Strongly Correlated Electron Systems a Dynamical Mean Field Perspective

1. Strongly Correlated Electron Systems a Dynamical Mean Field Perspective G. Kotliar Physics Department and Center for Materials Theory Rutgers ICAM meeting: Frontiers in Correlated Matter Snowmass September 2004

2. Strongly Correlated Electron Systems Display remarkable phenomena, that cannot be understood within the standard model of solids. Resistivities that rise without sign of saturation beyond the Mott limit, (e.g. H. Takagi’s work on Vanadates), temperature dependence of the integrated optical weight up to high frequency (e.g. Vandermarel’s work on Silicides). THE WHY Correlated electrons do “big things”, large volume collapses, colossal magnetoresitance, high temperature superconductivity . Properties are very sensitive to structure chemistry and stoichiometry, and control parameters large non linear susceptibilites,etc……….

3. Need non perturbative tool. THE HOW How to think about their electronic states ? How to compute their properties ? Mapping onto connecting their properties, a simpler “reference system”. A self consistent impurity model living on SITES, LINKS and PLAQUETTES...... DYNAMICAL MEAN FIELD THEORY. "Optimal Gaussian Medium " + " Local Quantum Degrees of Freedom " + "their interaction " is a good reference frame for understanding, and predicting physical properties of correlated materials. Focus on local quantities, construct functionals of those quantities, similarities with DFT.

4. What did we learn ? Schematic DMFT phase diagram and DOS of a partially frustrated integer filled Hubbard model and pressure driven Mott transition.

5. Pressure driven Mott transition.

6. How do we know there is some truth in this picture ? Qualitative Predictions Verified • Two different features in spectra. Quasiparticles bands and Hubbard bands. • Transfer of spectral weight which is non local in frequency. Optics and Photoemission. • Two crossovers, associated with gap closure and loss of coherence. Transport. • Mott transition endpoint, is Ising like, couples to all electronic properties. • An “exact numerical approach PRG “ recently found the first order line(M. Imada), C-DMFT offers a consistency check.

8. Ising critical endpoint found! In V2O3 P. Limelette et.al. (Science 2003)

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

10. Why does it work: Energy Landscape of a Correlated Material and a top to bottom approach to correlated materials. Single site DMFT. High temperature universality vs low temperature sensitivity to detail for materials near a temperature-pressure driven Mott transition Energy T Configurational Coordinate in the space of Hamiltonians

11. What did we gain? • Conceptual understanding of how the electronic structure evolves when the electron goes from localized to itinerant. • Uc1 Uc2, transfer of spectral weight, …. • A general methodology which was extended to clusters (non trivial!) and integrated into an electronic structure method, which allows us to incorporate structure and chemistry. Both are needed away from the high temperature universal region.

12. Mott transition across the 5f’s, a very interesting playground for studying correlated electron phenomena. • DMFT ideas have been extended into a framework capable of making first principles first principles studies of correlated materials. Pu Phonons. Combining theory and experiments to separate the contributions of different energy scales, and length scales to the bonding • In single site DMFT , superconductivity is an unavoidable consequence when we try to go move from a metallic state to a Mott insulator where the atoms have a closed shell (no entropy). Realization in Am under pressure ?

13. 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: connect bonding to energy and length scales. ( Dai, Savrasov, Kotliar,Ledbetter, Migliori, Abrahams, Science, 9 May 2003) (experiments from Wong et.al, Science, 22 August 2003)

14. Big question: will we be nearly as successful in our attemps to understand and predict (some ) physical properties of correlated materials, with DMFT, as we have been for weakly correlated materials using ( approximate DFT and perturbation theory in screened Coulomb interactions eg.GW )?

15. A rapidly convergent algorithm ? One dimensional Hubbard model 2 site (LINK) CDMFT compare with Bethe Anzats, [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.

16. Links, Ti2O3 : Coulomb and Pauling • LTS 250 K, HTS 750 K. C.E.Rice et all, Acta CrystB33, 1342 (1977)

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

18. U/t=16,t’= +0.9 U/t=8, t’= -0.3 Density= 0.88, 0.89, 0.9, 0.91, 0.922, 0.96, 0.986, 0.988, 0.989, 0.991, 0.993 Underlying normal state of the Hubbard model near the Mott transition, (force the Weiss field to its paramagnetic value), T=0 ED solution of the C-DMFT equations. M. Civelli, M. Capone, O. Parcollet and GK

19. Approaching the Mott transition: plaquette Cdmft. • 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. • Study the “normal state” of the Hubbard model. General phenomena, but the location of the cold regions depends on parameters. [Civelli Capone Parcollet and Kotliar ]

20. Where do we go now ? • One can study a large number of experimentally relevant problems within the single site framework. • Continue the methodological development, we need tools! • Solve the CDMFT Mott transition problem on the plaquette problem, hard, but it is a significant improvement, the early mean field theories while keeping its physical appeal. • Study material trends, make contact with phenomenological approaches, doped semiconductors (Bhatt and Sachdev), heavy fermions , 115’s(Nakatsuji, Pines and Fisk )……

21. Mott transition into an open (right) and closed (left) shell systems. In single site DMFT, superconductivity must intervene before reaching the Mott insulating state.[Capone et. al. ] AmAt room pressure a localised 5f6 system;j=5/2. S = -L = 3: J = 0 apply pressure ? S S .g T Log[2J+1] ??? Uc S=0 U U g ~1/(Uc-U)

22. Americium under pressure [J.C. Griveaux J. Rebizant G. Lander]

24. 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. Answer: cautiously optimistic yes, but it needs a lot of work. • Focus on short distance intermediate energy scale properties. [Method is designed for that] • Need analytic +numerical work. Connection with other approaches/DMRG • Need adaptive k space. • One can already do a lot with single site DMFT in many many many materials. • Plaquette equations are one order of magnitude harder to solve.

26. 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)

27. 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)

28. Epsilon Plutonium.

29. Phonon entropy drives the epsilon delta phase transition • Epsilon is slightly more delocalized than delta, has SMALLER volume and lies at HIGHER energy than delta at T=0. But it has a much larger phonon entropy than delta. • At the phase transition the volume shrinks but the phonon entropy increases. • Estimates of the phase transition following Drumont and G. Ackland et. al. PRB.65, 184104 (2002); (and neglecting electronic entropy). TC ~ 600 K.

30. Transverse Phonon along (0,1,1) in epsilon Pu in self consistent Born approximation.

31. Mott transition into an open (right) and closed (left) shell systems. In single site DMFT, superconductivity must intervene before reaching the Mott insulating state.[Capone et. al. ] AmAt room pressure a localised 5f6 system;j=5/2. S = -L = 3: J = 0 apply pressure ? S S .g T Log[2J+1] ??? Uc S=0 U U g ~1/(Uc-U)

32. Americium under pressure [J.C. Griveaux J. Rebizant G. Lander]

33. Overview of rho (p, T) of Am • Note strongly increasing resistivity as f(p) at all T. Shows that more electrons are entering the conduction band • Superconducting at all pressure • IVariation of rho vs. T for increasing p.

34. DMFT study in the fcc structure. S. Murthy and G. Kotliar fcc

35. LDA+DMFT spectra. Notice the rapid occupation of the f7/2 band.

36. One electron spectra. Experiments (Negele) and LDA+DFT theory (S. Murthy and GK )

37. Conclusion Am • Crude LDA+DMFT calculations describe the crude energetics of the material, eq. volume, even p vs V . • Superconductivity near the Mott transition. Tc increases first and the decreases as we approach the Mott boundary. Dramatic effect in the f bulk module. • What is going on at the Am I- Am II boundary ??? Subtle effect (bulk moduli do not change much ), but crucial modifications at low energy. • Mott transition of the f7/2 band ? Quantum critical point ?:

38. H.Q. Yuan et. al. CeCu2(Si2-x Gex). Am under pressure Griveau et. al.

39. Electronic states in weakly and strongly correlated materials • Simple metals, semiconductors. Fermi Liquid Description: Quasiparticles and quasiholes, (and their bound states ). Computational tool: Density functional theory + perturbation theory in W, GW method. • Correlated electrons. Atomic states. Hubbard bands. Narrow bands. Many anomalies. • Need tool that treats Hubbard bands, and quasiparticle bands, real and momentum space on the same footing. DMFT!

40. Weakly correlated electrons. FLT and DFT, and what goes wrong in correlated materials. • Fermi Liquid . . Correspondence between a system of non interacting particles and the full Hamiltonian. • A band structure is generated (Kohn Sham system).and in many systems this is a good starting point for perturbative computations of the spectra (GW).

41. DMFT Cavity Construction: A. Georges and G. Kotliar PRB 45, 6479 (1992). Figure from : G. Kotliar and D. Vollhardt Physics Today 57,(2004)http://www.physics.rutgers.edu/~kotliar/RI_gen.html The self consistent impurity model is a new reference system, to describe strongly correlated materials.

42. Dynamical Mean Field Theory (DMFT) Cavity Construction: A. Georges and G. Kotliar PRB 45, 6479 (1992).

43. 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)

44. Two paths for ab-initio 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.

45. LDA+DMFT V. Anisimov, A. Poteryaev, M. Korotin, A. Anokhin and G. Kotliar, J. Phys. Cond. Mat. 35, 7359 (1997). A Lichtenstein and M. Katsnelson PRB 57, 6884 (1988). • The light, SP (or SPD) electrons are extended, well described by LDA .The heavy, D (or F) electrons are localized treat by DMFT. • LDA Kohn Sham Hamiltonian already contains an average interaction of the heavy electrons, subtract this out by shifting the heavy level (double counting term) • Kinetic energy is provided by the Kohn Sham Hamiltonian (sometimes after downfolding ). The U matrix can be estimated from first principles of viewed as parameters. Solve resulting model using DMFT.

46. Functional formulation. Chitra and Kotliar (2001), Savrasov and Kotliarcond- matt0308053 (2003). Ir>=|R, r> Double loop in Gloc and Wloc

47. Impurity model representability of spectral density functional.

48. RVB phase diagram of the Cuprate Superconductors • P.W. Anderson. Baskaran Zou and Anderson. Connection between high Tc and Mott physics. • <b> coherence order parameter. • K, D singlet formation order paramters. G. Kotliar and J. Liu Phys.Rev. B 38,5412 (1988)

49. High temperature superconductivity is an unavoidable consequence of the need to connect with Mott insulator that does not break any symmetries to a metallic state. • Tc decreases as the quasiparticle residue goes to zero at half filling and as the Fermi liquid theory is approached. • Early on, accounted for the most salient features of the phase diagram. [d-wave superconductivity, anomalous metallic state, pseudo-gap state ]

50. Problems with the approach. • Numerous other competing states. Dimer phase, box phase , staggered flux phase , Neel order, • Stability of the pseudogap state at finite temperature. • Missing finite temperature . [ fluctuations of slave bosons , ] • Temperature dependence of the penetration depth [Wen and Lee , Ioffe and Millis ] Theory: • r[T]=x-Ta x2 , Exp: r[T]= x-T a. • Theory has uniform Z on the Fermi surface, in contradiction with ARPES.