1. Dynamical Mean Field Theory, Mott transition and Electronic Structure of Actinides Gabriel Kotliar Physics Department and Center for Materials Theory Rutgers University SCES 2001 Ann Arbor August 6 th -10 th 2001

2. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Outline Introduction to Pu Background: DMFT study of the Mott transition in a toy model DMFT as an electronic structure method. DMFT results for delta Pu, and some qualitative insights into the “Mott transition across the actinide series”

3. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Mott transition in the actinide series (Smith Kmetko phase diagram, Johanssen 1974)

4. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Phase transition with Large Volume changes! Small amounts of Ga stabilize the  phase (A. Lawson LANL)

5. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS The bonding problem o DFT in the LDA or GGA is a well established tool for the calculation of ground state properties. o Many studies (Freeman, Koelling 1972)APW methods. Full potential and ASA methods Soderlind et.al 1990, Kollar et.al 1997, Boettger et.al 1998, Wills et.al. 1999) give o an equilibrium volume of the  phase  Is 35% lower than experiment o This is the largest discrepancy ever known in DFT based calculations.

6. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Conventional viewpoint Alpha Pu is a simple metal, it can be described with LDA + pert. corrections. In contrast delta Pu is strongly correlated. Constrained LDA approach (Erickson, Wills, Balatzki, Becker). In Alpha Pu, all the 5f electrons are treated as band like, while in Delta Pu, one 5f electrons are band-like while four 5f electron is localized. LDA + U (Savrasov andGK Phys. Rev. Lett. 2000, Bouchet et.al 2000) predicts correct volume of Delta Pu with U=4,Alpha Pu has U =0.

7. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Problems with the conventional viewpoint of Pu U/W is not so different in alpha and delta LDA+U, LDA, constrained LDA are not good starting points to describe the transport and thermodynamics, Pu is a light heavy fermion. The specific heat of delta Pu, is only twice as big as that of alpha Pu. The susceptibility of alpha Pu is in fact larger than that of delta Pu. The resistivity of alpha Pu is comparable to that of delta Pu.

8. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Pu Specific Heat

9. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Anomalous Resistivity

10. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Outline Introduction to Pu Background: DMFT study of the Mott transition in a toy model DMFT as an electronic structure method. DMFT results for delta Pu, and some qualitative insights into the “Mott transition across the actinide series”

11. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Theoretical approach to the Mott transition problem Mean field approach to quantum many body systems, constructing equivalent impurity models embedded in a bath to be determined self consistently. Use and compare exact and approximate numerical techniques (QMC, RG, ED) as well as semianalytical approaches (interpolative schemes) to solve the self consistent impurity model. Formulation the DMFT equations as saddle points of a functional of the spectral function. Deeper understanding of the validity of the DMFT results.

12. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Schematic DMFT phase diagram one band Hubbard model (half filling, semicircular DOS, partial frustration) Rozenberg et.al PRL (1995)

13. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Phase Diagrams :V 2 O 3, Ni Se 2 -x S x Mc Whan et. Al 1971,. Czek et. al. J. Mag. Mag. Mat. 3, 58 (1976),

14. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Mott transition in layered organic conductors S Lefebvre et al. Ito et.al, Kanoda’s talk Bourbonnais talk Magnetic Frustration

15. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Insights from DMFT  The Mott transition is driven by transfer of spectral weight from low to high energy as we approach the localized phase. Fixed density.  Control parameters: doping, temperature,pressure…  The laws that govern the transfer of spectral weight can be formulated around special points in the phase diagram, where bifurcations take place

16. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Mott endpoint Transfer of spectral weight at fixed density. Anomalous transfer of spectral weight connected to the proximity to an Ising Mott endpoint (Kotliar Lange and Rozenberg PRL 84, 5180 (2000))

17. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Anomalous transfer of optical spectral weight in NiSeS(Miyasaka and Takagi 2000),Photoemission Matsuura et. Al. 1998

18. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Outline Introduction to Pu Background: DMFT study of the Mott transition in a toy model DMFT as an electronic structure method. DMFT results for delta Pu, and some qualitative insights into the “Mott transition across the actinide series”

19. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS LDA+DMFT The light, SP (or SPD) electrons are extended, well described by LDA The heavy, D (or F) electrons are localized,treat by DMFT. LDA 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

20. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Spectral Density Functional : effective action construction ( Chitra and GK 2000 ). 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  )]

21. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Spectral Density Functional The exact functional can be built in perturbation theory in the interaction (well defined diagrammatic rules )The functional can also be constructed from the atomic limit, but no explicit expression exists. DFT is useful because good approximations to the exact density functional  DFT  (r)] exist, e.g. LDA, GGA A useful approximation to the exact functional can be constructed, the DMFT +LDA functional. Motivated by LDA+U

22. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS LDA+DMFT functional  Baym Kadanoff functional of an ATOM. Sum of local 2PI graphs with local U matrix and local G

23. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS LDA+DMFT: Introduction of a Weiss field, mapping onto impurity models Weiss field

24. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Comments on LDA+DMFT Static limit of the LDA+DMFT functional, with  =  HF reduces to LDA+U Removes inconsistencies of this approach, Only in the orbitally ordered Hartree Fock limit, the Greens function of the heavy electrons is fully coherent Gives the local spectra and the total energy simultaneously, treating QP and H bands on the same footing. Luttinger theorem is obeyed.

25. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Outline Introduction to Pu Background: DMFT study of the Mott transition in a toy model DMFT as an electronic structure method. DMFT results for delta Pu, and some qualitative insights into the “Mott transition across the actinide series”

26. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Pu: DMFT total energy vs Volume(Savrasov et.al.2001)

27. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Dynamical Mean Field View of Pu ( Savrasov Kotliar and Abrahams, Nature 2001) Delta and Alpha Pu are both strongly correlated, the DMFT mean field free energy has a double well structure, for the same value of U. One where the f electron is a bit more localized (delta) than in the other (alpha). Is the natural consequence of the model Hamiltonian phase diagram once electronic structure is about to vary (see also Majumdar and Krishnmurthy 1995). This result resolves one of the basic paradoxes in the physics of Pu.

28. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Lda vs Exp Spectra

29. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Pu Spectra DMFT(Savrasov) EXP (Arko et. Al)

30. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS PU: (“cubic ALPHA” AND DELTA

31. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Double well structure and  Pu Qualitative explanation of negative thermal expansion Sensitivity to impurities which easily raise the energy of the  -like minimum.

32. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Minimum in melting curve and divergence of the compressibility at the Mott endpoint

33. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Superconductivity in Am Atomic state J=0 How to go from a metal to a closed shell insulator by increasing U. Entropy has to increase, as U increases, but the insulator has zero entropy! Something has to happen DMFT study of the problem (Capone Fabrizio and Tossatti) Superonductivity intervenes!

34. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Conclusion The character of the localization delocalization in simple( Hubbard) models within DMFT is now fully understood, nice qualitative insights.  This has lead to extensions to more realistic models, and a beginning of a first principles approach interpolating between atoms and band, encouraging results for simple elements, (Savrasov, Kotliar, Abrahams Nature 2001 Pu), Lichtenstein Katsenelson Kotliar (PRL 2001 Fe and Ni).  Outlook: compounds, C-DMFT ………..

35. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS References Review of DMFT :A. Georges, G. Kotliar, W. Krauth and M. Rozenberg Rev. Mod. Phys. 68,13 (1996)] LDA+DMFT: V. Anisimov, A. Poteryaev, M. Korotin, A. Anokhin and G. Kotliar, J. Phys. Cond. Mat. 35, 7359-7367 (1997). A Lichtenstein and M. Katsenelson Phys. Rev. B 57, 6884 (1988). S. Savrasov G.Kotliar funcional formulation for full self consistent implementation of a spectral density functional. Application to Pu S. Savrasov G. Kotliar and E. Abrahams (Nature 2001).

36. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Acknowledgements Useful discussions with A. Lichtenstein, J. Thompson and R. Schrieffer NSF-DMR, DOE (Basic Energy Sciences)

37. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS DMFT Review: A. Georges, G. Kotliar, W. Krauth and M. Rozenberg Rev. Mod. Phys. 68,13 (1996)] Weiss field

38. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Outlook Systematic improvements, short range correlations. Take a cluster of sites, include the effect of the rest in a G0 (renormalization of the quadratic part of the effective action). What to take for G0: DCA (M. Jarrell Krishnamurthy et.al), CDMFT ( GK Savrasov Palsson and Biroli) include the effects of the electrons to renormalize the quartic part of the action (spin spin, charge charge correlations) E. DMFT (Kajueter and GK, Si et.al)

39. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Outlook Extensions of DMFT implemented on model systems, carry over to more realistic framework. Better determination of Tcs………… First principles approach: determination of the Hubbard parameters, and the double counting corrections long range coulomb interactions E- DMFT Improvement in the treatement of multiplet effects in the impurity solvers, phonon entropies, …… …

40. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Wilson and Kadowaki Woods Ratio

41. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Vanadium Oxide

42. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS. ARPES measurements on NiS 2-x Se x Matsuura et. Al Phys. Rev B 58 (1998) 3690

43. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS T_MIT=.013 Rozenberg et.al 2001

44. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Realistic DMFT loop

45. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS V2O3

46. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Theoretical Foundations: functionals G. Kotliar and R. Chitra PRB 1999,2000 G. Kotliar and S. Savrasov 2001 LDA[ Fukuda et.al, Aliev and Fernando] LDA+U LDA+DMFT

47. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS LDA functional Conjugate field, V KS (r)

48. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Minimize LDA functional

49. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS LDA+U functional

50. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Double counting term (Lichtenstein et.al) Problem : What is the LDA+U functional, a functional of? What is n ab ?

51. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Functional Approach G. Kotliar EPJB (1999)

52. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Functional Approach The functional approach offers a direct connection to the atomic energies. One is free to add terms which vanish quadratically at the saddle point. Allows us to study states away from the saddle points, All the qualitative features of the phase diagram, are simple consequences of the non analytic nature of the functional. Mott transitions and bifurcations of the functional.

53. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Solving the impurity Multiorbital situation and several atoms per unit cell considerably increase the size of the space H (of heavy electrons). QMC scales as [N(N-1)/2]^3 N dimension of H Fast interpolation schemes (Slave Boson at low frequency, Roth method at high frequency, + 1 st mode coupling correction), match at intermediate frequencies. (Savrasov et.al 2001)

54. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Schematic DMFT phase diagram one band Hubbard model (half filling, semicircular DOS, partial frustration) Rozenberg et.al PRL (1995)

55. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Recent QMC phase diagram of the frustrated Half filled Hubbard model with semicircular DOS ( Joo and Udovenko 2001).

56. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Case study: IPT half filled Hubbard one band (Uc1) exact = 2.2+_.2 (Exact diag, Rozenberg, Kajueter, Kotliar PRB 1996), confirmed by Noack and Gebhardt (1999) (Uc1) IPT =2.6 (Uc2) exact =2.97+_.05(Projective self consistent method, Moeller Si Rozenberg Kotliar Fisher PRL 1995 ), (Confirmed by R. Bulla 1999) (Uc 2 ) IPT =3.3 (T MIT ) exact =.026+_.004 (QMC Rozenberg Chitra and Kotliar PRL 1999), (T MIT ) IPT =.045 (U MIT ) exact =2.38 +-.03 (QMC Rozenberg Chitra and Kotliar PRL 1999), (U MIT ) IPT =2.5 (Confirmed by Bulla 2001) For realistic studies errors due to other sources (for example the value of U, are at least of the same order of magnitude).

57. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS NiSeS

58. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Ising character of Mott endpoint Singular part of the Weiss field is proportional to  Max{ (p-pc) (T- Tc)} 1/   in mean field and 5 in 3d  couples to all physical quantities which then exhibit a kink at the Mott endpoint. Resistivity, double occupancy,photoemission intensity, integrated optical spectral weight, etc. Divergence of the specific heat.

59. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Mott transition endpoint Rapid variation has been observed in optical measurements in vanadium oxide and nises mixtures Experimental questions: width of the critical region. Ising exponents or classical exponents, validity of mean field theory Building of coherence in other strongly correlated electron systems. Unify concepts from different theoretical approaches, condensation of d and onset of coherence.

60. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Insights from DMFT  Low temperatures several competing phases. Their relative stability depends on chemistry and crystal structure  High temperature behavior around Mott endpoint, more universal regime, captured by simple models treated within DMFT

61. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Cerium

62. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Pu: Anomalous thermal expansion (J. Smith LANL)

63. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS MAGNETIC

64. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Specific heat and susceptibility.

65. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Remarks on the literature The qualitative features found by the Rutgers ENS groups were challenged in a series of publications: Logan and Nozieres (1987) S Kehrein Phys. Rev Lett. 3192 (1998),R. Noack and F. Gebhardt, Phys. Rev. Lett. 82, 1915 (1999), J. Schlipf et. al. Phys. Rev. Lett 82, 4890 (1999). These works missed subtle non perturbative aspects of the Mott metal to insulator transition such as the singular behavior of the self energy

66. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Two Roles for DMFT in calculations of the electronic structure of correlated materials Crystal Structure +atomic positions Correlation functions Total energies etc. Model Hamiltonian

67. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS the Mott phenomena Evolution of the electronic structure between the atomic limit and the band limit in an open shell situation. The “”in between regime” is ubiquitous central them in strongly correlated systems, gives rise to interesting physics. Example elemental plutonium [ B. Johanssen Phil Mag. 30,469 (1974)] Revisit the problem using a new insights and new techniques from the solution of the Mott transition problem within dynamical mean field theory in a simple model Hamiltonian (one band Hubbard, semicircular density of states). Use the ideas and concepts that resulted from this development to give physical insights into real materials. Turn the technology developed to solve the toy model into a practical electronic structure method.

68. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS More on DFT LSDA predicts magnetic long range (Solovyev et.al.0 Experimentally  Pu is not magnetic. If one treats the f electrons as part of the core LDA overestimates the volume by 30% LDA predicts correctly the volume of the  phase of Pu, when full potential LMTO (Soderlind Eriksson and Wills) is used. This is taken as an indication that  Pu is a weakly correlated system

69. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Mott transition in the actinide series (Smith Kmetko phase diagram)

70. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Minimum of the melting point Divergence of the compressibility at the Mott transition endpoint. Rapid variation of the density of the solid as a function of pressure, in the localization delocalization crossover region. Slow variation of the volume as a function of pressure in the liquid phase

71. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Anomalous Resistivity and Mott transition Ni Se 2-x S x Miyasaka and Tagaki (2000)

72. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS LDA+DMFT Self-Consistency loop DMFT U E