1. Dynamical Mean Field Theory (DMFT) Approach to Correlated Materials
G. Kotliar
Physics Department and Center for Materials Theory
Rutgers
Thanks for coming. No pizza.

2. Outline Introduction to the Dynamical Mean Field ideas and techniques.
Learning about materials with DMFT: (or Mott physics is everywhere ).
Kappa organics <sp>
The Mott transition across the actinide series , Pu- Am <5f>
Ti2O LixCoO3----Fe-Ni <3d>
Ce < 4f>
Correlated electrons. Faced with a difficult problem, physicist try to find a simpler “reference systems”
That they can study, and that can be connected to the system of interest. A reference frame,
For thinking and for computing physical properties of materials. DFT and DMFT.

3. Schematic DMFT phase diagram and DOS of a partially frustrated integer filled Hubbard model and pressure driven Mott transition.
Different way of thinking was generated by the study of the Mott transition at integer filling.
Universality and system specificity. . Bridge atomic physic and band physics. Crossovers with changing degrees of freedom.
Physics Today Vol 57, 53 (2004)

4. Outline, Collaborators, References
Introduction to extensions of DMFT for applications to electronic structure. [ S. Savrasov and Phys. Rev. B 69, (2004) ]
C-DMFTstudy of the Mott transition in kappa organics. [O. Parcollet G. Biroli and GK PRL, 92, (2004) ]
The Mott transition in Actinides Pu [Xi Dai S. Savrasov GK A Migliori H. Ledbetter E. Abrahams Science 300, 953 (2003)] and Am[J. C Griveaux J. Rebizant G. Lander and GK ][Sahana Murthy Ph. D].
Gabriel Kotliar and Dieter Vollhardt
Physics Today 57,(2004)
Rewritte with better reference to sessions. Clean.
Messages: Local Theory is Very Powerful and Flexible. While the method
Is under development it already allows to get quantitative results
And qualitative insights into interesting materials properties.
Illustrate with examples.

5. MIT in Ti2O3[S. Poteryaev S. Lichtenstein and GK cond-mat 0311319 ]
Alpha Gamma transition in Cerium.
K. Haule S. Savrasov V. Udovenko and GK cond-matt 2004.

6. Weakly correlated electrons
Weakly correlated electrons. FLT and DFT, and what goes wrong in correlated materials.
Fermi Liquid . . Correspondence between a system of non interacting particles and the full Hamiltonian.
A band structure is generated (Kohn Sham system).and in many systems this is a good starting point for perturbative computations of the spectra (GW).
Quantitative tools of the standard model. Kohn Sham reference system. Point where we can perturb around. Wrong for accessing the spectra of correlated materials!!!

7. A different paradigm: the area of influence of a quantum critical point

8. Energy Landscape of a Correlated Material and a top to bottom approach to correlated materials.
Configurational Coordinate in the space of Hamiltonians

9. DMFT Cavity Construction: A. Georges and G. Kotliar PRB 45, 6479 (1992). Figure from : G. Kotliar and D. Vollhardt Physics Today 57,(2004)
The self consistent impurity model is a new reference system, to describe strongly correlated materials.

10. Dynamical Mean Field Theory (DMFT) Cavity Construction: A
Dynamical Mean Field Theory (DMFT) Cavity Construction: A. Georges and G. Kotliar PRB 45, 6479 (1992).

11. EDMFT [H. Kajueter Rutgers Ph
EDMFT [H. Kajueter Rutgers Ph.D Thesis Si and Smith PRL77, 3391(1996) R. Chitra and G. Kotliar PRL84,3678 (2000)]
Animate. Stress the meaning of G0.

12. tˆ(K) hopping expressed in the superlattice notations.
Site Cell. Cellular DMFT. C-DMFT. G. Kotliar,S.. Savrasov, G. Palsson and G. Biroli, Phys. Rev. Lett. 87, (2001)
tˆ(K) hopping expressed in the superlattice notations.
Other cluster extensions (DCA Jarrell Krishnamurthy, Katsnelson and Lichtenstein periodized scheme, Nested Cluster Schemes Schiller Ingersent ), causality issues, O. Parcollet, G. Biroli and GK cond-matt (2003)

13. Correlation Functions Total Energies etc.
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.

14. LDA+DMFT V. Anisimov, A. Poteryaev, M. Korotin, A. Anokhin and G
LDA+DMFT V. Anisimov, A. Poteryaev, M. Korotin, A. Anokhin and G. Kotliar, J. Phys. Cond. Mat. 35, 7359 (1997). A Lichtenstein and M. Katsnelson PRB 57, 6884 (1988).
The light, SP (or SPD) electrons are extended, well described by LDA .The heavy, D (or F) electrons are localized treat by DMFT.
LDA Kohn Sham Hamiltonian already contains an average interaction of the heavy electrons, subtract this out by shifting the heavy level (double counting term)
Kinetic energy is provided by the Kohn Sham Hamiltonian (sometimes after downfolding ). The U matrix can be estimated from first principles of viewed as parameters. Solve resulting model using DMFT.

15. Ir>=|R, r> Double loop in Gloc and Wloc
Functional formulation. Chitra and Kotliar (2001), Savrasov and Kotliarcond- matt (2003).
Ir>=|R, r>
First insert the correct form of functional. Then write the Phi_EDMFT W_loc W_nloc G_loc G_nloc. Spectral density
Functional reference. Double look in G_loc and W_loc. Total energies. Spectra.
Double loop in Gloc and Wloc

16. Next Step: GW+EDMFT S. Savrasov and GK. (2001)
Next Step: GW+EDMFT S. Savrasov and GK.(2001). in New Theoretical Approaches to Strongly Correlated Systems, A.M. Tsvelik Ed., Kluwer Academic Publishers , (2001)) W W
.P Sun and G. Kotliar Phys. Rev. B 66, (2002) Phys. Rev. Lett. 91, (2003) Biermann et.al. PRL 90, (2003)
Put better references in here. Say we obtain sigma_nloc G_nloc perturbative. Say test only in models with short range interaction.

17. Impurity model representability of spectral density functional.

18. LDA+DMFT Self-Consistency loop. S. Savrasov and G
LDA+DMFT Self-Consistency loop. S. Savrasov and G. Kotliar (2001) and cond-matt E U
DMFT

19. Impurity Solvers. Hubbard I. Fye Hirsch Quantum Montecarlo.
Interpolative schemes for the self energy. H. Kajueter and G. Kotliar PRL (1996). cond-mat/ V. Oudovenko, K. Haule, S. Savrasov D. Villani and G. Kotliar.
Extensions of NCA. Th. Pruschke and N. Grewe, Z. Phys. B: Condens. Matter 74, 439, SUNCA K. Haule, S. Kirchner, J. Kroha, and P. W¨olfle, Phys. Rev. B 64, , (2001). K. Haule et. al. (2004)

20. How good is the local approximation ?
It becomes exact as the coordination number increases or in the limit of infinite dimensions introduced by Metzner and Vollhardt. PRL 62,34, (1989).
How good is it in low dimensions ? Promising recent developments from theory and experiments.

21. One dimensional Hubbard model
One dimensional Hubbard model . Compare 2 site cluster (in exact diag with Nb=8) vs exact Bethe Anzats, [V. Kancharla C. Bolech and GK PRB 67, (2003)][ [M. CaponeM.Civelli V Kancharla C.Castellani and GK Phys. Rev. B 69, (2004) ]
U/t=4.
Edit. LISA.

22. Applications of DMFT to materials : Goals of the research
Computations develop a first principles method , based on DMFT, capable of predicting physical properties of correlated materials.
Develop a physical picture of the f and spd electrons in Ce and Pu.
Test the theory against experiments.
Bring theory to the point that it plays an equal role in the field of correlated electron materials. Combining theory and experiment.

23. Experimental verifications
Finding the QP , the Hubbard band and the transfer of spectral weight between them in optics and photoemission in different materials.
Exploring the various regimes of the phase diagram, including the Mott endpoint using transport probes.

24. Recent Experiments support qualitative single site DMFT predictions
Limelette et. al.(2003)
Ito et. al. (1995)
Mo et al., Phys. Rev.Lett. 90, (2003).

26. Outline Introduction to the Dynamical Mean Field ideas and techniques.
Learning about materials with DMFT: (or Mott physics is everywhere ).
Kappa organics <sp>
The Mott transition across the actinide series , Pu- Am <5f>
Ti2O LixCoO3----Fe-Ni <3d>
Ce < 4f>
Correlated electrons. Faced with a difficult problem, physicist try to find a simpler “reference systems”
That they can study, and that can be connected to the system of interest. A reference frame,
For thinking and for computing physical properties of materials. DFT and DMFT.

27. k-(ET)2X are across Mott transition
Insulating
anion layer X-
Ground
State
U/t
t’/t
Cu2(CN)3
Mott insulator
8.2
1.06
Cu[N(CN)2]Cl
7.5
0.75
Cu[N(CN)2]Br SC
7.2
0.68
Cu(NCS)2
6.8
0.84
Cu(CN)[N(CN)2]
Ag(CN)2 H2O
6.6
0.60 I3
6.5
0.58
X-1
conducting
ET layer
[(ET)2]+1 t’ t
modeled to triangular lattice
Prof. Kanoda U. Tokyo

28. Mott transition in layered organic conductors S Lefebvre et al
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)

29. Theoretical issue: is there a Mott transition
in the integer filled Hubbard model, and is it
well described by the single site DMFT ?

30. Double Occupancy vs U Study frustrated t t’ model t’/t=.9
CDMFT Parcollet, Biroli GK PRL (2004)
Study frustrated t t’ model t’/t=.9

31. Evolution of the spectral function at low frequency.
If the k dependence of the self energy is weak, we expect to see contour lines corresponding to Ek = const and a height increasing as we approach the Fermi surface.

32. Evolution of the k resolved Spectral Function at zero frequency
Evolution of the k resolved Spectral Function at zero frequency. (Parcollet Biroli and GK)
U/D=2
U/D=2.25
Uc= , Tc/D=1/44

33. Near the transition k dependence is strong.
Qualitative effect, formation of hot regions!
D wave gapping of the single particle spectra as the Mott transition is approached. New paradigm for thinking about the approach to the Mott insulator.
Square symmetry is restored as we approched the insulator.
Experimental predictions! Photoemission ?

34. Lattice and cluster self energies

35. Mechanism for hot spot formation: nn self energy ! General phenomena.

36. Conclusion.
Mott transition survives in the cluster setting. Role of magnetic frustration.
Surprising result: formation of hot and cold regions as a result of an approach to the Mott transition. General result ?
Unexpected role of the next nearest neighbor self energy.
CDMFT a new window to extend DMFT to lower temperatures.

37. Mott transition in the actinide series (Smith-Kmetko phase diagram)
Unique position of Pu in the periodic table. Filling of the actinide shell. Johanssen (1970)-Mott transition.

38. Total Energy as a function of volume for Pu
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, 2001,410,793, 2001)

39. 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)

40. Mott transition in the actinide series (Smith-Kmetko phase diagram)
Unique position of Pu in the periodic table. Filling of the actinide shell. Johanssen (1970)-Mott transition.

41. At room pressure a localised 5f6 system;j=5/2. S = -L = 3: J = 0Am
At room pressure a localised 5f6 system;j=5/2. S = -L = 3: J = 0
J. Smith & R. Haire, Science (1978)
J. Smith, J. Phys. (1979)

42. Mott transition into an open (right) and closed (left) shell systems.
.5 g T2
Log[2J+1]
??? Uc
S=0 U U
g ~1/(Uc-U)

43. Approach the Mott transition, if the localized configuration has an OPEN shell the mass increases as the transition is approached.
Consistent theory, entropy increases monotonically as U  Uc .
Approach the Mott transition, if the localized configuration has a CLOSED shell. We have an apparent paradox. To approach the Mott transitions the bands have to narrow, but the insulator has not entropy.. SOLUTION: superconductivity intervenes.

44. Mott transition in systems with close shell.
Resolution: as the Mott transition is approached from the metallic side, eventually superconductivity intervenes to for a continuous transition to the localized side.
DMFT study of a 2 band model for Buckminster fullerines Capone et. al. Science 2002.
Mechanism is relevant to Americium.

45. Am under pressure. Lindbaum et.al. PRB 63,2141010(2001)

46. ITU [J.C. Griveaux J. Rebizant G. Lander]

47. Overview of rho (p, T) of Am
Note strongly increasing resistivity as f(p) at all T. Shows that more electrons are entering the conduction band
Superconducting at all pressure
IVariation of rho vs. T for increasing p.

48. DMFT study in the fcc structure. S. Murthy and G. Kotliar

49. LDA+DMFT spectra. Notice the rapid occupation of the f7/2 band.

50. One electron spectra. Experiments (Negele) and LDA+DFT theory (S
One electron spectra. Experiments (Negele) and LDA+DFT theory (S. Murthy and GK )

51. Conclusion Am
Crude LDA+DMFT calculations describe the crude energetics of the material, eq. volume, even p vs V .
Superconductivity near the Mott transition.
Tc increases first and the decreases as we approach the Mott boundary.
Dramatic effect in the f bulk module.
What is going on at the Am I- Am II boundary ??? Subtle effect (bulk moduli do not change much ), but crucial modifications at low energy.
Mott transition of the f7/2 band ? Quantum critical point ?:

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

53. Conclusion
DMFT mapping onto “self consistent impurity models” offer a new “reference frame”, to think about correlated materials and compute their physical properties.
Treats atomic excitations and band-like quasiparticle on the same footing. Can treat electrons near a localization delocalization boundary.

54. Conclusions Essential for many materials. New physics. Case studies.
Kappa organics , hot –cold regions.
Parcollet Biroli and Kotliar PRL 2004.
Mott transition across the actinide series. Pu and Am.
Interplay with dimerization, and Coulomb interactions. The two impurity model and Ti2O3.

55. Conclusions.
Alpha Gamma Cerium. Optics and Kondo scales. The mechanism revealed.
LixCoO3. Mott transition in your cell phone.[ C. Marianetti, G. Kotliar and G. Ceder Nature materials in press ]
Itinerant ferromagnets at high and low temperatures, Ni and Fe crossover from atomic to band physics.
Doping driven Mott transition in three dimensional materials: LaSrTiO3.
Other work, other materials, ENS, Stuttgart Augsburg, Ekaterinburg…………….

56. Ti2O3 : Coulomb or Pauling
LTS 250 K, HTS 750 K.
C.E.Rice et all, Acta Cryst B33, 1342 (1977)

57. Ti2O3.
Isostructural to V2-xCrxO3. Al lot of the qualitative physics of the high temperature part of the phase diagram of V2O3 can be understood within single site DMFT. Is this true in Ti2O3?
Band Structure Calculations good metal. L.F. Mattheiss, J. Phys.: Condens. Matter 8, 5987 (1996) .Unrestricted Hartree Fock calculations produce large antiferromagnetic gap. M. Cati, et. al. Phys. Rev. B. f55 , (1997).

58. 2site-Cluster DMFT with intersite Coulomb
U = 2, J = 0.5, W = 0.5
β = 20 eV-1, LT structure
U = 2, J = 0.5, W = 0.5
β = 10 eV-1, HT structure
A. Poteryaev

59. Pauling and Coulomb Ti2O3[S. Poteryaev S
Pauling and Coulomb Ti2O3[S. Poteryaev S. Lichtenstein and GK cond-mat ]
Dynamical Goodenough-Honing picture

60. Conclusion Ti2O3 2 site cluster DMFT describes the MIT in Ti2O3.
Different from V2O3 where single site DMFT works well, and cluster corrections are small [A. Poteryaev]
It requires the Coulomb interactions, and a frequency dependent enhancement of the a1g-a1g hopping, induced by the Coulomb interactions. [Haldane Ph.D thesis, Q Si and GK 1993 ].Dynamical Pauling-Goodenough mechanism is able to trigger the MIT at low enough temperatures.
Coulomb and Pauling synergistically cooperate.

61. Overview High T phase Low T phase  Various phases :
isostructural phase transition (T=298K, P=0.7GPa)
 (fcc) phase
[ magnetic moment
(Curie-Wiess law) ]
  (fcc) phase
[ loss of magnetic
moment (Pauli-para) ]
with large
volume collapse
v/v  15
( -phase a  5.16 Å
-phase a  4.8 Å)
volumes
exp.
LDA
LDA+U a
28Å3
24.7Å3 g
34.4Å3
35.2Å3
  -phase (localized):
High T phase
Curie-Weiss law (localized magnetic moment),
Large lattice constant
Tk around 60-80K
 -phase (delocalized:Kondo-physics):
Low T phase
Loss of Magnetism (Fermi liquid Pauli susceptibility) - completely screened magnetic moment
smaller lattice constant
Tk around K

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

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

64. LDA+DMFT:Ce spectra Successful calculations of thermodynamics.
M.B.Z¨olfl,I.A.NekrasovTh.Pruschke,V.I.Anisimov J. Keller,Phys.Rev. Lett 87, (2001).
K. Held, A.K. McMahan, and R.T. Scalettar, Phys. Rev.Lett. 87, (2001)
A.K.McMahan,K.Held,andR.T.Scalettar,Phys Rev. B 67, (2003).
Successful calculations of thermodynamics.

65. X.Zhang M. Rozenberg G. Kotliar (PRL 70, 1666(1993)).
Unfortunately photoemission cannot decide between the Kondo collapse picture and the Mott transition picture. Evolution of the spectra as a function of U , half filling full frustration, Hubbard model!!!!
X.Zhang M. Rozenberg G. Kotliar (PRL 70, 1666(1993)).
Three peak structure, HubbaRD BANDS AND quasiparticle bands. Trasfer of spectral weigth.
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

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

67. Photoemission&experiment
A. Mc Mahan K Held and R. Scalettar (2002)
Zoffl et. al (2002)
K. Haule V. Udovenko S. Savrasov and GK. (2004)

68. To resolve the conflict between the Mott transition and the volume collapse picture : Turn to Optics! Haule et.al.
Qualitative idea. The spd electrons have much larger velocities, so optics will be much more senstive to their behavior.
See if they are simple spectators (Mott transition picture ) or wether a Kondo binding unbinding takes pace (Kondo collapse picture).
General method, bulk probe.

69. Temperature dependence of the optical conductivity.

70. Theory: Haule et. al. cond-matt 04 Expt: J. W
Theory: Haule et. al. cond-matt 04 Expt: J.W. vanderEb PRL 886,3407 (2001)

72. Optical conductivity of Ce (expt. Van Der Eb et. al. theory Haule et
Optical conductivity of Ce (expt. Van Der Eb et.al. theory Haule et.al)
experiment
LDA+DMFT
K. Haule et.al.

73. Origin of the features.

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

75. Conclusion
DMFT mapping onto “self consistent impurity models” offer a new “reference frame”, to think about correlated materials and compute their physical properties.
Treats atomic excitations and band-like quasiparticle on the same footing. Can treat electrons near a localization delocalization boundary.

76. Conclusions Essential for many materials. New physics. Case studies.
Kappa organics , hot –cold regions.
Parcollet Biroli and Kotliar PRL 2004.
Mott transition across the actinide series. Pu and Am.
Interplay with dimerization, and Coulomb interactions. The two impurity model and Ti2O3.

77. Conclusions.
Alpha Gamma Cerium. Optics and Kondo scales. The mechanism revealed.
LixCoO3. Mott transition in your cell phone.[ C. Marianetti, G. Kotliar and G. Ceder Nature materials in press ]
Itinerant ferromagnets at high and low temperatures, Ni and Fe crossover from atomic to band physics.
Doping driven Mott transition in three dimensional materials: LaSrTiO3.
Other work, other materials, ENS, Stuttgart Augsburg, Ekaterinburg…………….

78. W (ev) vs (a.u. 27.2 ev) N.Zein G. Kotliar and S. Savrasov

79. Anomalous Resistivity
PRL 91, (2003)

80. Optical transfer of spectral weight

81. k-(ET)2X are across Mott transition
Insulating
anion layer X-
Ground
State
U/t
t’/t
Cu2(CN)3
Mott insulator
8.2
1.06
Cu[N(CN)2]Cl
7.5
0.75
Cu[N(CN)2]Br SC
7.2
0.68
Cu(NCS)2
6.8
0.84
Cu(CN)[N(CN)2]
Ag(CN)2 H2O
6.6
0.60 I3
6.5
0.58
X-1
conducting
ET layer
[(ET)2]+1 t’ t
modeled to triangular lattice

82. Controversy on the unfrustrated case
Controversy on the unfrustrated case. Comment on "Absence of a Slater Transition in the Two-Dimensional Hubbard Model"
B. Kyung, J.S. Landry, D. Poulin, A.-M.S. Tremblay Phys. Rev. Lett. 90, (2003)

83. k organics ET = BEDT-TTF=Bisethylene dithio tetrathiafulvalene
k organics = (ET)2 X
Increasing pressure ----- increasing t’ 
X X X X3
(Cu)(2CN)3 Cu(NCN)2 Cl Cu(NCN2)2Br Cu(NCS)2
Spin liquid Mott transition

84. Compare with single site results Rozenberg Chitra Kotliar PRL 2002

85. Mott transition in cluster (QMC)

86. Deviations from single site DMFT

87. W (ev) vs (a.u. 27.2 ev) N.Zein G. Kotliar and S. Savrasov