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 • Northwestern University January 2005

2. Outline • Introduction to strongly correlated electrons. • Introduction to Dynamical Mean Field Theory (DMFT) • The Mott transition problem. Theory and experiments. • Current Developments and future directions.

3. Electrons in a Solid:the Standard Model Electrons as a waves. Band Theory. Landau Fermi Liquid Theory. Maximum metallic resistivity 200 mohm cm • Quantitative Tools. Density Functional Theory+Perturbation Theory. Rigid bands , optical transitions , thermodynamics, transport………

4. LDA+GW: semiconducting gaps

5. 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)

6. Correlated Materials do big things • Huge resistivity changes V2O3. • Copper Oxides. .(La2-x Bax) CuO4 High Temperature Superconductivity.150 K in the Ca2Ba2Cu3HgO8 . • Uranium and Cerium Based Compounds. Heavy Fermion Systems,CeCu6,m*/m=1000 • (La1-xSrx)MnO3 Colossal Magneto-resistance.

7. Strongly Correlated Materials. • Large thermoelectric response in NaCo2-xCuxO4 • Huge volume collapses, Ce, Pu…… • Large and ultrafast optical nonlinearities Sr2CuO3 • …...................

8. Breakdown of standard model • Large metallic resistivities exceeding the Mott limit. • Breakdown of the rigid band picture. • Anomalous transfer of spectral weight in photoemission and optics. • The quantitative tools of the standard model fail.

9. Strong Correlation Anomalies cannot be understood within the standard model of solids, based on a RIGID BAND PICTURE,e.g.“Metallic “resistivities that rise without sign of saturation beyond the Mott limit, temperature dependence of the integrated optical weight up to high frequency C. Urano et. al. PRL 85, 1052 (2000)

10. Breakdown of standard model • Large metallic resistivities exceeding the Mott limit. • Breakdown of the rigid band picture. • Anomalous transfer of spectral weight in photoemission and optics. • LDA+GW loses its predictive power. • Need new starting point to do perturbation theory. • Has to treat real and momentum space on equal footing.

11. Localization vs Delocalization Strong Correlation Problem • Many interesting compounds do not fit within the “Sandard Model”. • Tend to have elements with partially filled d and f shells. • Need to incorporate a real space perspective, atomic physics (Mott). • Interesting compounds are not close to either the localized or itinerant limit. Excitations are different in both limits. Non perturbative problem. • DMFT, Simplest approach to electronic structure, which interpolates correctly between atoms and bands.

12. Hubbard model • U/t • Doping d or chemical potential • Frustration (t’/t) • T temperature Mott transition as a function of doping, pressure temperature etc.

13. 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) .

14. Pressure Driven Mott transition How does the electron go from the localized to the itinerant limit ?

15. One Particle Local Spectral Function and Angle Integrated Photoemission e • Probability of removing an electron and transfering energy w=Ei-Ef, f(w) A(w) M2 • Probability of absorbing an electron and transfering energy w=Ei-Ef, (1-f(w)) A(w) M2 • Theory. Compute one particle greens function and use spectral function. n n e A(w) is different in the localized and itinerant limit.

16. M. Rozenberg et. alf 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.

17. 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)

18. V2O3:Anomalous transfer of spectral weight bM 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)

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 kappa organics

21. Ising critical endpoint! In V2O3 P. Limelette et.al. Science 302, 89 (2003)

22. 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). .

23. 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 agree.

24. 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.

25. DMFT:Realistic Implementations • Focus on the “local “ spectral function A(w) (and of the local screened Coulomb interaction W(w) ) of the solid. • Write a functional of the local spectral function such that its stationary point, give the energy of the solid. • No explicit expression for the exact functional exists, but good approximations are available. LDA+DMFT. • The spectral function is computed by solving a local impurity model in a medium .Which is a new reference system to think about correlated electrons.

26. Pu in the periodic table actinides

27. Pu phases: A. Lawson Los Alamos Science 26, (2000) LDA underestimates the volume of fcc Pu by 30% Predicts magnetism in d Pu and gives negative shear Core-like f electrons overestimates the volume by 30 %

28. Small amounts of Ga stabilize the d phase (A. Lawson LANL)

29. 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)

30. Alpha and delta Pu

31. 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.

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

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

34. 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)

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

36. 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.

38. CDMFT vs single site DMFT

39. Superconductivity and Antiferromagnetism t’=0 M. Capone et. al. Explore connection with the Mott transition.

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. Break up of the Fermi surface as the Mott transition is approached • The quasiparticles in some regions of k space retain their coherence. • The quasiparticles in other regions of k space are strongly scattered and then they are gapped. • This is captured by the different behavior of the components of the cluster self energy.

45. 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.

46. Electron doped case t’=.9t U=16tn=.69 .92 .96 Breakup of the Fermi surface is a consequence of the approach to the Mott transition point, and not a result of a specifid form of long range order.

47. Goal of a good mean field theory • Provide a zeroth order picture of a physical phenomena. • Provide a link between a simple system (“mean field reference frame”) and the physical system of interest. • Formulate the problem in terms of local quantities (which we can compute better ). • Allows to perform quantitative studies, and predictions . Focus on the discrepancies between experiments and mean field predictions. • Generate useful language and concepts. Follow mean field states as a function of parameters. • Exact in some limit. • Can be made system specific, useful tool for material exploration.

48. Conclusion • DMFT: quantum impurity models in a self consistent medium, new reference system to understand the behavior of strongly correlated materials. • Realistic and cluster extensions of DMFT, increased of our understanding of correlated materials. Coupled to growing computer power will lead to greater predictive capabilities in the future. • Rational theoretically guided material design using the exotic properties of correlated materials ?

49. Collaborators 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)