Anomalous Resistivity PRL 91,061401 (2003)