1. Electronic Structure of Correlated Materials : a DMFT Perspective
Gabriel Kotliar
Physics Department and
Center for Materials Theory
Rutgers University
and KITP
Institute for Theoretical Physics
UCSB Santa Barbara
Realistic. System specific. Qualitative. Quantitative. Universal vs Non Universal.
Brookhaven National Laboratory
September 12th 2002
Supported by the NSF DMR

2. THE STATE UNIVERSITY OF NEW JERSEY
Outline
The Mott transition problem and electronic structure.
Dynamical Mean Field Theory
Model Hamiltonian Studies of the Mott transition. Universal aspects.
System specific studies of materials. LDA+DMFT. Some case studies.
Outlook
10 minutes intro
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3. Weakly correlated electrons:band theory.
Simple conceptual picture of the ground state, excitation spectra, transport properties of many systems (simple metals, semiconductors,….).
A methods for performing quantitative calculations. (Density functional theory, in various approximations).
Highly desirable, simple qualitative pictures of the states. Useful in predicting chemical trends.
Quantitative tools with great accuracy.
Modify and control weakly correlated materials. Electronics, is one the success stories
Of condensed matter physics of the past century.
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4. The electron in a solid: wave picture
Momentum Space (Sommerfeld)
Maximum metallic resistivity 200 mohm cm
See if I still have bands in ens talk.
Standard model of solids (Bloch, Landau) Periodic potential, waves form bands , k in Brillouin zone . Interactions renormalize away.
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5. Standard Model of Solids
Qualitative predictions: low temperature dependence of thermodynamics and transport.
Optical response, transition between the bands.
Qualitative predictions: filled bands give rise to insulting behavior. Compounds with odd number of electrons are metals.
Quantitative tools: Density Functional Theory with approximations suggested by the Kohn Sham formulation, (LDA GGA) is a successful computational tool for the total energy. Good starting point for perturbative calculation of spectra,eg. GW. Kinetic equations yield transport coefficients.
Put the qualitative idea of dft, functional of a density. Stress the power and the accuracy.
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6. Kohn Sham reference system
Excellent starting point for computation of spectra in perturbation theory in screened Coulomb interaction GW.
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7. Success story : Density Functional Linear Response
Tremendous progress in ab initio modelling of lattice dynamics
& electron-phonon interactions has been achieved
(Review: Baroni et.al, Rev. Mod. Phys, 73, 515, 2001)
(Savrasov, PRB 1996)

8. The electron in a solid: particle picture.
Array of hydrogen atoms is insulating if a>>aB. Mott: correlations localize the electron
e_ e_ e_ e_
Superexchange
Add a better picture of hydrogen atoms. See Motts book.
Think in real space , solid collection of atoms
High T : local moments, Low T spin-orbital order
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9. Mott : Correlations localize the electron
Low densities, electron behaves as a particle,use atomic physics, real space
One particle excitations: Hubbard Atoms: sharp excitation lines corresponding to adding or removing electrons. In solids they broaden by their incoherent motion, Hubbard bands (eg. bandsNiO, CoO MnO….)
H H H+ H H H motion of H+ forms the lower Hubbard band
H H H H- H H motion of H_ forms the upper Hubbard band
Quantitative calculations of Hubbard bands and exchange constants, LDA+ U, Hartree Fock. Atomic Physics.
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10. Photoemission spectroscopy.
Measures density of states for (BIS)adding and (PES) removing electrons
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11. Localization vs Delocalization Strong Correlation Problem
A large number of compounds with electrons in partially filled shells, are not close to the well understood limits (localized or itinerant). Non perturbative problem.
These systems display anomalous behavior (departure from the standard model of solids).
Neither LDA or LDA+U or Hartree Fock work well.
Dynamical Mean Field Theory: Simplest approach to electronic structure, which interpolates correctly between atoms and bands. Treats QP bands and Hubbard bands.
Limits do not match except for the case that we have filled atomic shells
In which case band theory solve the problem.
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12. Strong correlation anomalies
Metals with resistivities which exceed the Mott Ioffe Reggel limit.
Transfer of spectral weight which is non local in frequency.
Dramatic failure of DFT based approximations in predicting physical properties.
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13. Correlated Materials do big things
Huge resistivity changes V2O3.
Copper Oxides. .(La2-x Bax) CuO4 High Temperature Superconductivity.150 K in the Ca2Ba2Cu3HgO8 .
Uranium and Cerium Based Compounds. Heavy Fermion Systems,CeCu6,m*/m=1000
(La1-xSrx)MnO3 Colossal Magneto-resistance.
Put some materials formulae for quaternary
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14. Strongly Correlated Materials.
Large thermoelectric response in CeFe4 P12 (H. Sato et al. cond-mat ). Ando et.al.
NaCo2-xCuxO4 Phys. Rev. B 60, (1999).
Large and ultrafast optical nonlinearities Sr2CuO3 (T Ogasawara et.a Phys. Rev. Lett. 85, 2204 (2000) )
Get tokura osgawara PRL, and the science express, and ando prl , and formula units. Theory will play an important role in optimizing their physical properties
Ando et. Al. Phys. Rev. B 60, (1999).
Ogasawara Phys. Rev. Lett. 85, 2204 (2000)
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15. THE STATE UNIVERSITY OF NEW JERSEY
Mott transition in V2O3 under pressure or chemical substitution on V-site
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16. THE STATE UNIVERSITY OF NEW JERSEY
Mott transition in layered organic conductors S Lefebvre et al. cond-mat/ , Phys. Rev. Lett. 85, 5420 (2000)
Update reference Phys. Rev. Lett. 85, 5420 (2000)
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17. Failure of the Standard Model: NiSe2-xSx
Miyasaka and Takagi (2000)
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18. Mean-Field : Classical vs Quantum
Classical case
Quantum case
Animate, and expand.
A. Georges, G. Kotliar (1992)
Phys. Rev. B 45, 6497
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19. Limit of large lattice coordination
Metzner Vollhardt, 89
Muller-Hartmann 89
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20. DMFT Impurity cavity construction
animate
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21. THE STATE UNIVERSITY OF NEW JERSEY
E-DMFT references
H. Kajueter and G. Kotliar (unpublished and Kajuter’s Ph.D thesis).
Q. Si and Smith PRL [analysis of quantum critical points]
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22. C-DMFT: test in one dimension. (Bolech, Kancharla GK cond-mat 2002)
Gap vs U, Exact solution Lieb and Wu, Ovshinikov
Nc=2 CDMFT
vs Nc=1
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23. THE STATE UNIVERSITY OF NEW JERSEY
Insights from DMFT
Low temperature Ordered phases . Stability depends on chemistry and crystal structure
High temperature behavior around Mott endpoint, more universal regime, captured by simple models treated within DMFT. Role of magnetic frustration.
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24. Schematic DMFT phase diagram Hubbard model (partial frustration)
M. Rozenberg G. Kotliar H. Kajueter G Thomas D. Rapkikne J Honig and P Metcalf Phys. Rev. Lett. 75, 105 (1995)
Hydrogen atom of strong correlation problme.
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25. THE STATE UNIVERSITY OF NEW JERSEY
Kuwamoto Honig and Appell PRB (1980) M. Rozenberg G. Kotliar H. Kajueter G Thomas D. Rapkikne J Honig and P Metcalf Phys. Rev. Lett. 75, 105 (1995)
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26. THE STATE UNIVERSITY OF NEW JERSEY
Mott transition in layered organic conductors S Lefebvre et al. cond-mat/ , Phys. Rev. Lett. 85, 5420 (2000)
Update reference Phys. Rev. Lett. 85, 5420 (2000)
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27. THE STATE UNIVERSITY OF NEW JERSEY
Phase Diag: Ni Se2-x Sx
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28. THE STATE UNIVERSITY OF NEW JERSEY
Insights from DMFT
The Mott transition is driven by transfer of spectral weight from low to high energy as we approach the localized phase
Control parameters: doping, temperature,pressure…
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29. Evolution of the Spectral Function with Temperature
Anomalous transfer of spectral weight connected to the proximity to the Ising Mott endpoint (Kotliar Lange and Rozenberg Phys. Rev. Lett. 84, 5180 (2000)
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30. THE STATE UNIVERSITY OF NEW JERSEY
ARPES measurements on NiS2-xSex Matsuura et. al Phys. Rev B 58 (1998) Doniach and Watanabe Phys. Rev. B 57, 3829 (1998) .
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31. THE STATE UNIVERSITY OF NEW JERSEY
Mott transition in V2O3 under pressure or chemical substitution on V-site
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32. Anomalous transfer of optical spectral weight V2O3
:M Rozenberg G. Kotliar and H. Kajuter Phys. Rev. B 54, 8452 (1996).
M. Rozenberg G. Kotliar H. Kajueter G Tahomas D. Rapkikne J Honig and P Metcalf Phys. Rev. Lett. 75, 105 (1995)
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33. Anomalous Spectral Weight Transfer: Optics
Below energy
ApreciableT dependence found.
Schlesinger et.al (FeSi) PRL 71 ,1748 , (1993) B Bucher et.al. Ce2Bi4Pt3PRL 72, 522 (1994), Rozenberg et.al. PRB 54, 8452, (1996).
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34. THE STATE UNIVERSITY OF NEW JERSEY
ARPES measurements on NiS2-xSex Matsuura et. Al Phys. Rev B 58 (1998) Doniaach and Watanabe Phys. Rev. B 57, 3829 (1998) .
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35. THE STATE UNIVERSITY OF NEW JERSEY
Anomalous transfer of optical spectral weight, NiSeS. [Miyasaka and Takagi]
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36. Anomalous Resistivity and Mott transition Ni Se2-x Sx
Insights from DMFT: think in term of spectral
functions (branch cuts) instead of well defined
QP (poles )
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37. THE STATE UNIVERSITY OF NEW JERSEY
Qualitative phase diagram in the U, T , m plane (two band Kotliar Murthy Rozenberg PRL (2002).
Coexistence regions between localized and delocalized spectral functions.
Boundary in beyond which perturbation theory stops working.
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38. THE STATE UNIVERSITY OF NEW JERSEY
QMC calculationof n vs m (Kotliar Murthy Rozenberg PRL 2002, 2 band, U=3.0)
k diverges at generic Mott endpoints
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39. THE STATE UNIVERSITY OF NEW JERSEY
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40. THE STATE UNIVERSITY OF NEW JERSEY
Mott transition in layered organic conductors S Lefebvre et al. cond-mat/ , Phys. Rev. Lett. 85, 5420 (2000)
Update reference Phys. Rev. Lett. 85, 5420 (2000)
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41. THE STATE UNIVERSITY OF NEW JERSEY
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42. THE STATE UNIVERSITY OF NEW JERSEY
Insights from DMFT
Mott transition as a bifurcation of an effective action
Important role of the incoherent part of the spectral function at finite temperature
Physics is governed by the transfer of spectral weight from the coherent to the incoherent part of the spectra. Real and momentum space.
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43. Two roads for ab-initio calculation of electronic structure of strongly correlated materials
Crystal structure +Atomic positions
Model Hamiltonian
Correlation Functions Total Energies etc.
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44. THE STATE UNIVERSITY OF NEW JERSEY
Realistic Calculations of the Electronic Structure of Correlated materials
Combinining DMFT with state of the art electronic structure methods to construct a first principles framework to describe complex materials.
Anisimov Poteryaev Korotin Anhokin and Kotliar J. Phys. Cond. Mat. 35, 7359 (1997)
Savrasov Kotliar and Abrahams Nature 410, 793 (2001))
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45. THE STATE UNIVERSITY OF NEW JERSEY
Combining LDA and 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, subtract this out by shifting the heavy level (double counting term)
The U matrix can be estimated from first principles or viewed as parameters
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46. THE STATE UNIVERSITY OF NEW JERSEY
Materials…… Pu
Fe, Ni,
La1-x Srx TiO3
NiO
…………….
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47. THE STATE UNIVERSITY OF NEW JERSEY
Case study in f electrons, Mott transition in the actinide series. B. Johanssen 1974 Smith and Kmetko Phase Diagram 1984.
f electrons in Th Pr U Np are itinerant . From Am on they are localized. Pu is at the boundary.
Pu has a simple cubic fcc structure,the d phase which is easily stabilized over a wide region in the T,p phase diagram.
The d phase is non magnetic.
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48. Problems with density functional treatements of d Pu
DFT in the LDA or GGA is a well established tool for the calculation of ground state properties.
Many studies (APW Freeman, Koelling 1972, ASA and FP-LMTO, Soderlind et. al 1990, Kollar et.al 1997, Boettger et.al 1998, Wills et.al. 1999) show
an equilibrium volume of the d phase Is 35% lower than experiment
This is the largest discrepancy ever known in DFT based calculations.
LSDA predicts magnetic long range order which is not observed experimentally (Solovyev et.al.)
If one treats the f electrons as part of the core LDA overestimates the volume by 30%
Weak correlation picture for alpha phase.
short

49. THE STATE UNIVERSITY OF NEW JERSEY
Pu DMFT total energy vs Volume (Savrasov Kotliar and Abrahams Nature 410, 793 (2001)
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50. Lda vs Exp Spectra (Joyce et.al.)
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51. Pu Spectra DMFT(Savrasov) EXP (Joyce , Arko et.al)
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52. THE STATE UNIVERSITY OF NEW JERSEYPu
Alpha and delta are strongly correlated,
The coexistence in the toy model, give rise to
two distinct phases in the realistic calculation.
In progress: phonon spectrum, epsilon phase……
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53. THE STATE UNIVERSITY OF NEW JERSEY
LaxSr1-xO3
Adding holes to a Mott insulator in three dimensions.
Canonical example of a Brinkman Rice system.
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54. (Tokura et. Al. 1993)A doped Mott insulator:LaxSr1-xO3
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55. DMFT calculation U near the Mott transition, Rozenberg et.al 94
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56. Hall Coefficient, electron like.
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57. La1-xSrxTiO3 photoemission
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58. Evolution of spectra with doping U=4
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59. THE STATE UNIVERSITY OF NEW JERSEY
Optical conductivity
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60. Realistic Computation of Optical Properties : La1-xSrxTiO3
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61. THE STATE UNIVERSITY OF NEW JERSEY
Case study Fe and Ni
Archetypical itinerant ferromagnets
LSDA predicts correct low T moment
Band picture holds at low T
Main puzzle: at high temperatures c has a Curie Weiss law with a moment larger than the ordered moment.
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62. THE STATE UNIVERSITY OF NEW JERSEY
Iron and Nickel: crossover to a real space picture at high T (Lichtenstein, Katsnelson and Kotliar Phys Rev. Lett 87, , 2001)
Short Reference to PRL.
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63. THE STATE UNIVERSITY OF NEW JERSEY
Iron and Nickel:magnetic properties (Lichtenstein, Katsenelson,GK PRL 01)
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64. THE STATE UNIVERSITY OF NEW JERSEY
Ni and Fe: theory vs exp
meff / mB high T moment
Fe 3.1 (theory) 3.12 (expt)
Ni 1.5 (theory) (expt)
Curie Temperature Tc
Fe ( theory) (expt)
Ni (theory) (expt)
Short Bcc iron expt spin wave stiffness 314 mev/A^2
Fcc ni expt spin wave, 550 mev/A^2
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65. THE STATE UNIVERSITY OF NEW JERSEY
Fe and Ni
Consistent picture of Fe (more localized) and Ni
(more correlated)
Satellite in minority band at 6 ev, 30 % reduction of bandwidth, exchange splitting reduction .3 ev
Spin wave stiffness controls the effects of spatial flucuations, it is about twice as large in Ni and in Fe
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66. THE STATE UNIVERSITY OF NEW JERSEY
Photoemission Spectra and Spin Autocorrelation: Fe (U=2, J=.9ev,T/Tc=.8) (Lichtenstein, Katsenelson,Kotliar Phys Rev. Lett 87, , 2001)
Put correct reference
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67. Photoemission and T/Tc=.8 Spin Autocorrelation: Ni (U=3, J=.9 ev)
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68. Failures of lda for ground state properties.
NiO dielectric constant. LSDA:35.7 Exp:5.7
Lattice dynamics cannot be predicted:
Optical G-phonon in MnO within LSDA: 3.04 THz,
Experimentally: 7.86 THz (Massidda, et.al, PRL 1999)
Bulk modulus for metallic Plutonium is one order of
magnitude too large within LDA (214 GPa vs. 30 GPa)
Also elastic constants are off.
(Bouchet, et.al, J.Phys.C, 2001)

69. THE STATE UNIVERSITY OF NEW JERSEY
Functional approach allows computation of linear response.(S. Savrasov and GK 2002
Apply to NiO, canonical Mott insulator.
U=8 ev, J=.9ev
Simple Impurity solver Hubbard 1.
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70. Results for NiO: Phonons
Solid circles – theory, open circles – exp. (Roy et.al, 1976)

71. NiO U=8ev, J=1ev, Savrasov and GK (2002)
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72. THE STATE UNIVERSITY OF NEW JERSEY
Summary
Introduction to strongly correlated electrons
Dynamical Mean Field Theory
Model Hamiltonian Studies. Universal aspects insights from DMFT
System specific studies: LDA+DMFT
Outlook
10 minutes intro
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73. Very Partial list of application of realistic DMFT to materials
QP bands in ruthenides: A. Liebsch et al (PRL 2000)
N phase of Pu: S. Savrasov et al (Nature 2001)
MIT in V2O3: K. Held et al (PRL 2001)
Magnetism of Fe, Ni: A. Lichtenstein et al PRL (2001)
J-G transition in Ce: K. Held et al (PRL 2000); M. Zolfl et al PRL (2000).
3d doped Mott insulator La1-xSrxTiO3 (Anisimov et.al 1997, Nekrasov et.al. 1999, Udovenko et.al 2002)
………………..
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74. Acknowledgements: Development of DMFT
Collaborators: V. Anisimov, R. Chitra, V. Dobrosavlevic, D. Fisher, A. Georges, H. Kajueter, W.Krauth, E. Lange, A. Lichtenstein, G. Moeller, Y. Motome, G. Palsson, M. Rozenberg, S. Savrasov, Q. Si, V. Udovenko, X.Y. Zhang
Support: National Science Foundation.
Work on Pu: Departament of Energy and LANL.
Work on Fe and Ni: Office of Naval Research
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75. THE STATE UNIVERSITY OF NEW JERSEY
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76. THE STATE UNIVERSITY OF NEW JERSEY
Challenges
Short Range Magnetic Correlations without magnetic order.
Single Site DMFT does not capture these effects
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77. Pu: Anomalous thermal expansion (J. Smith LANL)
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78. THE STATE UNIVERSITY OF NEW JERSEY
Problems with LDA
DFT in the LDA or GGA is a well established tool for the calculation of ground state properties.
Many studies (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) give
an equilibrium volume of the d phase Is 35% lower than experiment
This is the largest discrepancy ever known in DFT based calculations.
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79. THE STATE UNIVERSITY OF NEW JERSEY
Problems with LDA
LSDA predicts magnetic long range order which is not observed experimentally (Solovyev et.al.)
If one treats the f electrons as part of the core LDA overestimates the volume by 30%
Notice however that LDA predicts correctly the volume of the a phase of Pu, when full potential LMTO (Soderlind Eriksson and Wills). This is usually taken as an indication that a Pu is a weakly correlated system
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80. Conventional viewpoint
Alpha Pu is a simple metal, it can be described with LDA + correction. 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, 4 5f electrons are band-like while one 5f electron is deloclized.
Same situation in LDA + U (Savrasov andGK Bouchet et. al. [Bouchet’s talk]) .Delta Pu has U=4,Alpha Pu has U =0.
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81. Problems with the conventional viewpoint of Pu
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.
Only the structural and elastic properties are completely different.
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82. THE STATE UNIVERSITY OF NEW JERSEY
Pu Specific Heat
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83. Anomalous Resistivity J. Smith LANL
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84. MAGNETIC SUSCEPTIBILITY
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85. THE STATE UNIVERSITY OF NEW JERSEY
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.
This result resolves one of the basic paradoxes in the physics of Pu.
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86. Pu: DMFT total energy vs Volume
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87. THE STATE UNIVERSITY OF NEW JERSEY
Lda vs Exp Spectra
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88. Pu Spectra DMFT(Savrasov) EXP (Arko et. Al)
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89. THE STATE UNIVERSITY OF NEW JERSEY
PU: ALPHA AND DELTA
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90. THE STATE UNIVERSITY OF NEW JERSEY
Case study Fe and Ni
Archetypical itinerant ferromagnets
LSDA predicts correct low T moment
Band picture holds at low T
But at high temperatures, they resemble more a collection of atoms with reduced moment.
The crossover from low T to high T requires DMFT.
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91. THE STATE UNIVERSITY OF NEW JERSEY
Iron and Nickel: crossover to a real space picture at high T (Lichtenstein, Katsnelson and GK)
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92. However not everything in low T phase is OK as far as LDA goes..
Magnetic anisotropy puzzle. LDA predicts the incorrect easy axis for Nickel .(instead of 111)
LDA Fermi surface has features which are not seen in DeHaas Van Alphen ( Lonzarich)
Use LDA+ U to tackle these refined issues, (WE cannot be resolved with DMFT, compare parameters with Lichtenstein’s )
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93. THE STATE UNIVERSITY OF NEW JERSEY
Some Earlier Work:
Kondorskii and E Straube Sov Phys. JETP 36, 188 (1973)
G. H Dallderop P J Kelly M Schuurmans Phys. Rev. B 41, (1990)
Trygg, Johansson Eriksson and Wills Phys. Rev. Lett (1995) Schneider M Erickson and Jansen J. Appl Phys (1997)
I Solovyev, Lichenstein Terakura Phys. Rev. Lett 80, 5758 (LDA+U +SO Coupling)…….
Present work : Imseok Yang, S Savrasov and GK
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94. Origin of Magnetic Anisotropy
Spin orbit coupling L.S
L is a variable which is sensitive to correlations, a reminder of the atomic physics
Crystal fields quench L, interactions enhance it,
T2g levels carry moment, eg levels do not any redistribution of these no matter how small will affect L.
Both J and U matter !
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95. Magnetic anisotropy of Fe and Ni LDA+ U
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96. THE STATE UNIVERSITY OF NEW JERSEY
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
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97. THE STATE UNIVERSITY OF NEW JERSEY
DMFT Review: A. Georges, G. Kotliar, W. Krauth and M. Rozenberg Rev. Mod. Phys. 68,13 (1996)]
Weiss field
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98. THE STATE UNIVERSITY OF NEW JERSEY
Resistivities
THE STATE UNIVERSITY OF NEW JERSEY
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99. THE STATE UNIVERSITY OF NEW JERSEY
Resistivities
THE STATE UNIVERSITY OF NEW JERSEY
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100. THE STATE UNIVERSITY OF NEW JERSEY
LiVO4
THE STATE UNIVERSITY OF NEW JERSEY
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101. THE STATE UNIVERSITY OF NEW JERSEY
Thermodynamics LiVO4
THE STATE UNIVERSITY OF NEW JERSEY
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102. Resistivity saturation
THE STATE UNIVERSITY OF NEW JERSEY
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103. Anomalous Resistivities: Doped Hubbard Model G. Palsson 1998
NCA
IPT
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THE STATE UNIVERSITY OF NEW JERSEY
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104. Anomalous Resistivities: Doped Hubbard Model (QMC)
Prushke and Jarrell 1993
THE STATE UNIVERSITY OF NEW JERSEY
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105. THE STATE UNIVERSITY OF NEW JERSEY
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106. THE STATE UNIVERSITY OF NEW JERSEY
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107. THE STATE UNIVERSITY OF NEW JERSEY
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 et.al) , CDMFT ( Savrasov Palsson and GK )
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)
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108. THE STATE UNIVERSITY OF NEW JERSEY
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, ………
THE STATE UNIVERSITY OF NEW JERSEY
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109. THE STATE UNIVERSITY OF NEW JERSEY
Ni moment
THE STATE UNIVERSITY OF NEW JERSEY
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110. THE STATE UNIVERSITY OF NEW JERSEY
Fe moment
THE STATE UNIVERSITY OF NEW JERSEY
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111. Magnetic anisotropy vs U , J=.95 Ni
THE STATE UNIVERSITY OF NEW JERSEY
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112. Magnetic anisotropy Fe J=.8
THE STATE UNIVERSITY OF NEW JERSEY
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113. THE STATE UNIVERSITY OF NEW JERSEY
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114. Conventional viewpoint
Alpha Pu is a simple metal, it can be described with LDA + correction. 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, 4 5f electrons are band-like while one 5f electron is deloclized.
Same situation in LDA + U (Savrasov andGK Bouchet et. al. [Bouchet’s talk]) .Delta Pu has U=4,Alpha Pu has U =0.
THE STATE UNIVERSITY OF NEW JERSEY
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115. THE STATE UNIVERSITY OF NEW JERSEY
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116. Delocalization-Localization across the actinide series
f electrons in Th Pr U Np are itinerant . From Am on they are localized. Pu is at the boundary.
Pu has a simple cubic fcc structure,the d phase which is easily stabilized over a wide region in the T,p phase diagram.
The d phase is non magnetic.
Many LDA , GGA studies ( Soderlind et. Al 1990, Kollar et.al 1997, Boettger et.al 1998, Wills et.al. 1999) give an equilibrium volume of the d phase Is 35% lower than experiment
This is one of the largest discrepancy ever known in DFT based calculations.
THE STATE UNIVERSITY OF NEW JERSEY
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117. Pu: Anomalous thermal expansion (J. Smith LANL)
THE STATE UNIVERSITY OF NEW JERSEY
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118. THE STATE UNIVERSITY OF NEW JERSEY
Problems with LDA
DFT in the LDA or GGA is a well established tool for the calculation of ground state properties.
Many studies (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) give
an equilibrium volume of the d phase Is 35% lower than experiment
This is the largest discrepancy ever known in DFT based calculations.
THE STATE UNIVERSITY OF NEW JERSEY
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119. THE STATE UNIVERSITY OF NEW JERSEY
Problems with LDA
LSDA predicts magnetic long range order which is not observed experimentally (Solovyev et.al.)
If one treats the f electrons as part of the core LDA overestimates the volume by 30%
Notice however that LDA predicts correctly the volume of the a phase of Pu, when full potential LMTO (Soderlind Eriksson and Wills). This is usually taken as an indication that a Pu is a weakly correlated system
THE STATE UNIVERSITY OF NEW JERSEY
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120. Conventional viewpoint
Alpha Pu is a simple metal, it can be described with LDA + correction. 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, 4 5f electrons are band-like while one 5f electron is deloclized.
Same situation in LDA + U (Savrasov andGK Bouchet et. al. [Bouchet’s talk]) .Delta Pu has U=4,Alpha Pu has U =0.
THE STATE UNIVERSITY OF NEW JERSEY
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121. Problems with the conventional viewpoint of Pu
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.
Only the structural and elastic properties are completely different.
THE STATE UNIVERSITY OF NEW JERSEY
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122. THE STATE UNIVERSITY OF NEW JERSEY
Pu Specific Heat
THE STATE UNIVERSITY OF NEW JERSEY
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123. Anomalous Resistivity J. Smith LANL
THE STATE UNIVERSITY OF NEW JERSEY
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124. MAGNETIC SUSCEPTIBILITY
THE STATE UNIVERSITY OF NEW JERSEY
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125. THE STATE UNIVERSITY OF NEW JERSEY
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.
This result resolves one of the basic paradoxes in the physics of Pu.
THE STATE UNIVERSITY OF NEW JERSEY
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126. THE STATE UNIVERSITY OF NEW JERSEY
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127. THE STATE UNIVERSITY OF NEW JERSEY
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128. F Sum of local 2PI graphs with local U matrix and local G
LDA+DMFT functional
F Sum of local 2PI graphs with local U matrix and local G
THE STATE UNIVERSITY OF NEW JERSEY
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129. THE STATE UNIVERSITY OF NEW JERSEY
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130. THE STATE UNIVERSITY OF NEW JERSEY
Comments on LDA+DMFT
Static limit of the LDA+DMFT functional , with F= FHF reduces to LDA+U
Removes inconsistencies and shortcomings of this approach. DMFT retain correlations effects in the absence of orbital ordering.
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.
THE STATE UNIVERSITY OF NEW JERSEY
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131. Anomalous Resistivities: Doped Hubbard Model (QMC)
Prushke and Jarrell 1993
THE STATE UNIVERSITY OF NEW JERSEY
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132. Anomalous Resistivities: Doped Hubbard Model G. Palsson 1998
NCA
IPT
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THE STATE UNIVERSITY OF NEW JERSEY
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133. DMFT: Methods of Solution
THE STATE UNIVERSITY OF NEW JERSEY
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134. Conjugate field, VKS(r)
LDA functional
Conjugate field, VKS(r)
Extraordinarily successful, in weakly correlated materials. Kohn Hohenberg. [nobel prize]
THE STATE UNIVERSITY OF NEW JERSEY
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135. Minimize LDA functional
Kohn Sham eigenvalues, auxiliary quantities.
THE STATE UNIVERSITY OF NEW JERSEY
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136. Anomalous transfer of spectral weight heavy fermions
THE STATE UNIVERSITY OF NEW JERSEY
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137. Anomalous transfer of spectral weight
THE STATE UNIVERSITY OF NEW JERSEY
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138. Anomalous transfer of spectral weigth heavy fermions
THE STATE UNIVERSITY OF NEW JERSEY
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139. THE STATE UNIVERSITY OF NEW JERSEY
V2O3 resistivity
THE STATE UNIVERSITY OF NEW JERSEY
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140. LDA+DMFT Self-Consistency loopU
DMFT
THE STATE UNIVERSITY OF NEW JERSEY
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141. THE STATE UNIVERSITY OF NEW JERSEY
DMFT Impurity cavity construction: A. Georges, G. Kotliar, PRB, (1992)]
animate
Weiss field
THE STATE UNIVERSITY OF NEW JERSEY
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142. THE STATE UNIVERSITY OF NEW JERSEY
Landau Functional
G. Kotliar EPJB (1999)
THE STATE UNIVERSITY OF NEW JERSEY
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143. THE STATE UNIVERSITY OF NEW JERSEY
Ni and Fe: theory vs exp
m( T=.9 Tc)/ mB ordered moment
Fe ( theory) (expt)
Ni (theory) (expt)
meff / mB high T moment
Fe 3.1 (theory) 3.12 (expt)
Ni 1.5 (theory) (expt)
Curie Temperature Tc
Fe ( theory) (expt)
Ni (theory) (expt)
Short Bcc iron expt spin wave stiffness 314 mev/A^2
Fcc ni expt spin wave, 550 mev/A^2
Expt .9tc 1.55 fe otc 2.2 ratio .7
Expt .9tc .35 ni t=0 .6 ratio .58
Multiply by get 2.5 theory vs 2.2 expt
And vs .6
THE STATE UNIVERSITY OF NEW JERSEY
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144. THE STATE UNIVERSITY OF NEW JERSEY
Problems with LDA
DFT in the LDA or GGA is a well established tool for the calculation of ground state properties.
Many studies (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) give
an equilibrium volume of the d phase Is 35% lower than experiment
This is the largest discrepancy ever known in DFT based calculations.
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS

145. THE STATE UNIVERSITY OF NEW JERSEY
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146. 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 expanding around the the atomic limit. No explicit expression exists.
DFT is useful because good approximations to the exact density functional GDFT[r(r)] exist, e.g. LDA, GGA
A useful approximation to the exact functional can be constructed, the DMFT +LDA functional. Savrasov Kotliar and Abrahams Nature 410, 793 (2001))
THE STATE UNIVERSITY OF NEW JERSEY
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147. Conjugate field, VKS(r)
LDA functional
Conjugate field, VKS(r)
Extraordinarily successful, in weakly correlated materials. Kohn Hohenberg. [nobel prize]
THE STATE UNIVERSITY OF NEW JERSEY
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148. Failure of the Standard Model: Anomalous Spectral Weight Transfer
Optical Conductivity o of FeSi for T=,20,20, and 250 K from Schlesinger et.al (1993)
Neff depends on T
THE STATE UNIVERSITY OF NEW JERSEY
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149. THE STATE UNIVERSITY OF NEW JERSEY
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150. Degenerate Hubbard model
Ising model of correlated electrons
U/t
Doping d or chemical potential
Frustration (t’/t)
T temperature
Mott transition as a function of doping, pressure temperature etc.
THE STATE UNIVERSITY OF NEW JERSEY
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151. THE STATE UNIVERSITY OF NEW JERSEY
Outlook
The Strong Correlation Problem:How to deal with a multiplicity of competing low temperature phases and infrared trajectories which diverge in the IR
Strategy: advancing our understanding scale by scale
Generalized cluster methods to capture longer range magnetic correlations
New structures in k space. Cellular DMFT
THE STATE UNIVERSITY OF NEW JERSEY
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152. Small amounts of Ga stabilize the d phase
THE STATE UNIVERSITY OF NEW JERSEY
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153. THE STATE UNIVERSITY OF NEW JERSEY
Insights from DMFT: think in term of spectral functions (branch cuts) instead of well defined QP (poles )
Resistivity near the metal insulator endpoint ( Rozenberg et.al 1995) exceeds the Mott limit
THE STATE UNIVERSITY OF NEW JERSEY
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154. THE STATE UNIVERSITY OF NEW JERSEY
res st9v9t9es
THE STATE UNIVERSITY OF NEW JERSEY
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155. Conjugate field, VKS(r)
LDA functional
Conjugate field, VKS(r)
Extraordinarily successful, in weakly correlated materials. Kohn Hohenberg. [nobel prize]
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS

156. THE STATE UNIVERSITY OF NEW JERSEY
Hubbard model
Ising model of correlated electrons
U/t
Doping d or chemical potential
Frustration (t’/t)
T temperature
Mott transition as a function of doping, pressure temperature etc.
THE STATE UNIVERSITY OF NEW JERSEY
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157. Dynamical Mean Field Theory, cavity construction
A. Georges G. Kotliar 92
Animate, and expand.
THE STATE UNIVERSITY OF NEW JERSEY
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158. THE STATE UNIVERSITY OF NEW JERSEY
DMFT Impurity cavity construction: A. Georges, G. Kotliar, PRB, (1992)]
animate
Weiss field
THE STATE UNIVERSITY OF NEW JERSEY
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159. Solving the DMFT equations
Wide variety of computational tools (QMC,ED….)Analytical Methods
Extension to ordered states.
Review: A. Georges, G. Kotliar, W. Krauth and M. Rozenberg Rev. Mod. Phys. 68,13 (1996)]
THE STATE UNIVERSITY OF NEW JERSEY
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160. THE STATE UNIVERSITY OF NEW JERSEY
Single site DMFT, functional formulation. Construct a functional of the observable ,local Greens function. Express in terms of Weiss field, (semicircularDOS) [G. Kotliar EBJB 99]
THE STATE UNIVERSITY OF NEW JERSEY
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161. THE STATE UNIVERSITY OF NEW JERSEY
Compressibilty divergence : One band case (Kotliar Murthy and Rozenberg 2002)
THE STATE UNIVERSITY OF NEW JERSEY
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162. THE STATE UNIVERSITY OF NEW JERSEY
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163. Spectral Evolution at T=0 half filling full frustration
X.Zhang M. Rozenberg G. Kotliar (PRL 1993)
THE STATE UNIVERSITY OF NEW JERSEY
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164. THE STATE UNIVERSITY OF NEW JERSEY
Spectral Density Functional : effective action construction (Fukuda, Valiev and Fernando , Chitra and Kotliar).
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. GDFT[r(r)]
Introduce local orbitals, caR(r-R)orbitals, and local GF
G(R,R)(i w) =
The exact free energy can be expressed as a functional of the local Greens function and of the density by introducing G[r(r),G(R,R)(iw)]
A useful approximation to the exact functional can be constructed, the DMFT +LDA functional. Savrasov Kotliar and Abrahams Nature 410, 793 (2001))
THE STATE UNIVERSITY OF NEW JERSEY
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165. Optical spectral weight
THE STATE UNIVERSITY OF NEW JERSEY
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166. LDA+DMFT Self-Consistency loopU
DMFT
THE STATE UNIVERSITY OF NEW JERSEY
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167. Parallel development: Fujimori et.al
THE STATE UNIVERSITY OF NEW JERSEY
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168. THE STATE UNIVERSITY OF NEW JERSEY
C-DMFT
C:DMFT The lattice self energy is inferred
from the cluster self energy.
THE STATE UNIVERSITY OF NEW JERSEY
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