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)
Will talk about a complex, and still very poorly understood element.
The Mott transition across the actinide series.

2. Collaborators, References
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 (LANL)
J. C. Griveau J Rebizant G. Lander (ITU) and G. Kotliar submitted to PRL .
S. Savrasov and G. Kotliar PRL (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).

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Pu phases Small amounts of Ga stabilize the d phase (A. Lawson LANL) Los Alamos Science,26, (2000).
Extraordinary sensitivity to impurities. Focus on delta mostly. Important for applications, ductile, stable. Any element (except Np stabilizes it) bigger or smaller.
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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
.g T
Log[2J+1]
??? Uc
S=0 U U
g ~1/(Uc-U)
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7. 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, (2000). and S. Savrasov cond-matt ]. Allows computation of total energy AND one electron spectra.
My own point of view, is very different. All the electrons are equivalent, and should be treated as such.
Pu is just one more example of strongly correlated electrons systems. Over the past ten years we
Have developed a new tool to treat them, DMFT. We have gain a lot of understanding into a basic problem.
The Mott transition. How the electron goes from being itinerant to localized. We have solved
A touch but well defined problem. The Hubbard model on a frustrated lattice in a lattice of large
Coordination. Everytime that we can solve mathematically models of complex phenomena, we gain
New insights (one dimensional systems). We have a new toy. It will give a deeper and more
Fundamental understanding of Pu, and a better way to compute physical properties, than the approaches cited above.
I need to give you the basic ideas of the method.
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8. 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.
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Dynamical Mean Field Theory (DMFT) Cavity Construction.
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Testing CDMFT (G.. Kotliar,S. Savrasov, G. Palsson and G. Biroli, Phys. Rev. Lett. 87, (2001) ) with two sites in the Hubbard model in one dimension.
[V. Kancharla C. Bolech and GK PRB 67, (2003)][[M.CaponeM.Civelli V Kancharla C.Castellani and GK P. R B 69, (2004) ]
U/t=4.
Edit. LISA.
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11. Mott transition, three peak structure and transfer of spectral weigth.
X.Zhang M. Rozenberg G. Kotliar (PRL 1993)
Evolution of the one particle spectral function in a frustrated Hubbard model at half filling.
Three peak structure, HubbaRD BANDS AND quasiparticle bands.
Comments on Ce, transfer of spectral weigth
impurity model, georges kotliar earlier work 1992.
The Mott transition is driven by transfer of spectra
l weight from low to high energy as we approach the localized phase.
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
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12. Qualitative phase diagram of a frustrated Hubbard model
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13. 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.
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G. Kotliar and S. Savrasov Strongly Correlated Systems, A. M. Tsvelik Ed Kluwer Academic conmat/ S. Y. Savrasov, G. Kotliar, Phys. Rev. B 69, (2004)
Full implementation in the context of a a one orbital model. P Sun and G. Kotliar Phys. Rev. B 66, (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, (2003) .
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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……….
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Start from functional of G and W (Chitra and Kotliar (2000), Ambladah et. al.
Make local or cluster approximation on F.
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) .
Truncate the W operator act on the H sector only. i.e.
Replace W(w) or V0(w) 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)
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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/
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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 mB, ,S=5/2 Ms=5 mB
Mtot=0
More realistic calculations, (GGA+U),itineracy, crystal fields G7 +G8, ML=-3.9 Mtot= 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
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19. Total Energy as a function of volume for Pu W (ev) vs (a.u. 27.2 ev)
Bistability of a material near the Mott transition. Model realization of the Johanssen ideas.
Central for understanding the physics of Pu.. New paradigm for thinking, about materials.
(Savrasov, Kotliar, Abrahams, Nature ( 2001)
Non magnetic correlated state of fcc Pu.
Zein Savrasov and Kotliar (2004)

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Ground State Theory of d-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.”
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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, 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, (2003) . There are large fluctuating (disordered moments) in Pu. Accounts for the correct volumes and bulk moduli across the actinide series.
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22. Electronic spectra and Total energy. LDA,LDA+U, LDA+DMFT
Method
E vs V
LDA
LDA+U LRO
DMFT

23. Double well structure and d Pu
Qualitative explanation of negative thermal expansion[ G. Kotliar J.Low Temp. Physvol.126, (2002)]See also A . Lawson et.al.Phil. Mag. B 82, 1837 ]
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24. A. C. Lawson et. al. LA UR 04-6008 F(T,V)=Fphonons+Finvar
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25. Invar model A. C. Lawson et. al. LA UR 04-6008
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27. Prediction of the Invar Model
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28. DMFT and the Invar Model
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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.
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31. DMFT Phonons in fcc d-Pu
Notice the agreement. Usefulness of theory. Notice the discreapancy. Scientfici opportunity.
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
( Dai, Savrasov, Kotliar,Ledbetter, Migliori, Abrahams, Science, 9 May 2003)
(experiments from Wong et.al, Science, 22 August 2003)

32. 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.
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33. 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.
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Epsilon Plutonium.
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35. 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, (2002); (and neglecting electronic entropy). TC ~ 600 K.
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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.
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Alpha and delta Pu Photoemission Spectra DMFT(Savrasov et.al.) EXP (Arko Joyce Morales Wills Jashley PRB 62, 1773 (2000))
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38. J. Tobin et. al. PHYSICAL REVIEW B 68, 155109 ,2003
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39. K. Haule , Pu- photoemission with DMFT using vertex corrected NCA.
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40. 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. Optical Conductivity Temperature dependence.
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Origin of the features. K. Held, A.K. McMahan, and R.T. Scalettar, Phys. Rev.Lett. 87, (2001); A.K. McMahan, K. Held, and R.T. Scalettar, Phys. Rev. B 67, (2003). K. Haule et. al. (2004)
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44. Americium under pressure (Lindbaum et. al. PRB 2003)
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45. LDA+DMFT calculations for fcc Americium S. Murthy (2003)
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47. LDA+DMFT spectra. Notice the rapid occupation of the f7/2 band, (5f)7
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49. 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.
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J. C. Griveau et. al. (2004)
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51. Mott transition in open (right) and closed (left) shell systems.
g T Tc
Log[2J+1]
??? Uc
S=0 U U
g ~1/(Uc-U)
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Griveau et.al. (2004)
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55. H.Q. Yuan et. al. CeCu2(Si2-x Gex). Am under pressure Griveau et. al.
Superconductivity due to valence fluctuations ?
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56. 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.
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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.
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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.
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a 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.
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Pu is not MAGNETIC, alpha and delta have comparable susceptibility and specifi heat.
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61. Anomalous Resistivity
PRL 91, (2003)
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64. 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.
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LDA+DMFT References
Anisimov Poteryaev Korotin Anhokin and Kotliar J. Phys. Cond. Mat. 35, 7359 (1997).
Lichtenstein and Katsenelson PRB (1998).
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).
A. Lichtenstein M. Katsnelson and G. Kotliar (2002)
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66. 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 .
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67. THE STATE UNIVERSITY OF NEW JERSEY
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.!
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68. 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.
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Transverse Phonon along (0,1,1) in epsilon Pu in self consistent Born approximation.
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70. 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. LDA+DMFT Self-Consistency loopU
DMFT
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Further approximations, use approximate impuirity solvers rational Interpolative Perturbative Theory. Savrasov Udovenko Villani Haule and Kotliar . Cond-matt
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74. Benchmarking SUNCA, V. Udovenko and K. Haule
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