Dynamical Mean Field Theory (DMFT) Approach to Correlated Solids

1. Dynamical Mean Field Theory (DMFT) Approach to Correlated Solids Gabriel Kotliar and Center for Materials Theory 1 $upport : NSF -DMR DOE-Basic Energy Sciences KaohsiungDecember 18th 2009

2. Outline • Weakly and Strongly Correlated Solids. • Dynamical Mean Field Theory. • Examples from across the periodic table, 3d’s , 4f’s , 5f’s . Theoretical/computational spectroscopy of correlated materials in the making ? • Conclusion and outlook Collaborators : K. Haule (Rutgers) A. Kutepov (Rutgers) S. Savrasov (Davis ) Ref: Electronic structure calculations with dynamical mean-field theory: G. Kotliar, S. Savrasov, K. Haule, V. Oudovenko, O. Parcollet, and C. Marianetti, Rev. Mod. Phys. 78, 000865 (2006)

3. Weakly Correlated Electrons in Solids Band Theory: electrons as waves. Landau Fermi Liquid Theory. n band index, e.g. s, p, d,,f Rigid bands , optical transitions , thermodynamics, transport……… • Quantitative Tools. Density Functional Theory • Kohn Sham (1964) Static Mean Field Theory. Powerful implementations in advanced basis sets!! 2

4. Kohn Sham Eigenvalues and Eigenstates: Excellent starting point for perturbation theory in the screened interactions GW approximation : 1st order PT in screened Coulomb interactions L. Hedin, Phys. Rev. 139, A796 (1965)M. S. Hybertsen and S. G. Louie PRL. 55, 1418 (1985) -Vxc W screended Coulomb interaction G Greens function T. Kotani, M. van Schilfgaarde, and S. V. Faleev, PRB 76, 165106 2007. Succesful description the excitation spectra of a large number of simple metals semiconductors and insulators. Optics requires more graphs [BS-VC] 4

5. Total Energies Fully Self Consistent GW. Electron gas GWHolm and U. von Barth, PRB57, 2108 (1998). • Total energy of HEG vs Rs : HF(+) , GW (------) , and QMC x • QMC D. M. Ceperley and B. J. Alder, Phys. Rev. Lett. 45, 566 1980. • (fully self consistent GW in an LAPW basis set ) A. Kutepov S. Savrasov and Kotliar PRB B 80, 041103R (2009) Brillouin-zone meshes 8x8x8 Na and Al and 555 Si. 4

6. Strong vs Weak Correlation • Lowest order perturbation theory around a non interacting theory fails when the spectra has multiple peak structures !! Correlated electron materials display new features beyond the single particle description: • ATOMIC MULTIPLET STRUCTURE, LARGE MASS RENORMALIZATIONS, LARGE INCOHERENT SPECTRAL WEIGHTS, KONDO PHYSICS, SPIN-ELECTRON BOUND STATES, BEYOND QUASIPARTICLES!! , STRONGLY TEMPERATURE DEPENDENT ELECTRONIC STRUCTURE, ……………. 5

7. Strongly correlated materials do “big” things Potential applications • Huge volume collapses lanthanides actinides, eg. Ce , Pu, ……. • Metal insulator transitions as a function of pressure in vanadium oxides, VO2 V2O3………………. • Quasiparticles with large masses m* =1000 mel in Ce and U based heavy fermions. • Colossal Magneto-resistance in La1-xSrxMnO3 • High Temperature Superconductivity. 150 K Ca2Ba2Cu3HgO8 . • Large thermoelectric response in NaxCo2O4 • 50K superconductivity in SmO1-xFxFeAs • Many many others…… Discovered by accident!! 6

8. Dynamical Mean Field Theory DMFT. Hubbard (toy)model A(w) A. Georges and G. Kotliar PRB 45, 6479 (1992). 7

9. + Atomic levels Impurity Solver Exact Diagonalization DeterminantalMontecarlo Diagrammatic Continuous Time Montecarlo DMRG, Semi-nalytic Methods, Perturbative Methods, NRG, ………. SOLVING IMPURITIES IS EASIER THAN THE FULL LATTICE PROBLEM!! Quantifying the degree of localization/delocalization Reviews : A.Georges, G. Kotliar, W. Krauth and M. J. Rozenberg, R.M.P. 68, 13 (1996). D. Vollhardt and G. Kotliar Physics Today 57, 53, (2004) 8

10. DMFT Can study different broken symmetries. Different mean field states (phases) Compare free energies. • Reference frame can be cluster of sites Many cluster schemes. DCA (Dynamical Cluster Approximation) M.H. Hettler, A.N. Tahvildar-Zadeh, M. Jarrell, T. Pruschke, and H.R. Krishnamurthy, Phys. Rev. B 58 7475, CDMFT (Cellular cluster mean field theory ) Kotliar Savrasov Palsson and Biroli PRL 87, 186401 (2001) 9

11. Making it Realistic : LDA+DMFT. V. Anisimov, A. Poteryaev, M. Korotin, A. Anokhin and G. Kotliar, J. Phys. Cond. Mat. 35, 7359 (1997). Spectra=- Im G(k,w) Determine energy and and S self consistently from extremizing a functional : the spectral density functional . Savrasov and Kotliar PRB 69, 245101, (2001) 10

12. LDA+DMFT Self-Consistency loop Edc U DMFT 11 REVIEWS : G. Kotliar et.al. RMP 78, 865 (2006)K. Held Advances in Physics 56, 829 (2007) .

13. Main DMFT Concepts Local Self Energies -Correlated Bands Valence Histograms. Describes the history of the “atom” in the solid, multiplets! Weiss Weiss field, collective hybridizationfunction, quantifies the degree of localization Functionals of density and spectra : spectral density functionals 12

14. 1d : Far away from the Mezner Vollhardt limit of infinite d CDMFT-ED 13

15. 2d :arXiv:0907.0863Diagrammatic Monte Carlo for the Hubbard Model :E. Kozik, K. Van Houcke, E. Gull, L. Pollet, N. Prokof'ev, B. Svistunov, M. Troyer 14

16. 3d LDA+DMFT Tests in Cold Atom Traps At larger U’s the accuracy of DMFT should increase but the Hubbard model is no longer a good model of the trap, because more bands come into play. 15

17. Crystal structure and resistivity of 115’s CeMIn5 M=Co, Ir, Rh Ir In Ce In Ce In Ir (5d) Rh(4d) Co (3d) Tetragonal crystal structure IrIn2 layer 3.27au 4 in plane In neighbors 3.3 au CeIn3 layer IrIn2 layer 8 out of plane in neighbors Onset of coherence at extremely low temperatures 16

18. Optical conductivity in LDA+DMFT OCA Experiments: Basov’s group VanderMarel group • J.H. Shim, Khaule , and G. Kotliar, Science 318, 1618 (2007). 17

19. 10K In eV Ce In Multiple hybridization gaps non-f spectra 300K • Larger gap due to hybridization with out of plane In • Smaller gap due to hybridization with in-plane In 18

20. Ce 115’s • Protracted Screening. Fermi Liquid Regime is not reached until very low temperatures. • . • Crossover is slower than in single impurity because of the • self consistency condition feedback. • Structure Property Relation. Out of plane In site controls • hybridization. Confirmed by NMR. (N. Curro) • Non Monotonic behavior in the periodic table. • Predictions for ARPES! • Accounts for Co 3d –Rh 4d -Ir 5d (Haule Yee and Kim 2009) 19

21. Localization Delocalization in Pu Mott Transition d Pu a a Modern understanding : LDA+DMFT. 20

22. 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) 21

23. Gouder , Havela PRB 2002, 2003 alpa->delta volume collapse transition F0=4,F2=6.1 Photoemission 22

24. Expanding the lattice and freeing the moments. C Marianetti, K Haule G. Kotliar and M. Fluss LDA+DMFT+CTQMC Phys. Rev. Lett. 101, 056403 (2008) 23

25. LDA+DMFT Pu • Pu an element near the Mott transition • Total Energies. Theoretical Prediction of Phonon Spectra. • Quasiparticle Multiplets: Remanent of Atomic Physics in the Quasiparticle Spectra! Different from Hubbard bands, i.e. Am Material Specific Fingerprint of Strong Correlations • Very large expansion should liberate moments. 24

26. Hidden Order Discovered in (1985) U Ru Si URu2Si2 No detection yet ! 25 . T. M. Palstra et.al. PRL 55, 2727 (1985)

27. “Adiabatic continuity” between HO & AFM phase • Similar T0 and TN • Almost identical thermodynamic • quantities (jump in Cv) E. Hassinger et.al. PRL 77, 115117 (2008) 26

28. Two Broken Symmetry Solutions Weiss field Hidden Order LMA K. Haule and GK Nature Physics 2009 27

29. Valence Histogram Hidden order parameter Paramagnetic phase low lying singlets f^2 Order parameter: Different orientation gives different phases: “adiabatic continuity” explained. Hexadecapole order testable by resonant X-ray In the atomic limit: 28

30. Simplified model and phase diagram mean field theory Mean field Exp. by E. Hassinger et.al. PRL 77, 115117 (2008) Difficult Experiments to detect staggered hexadecapole Under Way 29

31. LDA+DMFT URu2Si2 • Kondo effect in URu2Si2 is arrested • below crystal field splitting energy. Gives room to • ordered states, either AFM state or orbital order. • AFM state and hidden order state have the same order parameter: mixing between atomic singlet states. • Orientation of the order parameter decided which state is stabilized. • Mystery of URu2Si2 hidden order solved by theoretical spectroscopy ? ? • Hexadecupolar order may be detectable resonant and non resonant X ray scattering! 30

32. Electron and Hole DopedCuprates “Pseudo-gap” Review of electron doped cuprates. P. Armitage Fournier R. Green RMP (arXiv:0906.2931 ) Antiferromagnetic Insulator superconductor Underdoped Overdoped Optimal doping t-t’ effects C. T. Shih, Y. C. Chen, C. P. Chou, and T. K. Lee, Phys. Rev. B 70, 220502(R) (2004). T-J variational wave functions. 31

33. LDA+ Single Site DMFT- CTQMC and LDA+ 2 site CDMFT -CTQMC Phase Diagram C. Weber K. Haule and GK (in preparation). LSCO and NCCO 32

34. Doping NCCO N. L. Wang, G. Li, D. Wu, X. H. Chen, C. H. Wang, and H. Ding, Phys. Rev. B 73, 184502 (2006). Y. Onose et al., Phys. Rev. B, 69, 024504 (2004) .03 ev 33 .2 ev

35. Comparing the AF and the “underlying PM state “ <K>sdw -<K>pm LCCO DEkin NCCO NCCO magnetizes to lower its double occupancy . LSCO increases double occupancy when it magnetizes LSCO gains kinetic energy when it magnetizes. NCCO pays kinetic energy 33

36. LDA+DMFT LSCO vs NNCO • Strength of correlations as quantified by single site DMFT, a fundamental difference between NCCO and LSCO compounds. • Size (of the correlation) does matter. • Differences in the metallization process and its connection to the magnetism and superconductivity. • No need to use x dependent values of the interaction in electron doped cuprates, (which was done in previous studies ) 34

37. Perspectives and Outlook • DMFT: exploiting a notion of locality to make the quantum many body theory of solids computationally tractable. • Reduction of the full problem to a Quantum Impurity Model in a mdeium. Powerful as conceptual tool and computational tool. • Some evidence that (in some cases) (and for some quantities) it works (unreasonably) well! • Still much work is needed to make it into a completely systematic, and user friendly tool. 35

38. Interface with realistic electronic structure methods (e.g. LDA+DMFT ) . Connects structure with physical property. • Example from 3d’s, 4f’s, 5f’s, many many other examples. Strongly correlated materials are ubiquitous, many more will be man made. • DMFT: conceptual tool –computatational tool for materials • Theoretical-Computatational spectroscopies. • Many groups contributing around the world, lots of materials, codes, basis sets, imp-solvers………….. : G. Kotliar et.al. RMP 78, 865 (2006)K. Held Advances in Physics 56, 829 (2007).. 36

39. Many improvements are needed/possible. • GW+DMFT [ Sun and Kotliar PRB P Sun and G. KotliarPhys. Rev. B 66, 85120 (2002). Biermann et. al. Phys. Rev. Lett. 90, 086402 (2003) ]. More stable/faster LDA+DMFT implementations. • Many interesting material related problems can be attacked with the current method as is • Field of correlated electrons is still full of promise and suprises. Computational physics set to play a leading role. • Material design with correlated electrons ?? 37

40. Thanks for your Attention!! Post-Doctoral Position in this field available! Please contact me if you are interested or know of an excellent candidate. kotliar@physics.rutgers.edu

44. Toy model Hamiltonian Hubbard model BaymKadanoff Functional, F sum over two particle irreducible graphs. sum ALL LOCAL graphs only ! Reviews : A.Georges, G. Kotliar, W. Krauth and M. J. Rozenberg, R.M.P. 68, 13 (1996). D. Vollhardt and G. Kotliar Physics Today (2004)

45. Neutron Scattering

46. 1d - CDMFT (2 sites) vs exact solution (Bethe Anzats) density n vs chemical potentialμ Gap vs U at half filling V. Kancharla C. Bolech and GK PRB 67, 075110 (2003)][[M.CaponeM.Civelli V KancharlaC.Castellani and GK PR B 69,195105 (2004) ]

48. Consequences of the Approximate U(1) Symmetry

50. Still looking for moments. Pu under (negative ) pressure. C Marianetti, K Haule GK and M. Fluss