1. Electronic Structure of Actinides at the Mott Boundary: A Dynamical Mean Field Theory Perspective Gabriel Kotliar Physics Department and Center for Materials Theory Rutgers University Livermore Ca Oct 20 (2004)

2. Collaborators, References S. Savrasov and G. Kotliar PRL 84 3670 (2000). S.Savrasov G. Kotliar and E. Abrahams, Nature 410,793 (2001). X. Dai,S. Savrasov, G. Kotliar,A. Migliori, H. Ledbetter, E. Abrahams Science, Vol300, 954 (2003). S. Murthy Rutgers Ph.D Thesis (2004). The Mott transition in the actinide series. Dynamical Mean Field Theory. DMFT studies of Plutonium. Connection with invar model ? Americium under pressure. New experiments and DMFT results. Conclusions and Outlook. A. Lawson et. al. LA UR 04-6008 (LANL) J. C. Griveau J Rebizant G. Lander (ITU) and G. Kotliar submitted to PRL.

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4. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Pu phases Small amounts of Ga stabilize the  phase (A. Lawson LANL) Los Alamos Science,26, (2000 ).

5. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS 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. ] Am At room pressure a localised 5f6 system;j=5/2. S = -L = 3: J = 0 apply pressure ? S S U U  T Log[2J+1] Uc  ~1/(Uc-U) S=0 ???

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7. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Dynamical Mean Field Theory Basic idea: reduce the quantum many body problem to a one site or a cluster of sites, in a medium of non interacting electrons obeying a self consistency condition.[A. Georges and GK Phys. Rev. B 45, 6497, 1992]. Atom in a medium = Quantum impurity model. Solid in a frequency dependent potential. Basic idea: instead of using functionals of the density, use more sensitive functionals of the one electron spectral function. [density of states for adding or removing particles in a solid, measured in photoemission] [GK R. Chitra GKPhys. Rev. B62, 12715 (2000). and S. Savrasov cond-matt 0308053]. Allows computation of total energy AND one electron spectra.

8. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Two paths 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.

9. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Dynamical Mean Field Theory (DMFT) Cavity Construction.

10. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS [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. Testing CDMFT (G.. Kotliar,S. Savrasov, G. Palsson and G. Biroli, Phys. Rev. Lett. 87, 186401 (2001) ) with two sites in the Hubbard model in one dimension.

11. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS X.Zhang M. Rozenberg G. Kotliar (PRL 1993) Mott transition, three peak structure and transfer of spectral weigth. Evolution of the one particle spectral function in a frustrated Hubbard model at half filling.

12. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Qualitative phase diagram of a frustrated Hubbard model

13. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Two paths 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.

14. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Full implementation in the context of a a one orbital model. P Sun and G. Kotliar Phys. Rev. B 66, 85120 (2002). After finishing the loop treat the graphs involving Gnonloc Wnonloc in perturbation theory. P.Sun and GK PRL (2004). Related work, Biermann Aersetiwan and Georges PRL 90,086402 (2003). G. Kotliar and S. Savrasov Strongly Correlated Systems, A. M. Tsvelik Ed. 2001 Kluwer Academic 259-301. conmat/0208241 S. Y. Savrasov, G. Kotliar, Phys. Rev. B 69, 245101 (2004)

15. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Approximations Electronic structure. LMTO’s ASA, LMTO full potential. Crystal field splitting in the self energies is neglected. W(r,r’) (w) (or V0(w) replaced by U on the f electrons. 4 ev. Americium U 4.5 ev. Neglect multiplet splittings. Non perturbative treatment of spin orbit coupling. Approximate Impurity Solver. Interpolative Perturbation Theory, NCA, Hubbard I……….

16. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS o Start from functional of G and W (Chitra and Kotliar (2000), Ambladah et. al. o Make local or cluster approximation on  o FURTHER APPROXIMATIONS: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). o Truncate the W operator act on the H sector only. i.e. Replace W(  ) or V 0 (  ) by a static U. This quantity can be estimated by a constrained LDA calculation or by a GW calculation with light electrons only. e.g. M.Springer and F.Aryasetiawan,Phys.Rev.B57,4364(1998) T.Kotani,J.Phys:Condens.Matter12,2413(2000). FAryasetiawan M Imada A Georges G Kotliar S Biermann and A Lichtenstein cond-matt (2004)

17. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS or the U matrix can be adjusted empirically. At this point, the approximation can be derived from a functional (Savrasov and Kotliar 2001) FURTHER APPROXIMATION, ignore charge self consistency, namely set LDA+DMFT V. Anisimov, A. Poteryaev, M. Korotin, A. Anokhin and G. Kotliar, J. Phys. Cond. Mat. 35, 7359 (1997) See also. A Lichtenstein and M. Katsnelson PRB 57, 6884 (1988). Reviews: Held, K., I. A. Nekrasov, G. Keller, V. Eyert, N. Blumer, A. K. McMahan, R. T. Scalettar, T. Pruschke, V. I. Anisimov, and D. Vollhardt, 2003, Psi-k Newsletter #56, 65. Lichtenstein, A. I., M. I. Katsnelson, and G. Kotliar, in Electron Correlations and Materials Properties 2, edited by A. Gonis, N. Kioussis, and M. Ciftan (Kluwer Academic, Plenum Publishers, New York), p. 428. Georges, A., 2004, Electronic Archive,.lanl.gov, condmat/ 0403123.

18. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS DMFT : What is the dominant atomic configuration,what is the fate of the atomic moment ? Snapshots of the f electron Dominant configuration:(5f) 5 Naïve view Lz=-3,-2,-1,0,1 ML=-5  B,, S=5/2 Ms=5  B Mtot=0 More realistic calculations, (GGA+U),itineracy, crystal fields     ML=-3.9 Mtot=1.1. S. Y. Savrasov and G. Kotliar, Phys. Rev. Lett., 84, 3670 (2000) This moment is quenched or screened by spd electrons, and other f electrons. (e.g. alpha Ce).  Contrast Am:(5f) 6

19. Total Energy as a function of volume for Pu 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)

20. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Ground State Theory of  -Pu S.Y. Savrasov and G. Kotliar PRL 84, (2000). “In conclusion, using a realistic value of the Hubbard U = 4 eV we have been able to describe ground state properties of d-Pu in good agreement with experimental data. This theory correctly predicts the equilibrium volume of the d phase and suggests that nearly complete cancellation occurs between spin and orbital moments. The main shortcoming of the present calculation is the assumed long-range spin and orbital order. This is the essential limitation of the LDA + U approach (or of any static mean field theory): in order to capture the effects of correlations it has to impose some form of long-range order. Static mean field theories are unable to capture subtle many-body effects such as the formation of local moments and their subsequent quenching via the Kondo effect. These deficiencies will be removed by ab initio dynamical mean field calculations for which codes are currently being developed.”

21. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Dual model, Zwignagl and Fulde.Erickson Becker Balatzki and J. Wills. J. Alloys and Compounds 287, (1995) 1-5. Part of the f electrons are in a core like (5f)4 configuration (non magnetic )and 1 5f electron is itinerant. GGA+ Orbital Polarization. Soderlind and Sadigh PRL 92, 1857021. Correct volume of all phases of Pu. Ordered Orbital and Spin moments in all phases of Pu. Disordered Local Moment approach. A. Niklasson, J M. Wills, M I. Katsnelson, I.A. Abrikosov, O. Eriksson, and B. Johansson Phys Rev B 67, 235105 (2003). There are large fluctuating (disordered moments) in Pu. Accounts for the correct volumes and bulk moduli across the actinide series.

22. Electronic spectra and Total energy. LDA,LDA+U, LDA+DMFT LDA LDA+U  LRO DMFT Spectra Method E vs V

23. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Double well structure and  Pu Qualitative explanation of negative thermal expansion[ G. Kotliar J.Low Temp. Phys vol.126, 1009 27. (2002)]See also A. Lawson et.al.Phil. Mag. B 82, 1837 ]

24. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS A. C. Lawson et. al. LA UR 04-6008 F(T,V)=Fphonons+Finvar

25. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Invar model A. C. Lawson et. al. LA UR 04-6008

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27. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Prediction of the Invar Model

28. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS DMFT and the Invar Model

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30. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS 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.

31. DMFT Phonons in fcc  -Pu C 11 (GPa) C 44 (GPa) C 12 (GPa) C'(GPa) Theory 34.56 33.03 26.81 3.88 Experiment 36.28 33.59 26.73 4.78 ( Dai, Savrasov, Kotliar,Ledbetter, Migliori, Abrahams, Science, 9 May 2003) (experiments from Wong et.al, Science, 22 August 2003)

32. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Comparison of theory and experiment. Good agreement over the majority of the Brillouin zone, is significant. The phonon frequencies depend on the forces acting on the atoms as a result of their displacement. Ability to compute forces, is a first step to derive potentials, and do molecular dynamics. Discrepancies along (111) are significant. Role of temperature ? Improve the impurity solver ? Non local corrections, and deviations from DMFT. Spectral Density Functional. Connection between spectra and bonding. Microscopic theory of Pu, connecting its anomalies to the vicinity of a Mott point. Combining theory and experiment we can more than the sum of the parts. Next step in Pu, much better defined problem, discrepancy in (111 ) zone boundary, may be due to either the contribution of QP resonance, or the inclusion of nearest neighbor correlations. Both can be individually studied.

33. 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? LDA+DMFT functional computes total energies opens the way to the computation of phonon frequencies in correlated materials (S. Savrasov and G. Kotliar 2002). Combine linear response and DMFT.

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

35. 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 G. Ackland et. al. PRB.65, 184104 (2002); (and neglecting electronic entropy). TC ~ 600 K.

36. 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 earlier studies of the Mott transition phase diagram once electronic structure is about to vary.

37. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Alpha and delta Pu Photoemission Spectra DMFT(Savrasov et.al.) EXP ( Arko Joyce Morales Wills Jashley PRB 62, 1773 (2000))

38. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS J. Tobin et. al. PHYSICAL REVIEW B 68, 155109,2003

39. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS K. Haule, Pu- photoemission with DMFT using vertex corrected NCA.

40. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS How to track the origin of the resonance ?Turn to Optics! 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).

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42. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Optical Conductivity Temperature dependence.

43. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Origin of the features. K. Held, A.K. McMahan, and R.T. Scalettar, Phys. Rev.Lett. 87, 276404 (2001); A.K. McMahan, K. Held, and R.T. Scalettar, Phys. Rev. B 67, 075108 (2003). K. Haule et. al. (2004)

44. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Americium under pressure (Lindbaum et. al. PRB 2003)

45. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS LDA+DMFT calculations for fcc Americium S. Murthy (2003)

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47. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS LDA+DMFT spectra. Notice the rapid occupation of the f7/2 band, (5f) 7

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49. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS New picture of the electronic structure of Am. The traditional picture of Am, views its f electrons as a closed shell (5f) 6 As a result the spd electrons are free electron-like. This resulted in the early prediction that Am should be a superconductor. Theoretical calculations and experiments shows that Am is very close to a mixed valence situation that can be induced by a small amount of pressure!! At larger pressures a Mott transition and a Tc vs V with a dome- like shape results.

50. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS J. C. Griveau et. al. (2004)

51. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Mott transition in open (right) and closed (left) shell systems. S S U U  T Log[2J+1] Uc  ~1/(Uc-U) S=0 ??? Tc

52. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Griveau et.al. (2004)

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55. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS H.Q. Yuan et. al. CeCu2(Si 2-x Ge x ). Am under pressure Griveau et. al. Superconductivity due to valence fluctuations ?

56. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Conclusions and Outlook Motivation: Mott transition in Americium and Plutonium. In both cases theory (DMFT) and experiment suggest a more gradual transformation than postulated in earlier theories. DMFT: Physical connection between spectra and structure. Studied the Mott transition from both ends. DMFT: method under construction, but it already gives quantitative results and qualitative insights. It CAN be systematically improved in many directions. Interactions between theory and experiments. PU: simple picture of alpha (no quantitative study yet) delta and epsilon. Many approaches give the correct energetics and some of the spectra with very different pictures. Mixed level model, DLM, ordered sates. Look at some experiments and computations. Am: Rich physics, mixed valence under pressure ? Superconductivity under pressure.

57. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Conclusions DMFT produces non magnetic state, around a fluctuating (5f)^5 configuration with correct volume the qualitative features of the photoemission spectra, quasiparticle resonance and Hubbard band, and a double minima structure in the E vs V curve. Correlated view of the alpha and delta phases of Pu. Interplay of correlations and electron phonon interactions account for delta-epsilon transition. Anomalous phonons in epsilon Pu. Calculations can be refined, include multiplets, better impurity solvers, frequency dependent U’s, electronic entropy. User friendly interfaces.

58. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Experiments and Theory are needed to test the different pictures of the electronic structure of Pu Model of Erickson and Wills : 4 (5f) electrons are core-like and 1 is delocalized. DMFT picture: all the 5 (5f) electrons are equivalent, they are localized over short time scales and itinerant over long time scales resulting in Hubbard band and quasiparticle resonance in the spectra. Both pictures require strong correlations in the delta phase but how to differentiate between them experimentally ? Focus on the alpha phase. Resonant Photoemission Probe unoccupied states. Upper Hubbard band, BIS. Optics. X ray absortion. Etc.. Fermi Surface Probes. Different Fermi surfaces.

59. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS  Pu and delta Pu differ electronically by the distribution of spectral weight in the resonance and the Hubbard band. U/W is not so different in alpha and delta 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 and near the Mott limit.

60. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Pu is not MAGNETIC, alpha and delta have comparable susceptibility and specifi heat.

61. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Anomalous Resistivity PRL 91,061401 (2003)

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64. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Search for ordered phases and DLM On the possibility of magnetic moments in plutonium J. C. Lashley, A. Lawson, R. J. McQueeney, and G. H. Lander. (2004) Elastic Neutron Scattering. Inelastic Neutron Scattering. Magnetic Susceptibility. Specific heat measurements in a magnetic field. No indication whatsoever of ordered or disordered moments in either alpha or delta Pu.

65. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS LDA+DMFT References Anisimov Poteryaev Korotin Anhokin and Kotliar J. Phys. Cond. Mat. 35, 7359 (1997). Lichtenstein and Katsenelson PRB (1998). Kotliar, Savrasov, in New Theoretical approaches to strongly correlated systems, Edited by A. Tsvelik, Kluwer Publishers, (2001). Reviews: Kotliar, Savrasov, in New Theoretical approaches to strongly correlated systems, Edited by A. Tsvelik, Kluwer Publishers, (2001). Held Nekrasov Blumer Anisimov and Vollhardt et.al. Int. Jour. of Mod PhysB15, 2611 (2001). Held Nekrasov Blumer Anisimov and Vollhardt et.al. Int. Jour. of Mod PhysB15, 2611 (2001). A. Lichtenstein M. Katsnelson and G. Kotliar (2002)

66. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Plutonium and 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 Mott transition across the actinide series [ B. Johansson 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 the model Hamiltonian context. Use the ideas and concepts that resulted from this development to give physical qualitative insights into real materials. Turn the technology developed to solve simple models into a practical quantitative electronic structure method.

67. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Important Physics Proximity to the Mott Transition. Redistribution of spectral weight. Simultaneous description of band physics and atomic physics. All captured by DMFT in the approximations used.!

68. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Insights into the anomalous properties of Pu Physical anomalies, are the result of the unique position of Pu in the periodic table, where the f electrons are near a localization delocalization transition. The Mott transition across the actinide series [ B. Johansson Phil Mag. 30,469 (1974)] concept has finally been worked out!.We learned how to think about this unusual situation using DMFT, Weiss fields, local spectral functions etc.

69. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Transverse Phonon along (0,1,1) in epsilon Pu in self consistent Born approximation.

70. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Negative thermal expansion of Pu revisited. The distortion described by C' is very soft, nearly like a liquid,. C' measures the rigidity against the volume conserving tetragonal deformation. This is in fact the deformation from fcc towards a bcc along a Bain path. Previous LDA+ U study [Bouchet et. al. ] and our DMFT study show that the total energy difference between  phase and  phases is quite small and is around 1000K. Soft behavior along the Bain path.  Pu can sample the bcc structure, which has lower volume by the thermal fluctuation along Bain path.

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

73. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Further approximations, use approximate impuirity solvers rational Interpolative Perturbative Theory. Savrasov Udovenko Villani Haule and Kotliar. Cond-matt 0401539

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