1. Dynamical Mean Field Theory and Electronic Structure Calculations Gabriel Kotliar Center for Materials Theory Rutgers University

2. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Outline Incorporating electronic structure methods in DMFT. C-DMFT. [M. Capone, M. Civelli ] Why do we need k-sum to do optics. Cerium puzzles. [K. Haule V. Udovenko ] Why do we need functionals to do total energies. Phonons and plutonium puzzles. [X. Dai S. Savrasov ] Physics Today Vol 57, 53 (2004) Gabriel Kotliar and Dieter VollhardtPhysics Today Vol 57, 53 (2004)

3. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Two roads for ab-initio calculation of electronic structure of strongly correlated materials Correlation Functions Total Energies etc. Model Hamiltonian Crystal structure +Atomic positions DMFT ideas can be used in both cases.

4. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS 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, substract this out by shifting the heavy level (double counting term)  The U matrix can be estimated from first principles of viewed as parameters. Solve resulting model using DMFT.

5. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Single site DMFT Impurity cavity construction: A. Georges, G. Kotliar, PRB 45, 6497 (1992)] Weiss field

6. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS EDMFT [H. Kajueter Rutgers Ph.D Thesis 1995 Si and Smith PRL77, 3391(1996) R. Chitra and G. Kotliar PRL84,3678 (2000)]

7. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Realistic DMFT loop: matrix inversion-tetrahedron method

8. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Site  Cell. Cellular DMFT. C-DMFT. G. Kotliar,S.. Savrasov, G. Palsson and G. Biroli, Phys. Rev. Lett. 87, 186401 (2001) tˆ(K) is the hopping expressed in the superlattice notations. Other cluster extensions (DCA, Katsnelson and Lichtenstein periodized scheme, nested cluster schemes, PCMDFT ), causality issues, O. Parcollet, G. Biroli and GK cond-matt 0307587 (2003)

9. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS N vs mu in one dimensional Hubbard model. Compare 2 site cluster (in exact diag with Nb=8) vs exact Bethe Anzats, [M. Capone M.Civelli C. Castellani V Kancharla and GK 2004]

10. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Two roads for ab-initio calculation of electronic structure of strongly correlated materials Correlation Functions Total Energies etc. Model Hamiltonian Crystal structure +Atomic positions

11. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Spectral Density Functional : Effective action construction R. Chitra G. Kotliar PRB 62,12715. Kotliar Savrasov in New Theoretical Approaches to Strongly Correlated Systems, A. M. Tsvelik ed. (2001) Kluwer Academic Publishers. 259-301 ; cond-mat/0208241. S Savrasov G Kotliar cond- mat0308053.cond-mat/0208241 DFT, consider the exact free energy as a functional of an external potential. Express the free energy as a functional of the density by Legendre transformation.  DFT  (r)] Introduce local orbitals,   R (r-R)orbitals, and local GF G(R,R)(i  ) = The exact free energy can be expressed as a functional of the local Greens function and of the density by introducing sources for  (r) and G and performing a Legendre transformation,  (r),G(R,R)(i  )] Allows computation of total energy, phonons!!!!

12. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS LDA+DMFT Self-Consistency loop. See also S. Savrasov and G. Kotliar cond-matt 0308053 DMFT U E

13. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Impurity Solvers. Hubbard I. Quantum Montecarlo. Rational Approximations to the self energy, constructed with slave bosons. cond-mat/0401539 V. Oudovenko, K. Haule, S. Savrasov D. Villani and G. Kotliar. Extensions of NCA. Th. Pruschke and N. Grewe, Z. Phys. B: Condens. Matter 74, 439, 1989. SUNCA K. Haule, S. Kirchner, J. Kroha, and P. W¨olfle, Phys. Rev. B 64, 155111, (2001).

14. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Application to Materials Cerium: Alpha to Gamma Transition. Plutonium : Alpha-Delta-Epsilon.

15. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Overview  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) volumesexp.LDALDA+U  28Å 3 24.7Å 3  34.4Å 3 35.2Å 3   -phase (localized) : High T phase  Curie-Weiss law (localized magnetic moment),  Large lattice constant  Tk around 60-80K   -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   -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

16. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Qualitative Ideas. B. Johansson, Philos. Mag. 30, 469 (1974). Mott transition of the f electrons as a function of pressure. Ce alpha gamma transition. spd electrons are spectators. Mathematical implementation, “metallic phase” treat spdf electrons by LDA, “insulating phase” put f electron in the core.  J.W. Allen and R.M. Martin, Phys. Rev. Lett. 49, 1106 (1982); Kondo volume collapse picture. The dominant effect is the spd-f hybridization.

17. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Qualitative Ideas alpha phase Kondo effect between spd and f takes place. “insulating phase” no Kondo effect (low Kondo temperature). Mathematical implementation, Anderson impurity model in the suplemented with elastic terms. (precursor of realistic DMFT ideas, but without self consistency condition). J.W. Allen and L.Z. Liu, Phys. Rev. B 46, 5047 (1992).

18. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS LDA+DMFT:Ce spectra M.B.Z¨olfl,I.A.NekrasovTh.Pruschke,V.I.Anisimov J. Keller,Phys.Rev. Lett 87, 276403 (2001). K. Held, A.K. McMahan, and R.T. Scalettar, Phys. Rev.Lett. 87, 276404 (2001) A.K.McMahan,K.Held,andR.T.Scalettar,Phys Rev. B 67, 075108 (2003). Successful calculations of thermodynamics.

19. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS X.Zhang M. Rozenberg G. Kotliar (PRL 70, 1666(1993)). 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!!!!

20. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS The schematic phase diagram of cannot distinguish between the two scenarios. J.W. Allen and L.Z. Liu, Phys. Rev. B 46, 5047 (1992). Kondo impurity model + elastic terms. DMFT phase diagram of a Hubbard model at integer filling, has a region between Uc1(T) and Uc2(T) where two solutions coexist. A. Georges G. Kotliar W. Krauth and M Rozenberg RMP 68,13,(1996). Coupling the two solutions to the lattice gives a phase diagram akin to alpha gamma cerium. Majumdar and Krishnamurthy PRL 73 (1994).

21. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Photoemission&experiment A. Mc Mahan K Held and R. Scalettar (2002) Zoffl et. al (2002) K. Haule V. Udovenko S. Savrasov and GK. (2004)

22. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS To resolve the conflict between the Mott transition and the volume collapse picture : Turn to Optics! Haule et.al. 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 wether a Kondo binding unbinding takes pace (Kondo collapse picture). General method, bulk probe.

23. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Optics formula double pole single pole One divergence integrated out!

24. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Temperature dependence of the optical conductivity.

25. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS The volume of alpha is 28.06°A and the temperature 580K. The volume of the gamma phase is 34.37°A and T = 1160K. Experiments : alpha at 5 K and gamma phase at 300 K. Theory: Haule et. al. cond-matt 04 Expt: J.W. vanderEb PRL 886,3407 (2001)

26. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Optical conductivity of Ce (expt. Van Der Eb et.al. theory Haule et.al) experiment LDA+DMFT K. Haule et.al.

27. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Origin of the features.

28. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Conclusion: Cerium Qualitatively good agreement with existing experiment. Some quantitative disagreement, see however. Experiments should study the temperature dependence of the optics. Optics + Theory can provide a simple resolution of the Mott vs K-Collapse conundrum.

29. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Phases of Pu (A. Lawson LANL) Los Alamos Science 26, (2000)

30. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Delta phase of Plutonium: Problems with LDA an equilibrium volume of the  phase  Is 35% lower than experiment o Many studies and implementations.(Freeman, Koelling 1972)APW methods, ASA and FP-LMTO Soderlind et. Al 1990, Kollar et.al 1997, Boettger et.al 1998, Wills et.al. 1999).all give an equilibrium volume of the  phase  Is 35% lower than experiment this is the largest discrepancy ever known in DFT based calculations. LSDA predicts magnetic long range (Solovyev et.al.) Experimentally  Pu is not magnetic. If one treats the f electrons as part of the core LDA overestimates the volume by 30%

31. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Pu: DMFT total energy vs Volume ( Savrasov Kotliar and Abrahams 2001)

32. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS DMFT studies of Pu. Savrasov, S. Y., and G. Kotliar, 2003, Phys. Rev. Lett. 90(5), 056401/1. Savrasov, S. Y., and G. Kotliar, 2003, cond- mat/0308053. Savrasov, S. Y., G. Kotliar, and E. Abrahams, 2001, Nature 410, 793 Dai X. Savrasov S.Y. Kotliar G. Migliori A. Letbetter H, Abrahams A. Science 300, 953, (2003)

33. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS DFT Studies of Pu DFT in GGA predicts correctly the volume of the  phase of Pu, when full potential LMTO (Soderlind Eriksson and Wills) is used. This is usually taken as an indication that  Pu is a weakly correlated system The shear moduli in the delta phase were calculated within LDA and GGA by Bouchet et. al. (2000) and c’ is negative!.

34. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS X.Zhang M. Rozenberg G. Kotliar (PRL 70, 1666(1993)). Evolution of the spectra as a function of U, half filling full frustration.

35. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Alpha and delta Pu : Expt. Arko et.al. PRB 62, 1773 (2000). DMFT: Savrasov and Kotliar

36. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Phonon freq (THz) vs q in delta Pu X. Dai et. al. Science vol 300, 953, 2003

37. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Expts’ Wong et. al. Science 301. 1078 (2003) Theory Dai et. al. Science 300, 953, (2003)

38. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS The delta –epsilon transition The high temperature phase, (epsilon) is body centered cubic, and has a smaller volume than the (fcc) delta phase. What drives this phase transition? Having a functional, that computes total energies opens the way to the computation of phonon frequencies in correlated materials (S. Savrasov and G. Kotliar 2002)

39. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Phases of Pu (A. Lawson LANL) Los Alamos Science 26, (2000)

40. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Epsilon Plutonium.

41. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS 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 Ackland et. al. PRB.65, 184104 (2002); (and neglecting electronic entropy). TC ~ 600 K.

42. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Phonons epsilon

43. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Summary Incorporating electronic structure methods in DMFT. C-DMFT. [M. Capone, M. Civelli ] Why do we need k-sum to do optics. Cerium puzzles. [K. Haule V. Udovenko ] Why do we need functionals to do total energies. Phonons and plutonium puzzles. [X. Dai S. Savrasov ]

44. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

45. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Why is optics calculation not completely trivial? Analytic tetrahedron method: Integral is analytic and simple (combination of logarithms)  Energies linearly interpolated no simple analytic expression  Product of two energies linearly interpolatedATM applicable but numerically very unstable because of quadratic pole 1D example: Parabola has 2 zeros (2poles) Line has no zeros (no poles)

46. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS LDA+DMFT functional  Sum of local 2PI graphs with local U matrix and local G

47. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Schematic DMFT phase diagram one band Hubbard model. Rozenberg et. al. 1996. Introduce coupling to the lattice will cause a volume jump across the first order transition. (Majumdar and Krishnamurthy ).

48. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Shear anisotropy. Expt. vs Theory C’=(C11-C12)/2 = 4.78 GPa C’=3.9 GPa C44= 33.59 GPa C44=33.0 GPa C44/C’ ~ 7 Largest shear anisotropy in any element! C44/C’ ~ 8.4

49. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Benchmarking SUNCA, V. Udovenko and K. Haule