Strongly Correlated Electron Materials : a DMFT Perspective

1
Strongly Correlated Electron Materials a DMFT
Perspective

• Gabriel Kotliar
• Physics Department and
• Center for Materials Theory
• Rutgers University

2
Outline

• Introduction to the strong correlation problem.

• Essentials of DMFT

• Applications to the Mott transition problem some
  insights from studies of models.

• Towards an electronic structure method
  applications to materials.

3
The electron in a solid wave picture
Momentum Space (Sommerfeld)

Maximum metallic resistivity 200 mohm cm
Standard model of solids Periodic potential,
waves form bands , k in Brillouin zone
Landau Interactions renormalize away
4
Standard Model of Solids

• Qualitative predictions low temperature
  dependence of thermodynamics and transport

Optical response, transitions between bands.
Qualitative predictions. Filled bands-Insulators,
Unfilled bands metals. Odd number of electrons
metallicity.
Quantitative tools DFT, LDA, GGA, total
energies,good starting point for spectra, GW,and
transport

5
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)
6
The electron in a solid particle picture.

• NiO, MnO, Array of atoms is insulating if
  agtgtaB. Mott correlations localize the electron
• e_ e_ e_
  e_

• Superexchange

• Think in real space , solid collection of atoms
• High T local moments, Low T spin-orbital order

7
Mott Correlations localize the electron

• Low densities, electron behaves as a particle,use
  atomic physics, work in 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.

8
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 LDAU 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.

9
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/m1000
• (La1-xSrx)MnO3 Colossal Magneto-resistance.

10
Strongly Correlated Materials.

• Large thermoelectric response in CeFe4 P12 (H.
  Sato et al. cond-mat 0010017). Ando et.al.
  NaCo2-xCuxO4 Phys. Rev. B 60, 10580 (1999).
• Huge volume collapses, Ce, Pu
• Large and ultrafast optical nonlinearities
  Sr2CuO3 (T Ogasawara et.a Phys. Rev. Lett. 85,
  2204 (2000) )

11
The Mott transition

• Electronically driven MIT.
• Forces to face directly the localization
  delocalization problem.
• Relevant to many systems, eg V2O3
• Techniques applicable to a very broad
• range or problems.

12
Mott transition in V2O3 under pressure or
chemical substitution on V-site
13
Universal and non universal features. Top to
bottom approach to correlated materials.

• Some aspects at high temperatures, depend weakly
  on the material (and on the model).
• Low temperature phase diagram, is very sensitive
  to details, in experiment (and in the theory).

14
Mott transition in layered organic conductors
S Lefebvre et al. cond-mat/0004455, Phys. Rev.
Lett. 85, 5420 (2000)
15
Failure of the Standard Model NiSe2-xSx
Miyasaka and Takagi (2000)
16
Phase Diagrams V2O3, Ni Se2-x Sx Mc Whan et. Al
1971,. Czek et. al. J. Mag. Mag. Mat. 3, 58
(1976),
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Outline

• Introduction to the strong correlation problem
  and to the Mott transition.
• Summary of the essential concepts of DMFT
• Applications to the Mott transition problem some
  insights from studies of models.
• Towards an electronic structure method
  applications to materials Pu, Fe, Ni, LaSrTiO3,
  NiO,.
• Outlook

18
Hubbard model

• U/t
• Doping d or chemical potential
• Frustration (t/t)
• T temperature

Mott transition as a function of doping, pressure
temperature etc.
19
Mean-Field Classical vs Quantum
Classical case
Quantum case
A. Georges, G. Kotliar (1992)
Phys. Rev. B 45, 6497
20
Limit of large lattice coordination
Metzner Vollhardt, 89
Muller-Hartmann 89
21
DMFT Effective Action point of view.R. Chitra
and G. Kotliar Phys Rev. B.(2000).

• Identify observable, A. Construct an exact
  functional of ltAgta, G a which is stationary at
  the physical value of a.
• Example, density in DFT theory. (Fukuda et. al.)
• When a is local, it gives an exact mapping onto a
  local problem, defines a Weiss field.
• The method is useful when practical and accurate
  approximations to the exact functional exist.
  Example LDA, GGA, in DFT.

22
Example DMFT for lattice model (e.g. single band
Hubbard).Muller Hartman 89, Chitra and GK 99.

• Observable Local Greens function Gii (w).
• Exact functional G Gii (w) .
• DMFT Approximation to the functional.

23
Extensions of DMFT.

• Renormalizing the quartic term in the local
  impurity action.
• EDMFT.
• Taking several sites (clusters) as local entity.
  
• CDMFT
• Combining DMFT with other methods.
• LDADMFT, GWU.

24
Outline

• Introduction to the strong correlation problem.
• Essentials of DMFT
• Applications to the Mott transition problem some
  insights from studies of models.
• Towards an electronic structure method
  applications to materials Pu, Fe, Ni, Ce
  LaSrTiO3, NiO.
• Outlook

25
Schematic DMFT phase diagram Hubbard model
(partial frustration)
M. Rozenberg G. Kotliar H. Kajueter G Thomas D.
Rapkine J Honig and P Metcalf Phys. Rev. Lett.
75, 105 (1995)
26
Schematic DMFT phase diagram one band Hubbard
model (half filling, semicircular DOS, partial
frustration) Rozenberg et.al PRL (1995)
27
Spectral Evolution at T0 half filling full
frustration. Three peak structure.
X.Zhang M. Rozenberg G. Kotliar (PRL 1993)
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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)
29
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

30
Parallel development Fujimori et.al
31
Mott transition in V2O3 under pressure or
chemical substitution on V-site
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)

33
Anomalous transfer of spectral weight in v2O3
34
Anomalous transfer of spectral weight in v2O3
35
ARPES measurements on NiS2-xSexMatsuura et. al
Phys. Rev B 58 (1998) 3690. Doniach and Watanabe
Phys. Rev. B 57, 3829 (1998)
.
36
Anomalous transfer of optical spectral weight,
NiSeS. Miyasaka and Takagi
37
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 )
38
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.

39
Conclusions generic aspects

• Three peak structure, quasiparticles and Hubbard
  bands.
• Non local transfer of spectral weight.
• Large resistivities.

40
Insights from DMFT

• .
• 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 pictures
  are needed as synthesized in DMFT.

41
Outline

• Introduction to the strong correlation problem.
• Essentials of DMFT
• Applications to the Mott transition problem some
  insights from studies of models.
• Towards an electronic structure method
  applications to materials
• Outlook

42
Interface DMFT with electronic structure.

• Derive model Hamiltonians, solve by DMFT
• (or cluster extensions).
• Full many body aproach, treat light electrons by
  GW or screened HF, heavy electrons by DMFT .
• Treat correlated electrons with DMFT and light
  electrons with DFT (LDA, GGA DMFT)

43
Spectral Density Functional effective action
construction (Chitra and GK).

• 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 sources for r(r) and G
  and performing a Legendre transformation,
  Gr(r),G(R,R)(iw)
• Approximate functional using DMFT insights.

44
Very Partial list of application of realistic
DMFT to materials

• QP bands in ruthenides A. Liebsch et al (PRL
  2000)
• N phase of Pu Savrasov GK and Abrahams (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)
• Paramagnetic Mott insulators. NiO MnO, Savrasov
  and GK

45

Case study in f electrons, Mott transition in
the actinide series. B. Johanssen 1974 Smith and
Kmetko Phase Diagram 1984.
46
Pu DMFT total energy vs Volume (Savrasov
Kotliar and Abrahams 2001)
47
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.

48
Iron and Nickel crossover to a real space
picture at high T (Lichtenstein, Katsnelson and
Kotliar Phys Rev. Lett 87, 67205 , 2001)
49
Iron and Nickelmagnetic properties
(Lichtenstein, Katsenelson,GK PRL 01)
50
Ni and Fe theory vs exp

• meff / mB high T
  moment
• Fe 3.1 (theory) 3.12 (expt)
• Ni 1.5 (theory) 1.62 (expt)
• Curie Temperature Tc
• Fe 1900 ( theory) 1043(expt)
• Ni 700 (theory) 631 (expt)

51
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

52
Photoemission Spectra and Spin Autocorrelation
Fe (U2, J.9ev,T/Tc.8) (Lichtenstein,
Katsenelson,Kotliar Phys Rev. Lett 87, 67205 ,
2001)
53
Photoemission and T/Tc.8 Spin Autocorrelation
Ni (U3, J.9 ev)
54
Outline

• Introduction to the strong correlation problem.
• Essentials of DMFT
• Applications to the Mott transition problem some
  insights from studies of models.
• Towards an electronic structure method
  applications to materials Pu, Fe, Ni, LaSrTiO3,
  NiO, Ce, LaSrMnO3
• Outlook

55
Outlook

• Local approach to strongly correlated electrons.
• Many extensions, make the approach suitable for
  getting insights and quantitative results in
  correlated materials.

56
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 to the electronic structure of
  correlated materials.

57
Outlook

• Systematic improvements, short range
  correlations, cluster methods, improved mean
  fields.
• Improved interfaces with electronic structure.
• Exploration of complex strongly correlated
  materials.

58
Acknowledgements Development of DMFT
Collaborators V. Anisimov, R. Chitra, V.
Dobrosavlevic, X. Dai, 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, I. Yang, X.Y. Zhang
Support NSF DMR 0096462 Support
Instrumentation. NSF DMR-0116068 Work on Fe
and Ni ONR4-2650 Work on Pu DOE
DE-FG02-99ER45761 and LANL subcontract No.
03737-001-02
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E-DMFT references

• H. Kajueter and G. Kotliar (unpublished and
  Kajuters Ph.D thesis (1995)).
• Q. Si and J L Smith PRL 77 (1996)3391 .
• R. Chitra and G.Kotliar Phys. Rev. Lett 84,
  3678-3681 (2000 )
• Y. Motome and G. Kotliar. PRB 62, 12800 (2000)
• R. Chitra and G. Kotliar Phys. Rev. B 63, 115110
  (2001)
• S. Pankov and G. Kotliar PRB 66, 045117 (2002)

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DMFT Impurity cavity construction
63
Cluster extensions of DMFT

• Two impurity method. A. Georges and G. Kotliar
  (1995 unpublished ) and RMP 68,13 (1996) , A.
  Schiller PRL75, 113 (1995)
• M. Jarrell et al Dynamical Cluster Approximation
  Phys. Rev. B 7475 1998
• Periodic cluster M. Katsenelson and A.
  Lichtenstein PRB 62, 9283 (2000).
• G. Kotliar S. Savrasov G. Palsson and G. Biroli
  Cellular DMFT PRL87, 186401 2001

64
C-DMFT
CDMFT The lattice self energy is inferred from
the cluster self energy.

Alternative approaches DCA (Jarrell et.al.)
Periodic clusters (Lichtenstein and Katsnelson)
65
C-DMFT test in one dimension. (Bolech, Kancharla
GK cond-mat 2002)
Gap vs U, Exact solution Lieb and Wu, Ovshinikov
Nc2 CDMFT vs Nc1
66
DMFT plus other methods.

• DMFT LDA , V. Anisimov, A. Poteryaev, M.
  Korotin, A. Anokhin and G. Kotliar, J. Phys.
  Cond. Mat. 35, 7359-7367 (1997).
• A Lichtenstein and M. Katsenelson Phys. Rev. B
  57, 6884 (1988).
• S. Savrasov and G.Kotliar, funcional
  formulation for full self consistent
  implementation. Savasov Kotliar and Abrahams .
  Application to delta Pu Nature (2001)
• Combining EDMFT with GW. Ping Sun and Phys. Rev.
  B 66, 085120 (2002). G.  Kotliar and S. Savrasov
  in New Theoretical Approaches to Strongly
  Correlated Systems, A. M. Tsvelik Ed. 2001 Kluwer
  Academic Publishers. 259-301 . cond-mat/0208241

67
ARPES measurements on NiS2-xSexMatsuura et. Al
Phys. Rev B 58 (1998) 3690. Doniaach and Watanabe
Phys. Rev. B 57, 3829 (1998)
.
68
LDADMFT Self-Consistency loop
E
U
DMFT
69
Recent exps. Suga, Allen, Vollhardt
70
Ni and Fe theory vs exp

• m( T.9 Tc)/ mB ordered moment
• Fe 1.5 ( theory) 1.55 (expt)
• Ni .3 (theory) .35 (expt)
• meff / mB high T moment
• Fe 3.1 (theory) 3.12 (expt)
• Ni 1.5 (theory) 1.62 (expt)
• Curie Temperature Tc
• Fe 1900 ( theory) 1043(expt)
• Ni 700 (theory) 631 (expt)

71
LDADMFT Self-Consistency loop
E
U
DMFT
72
LDADMFT functional
F Sum of local 2PI graphs with local U matrix and
local G
73
Anomalous Spectral Weight Transfer Optics
Below energy
AppreciableT 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).
74
Comments on LDADMFT

• Static limit of the LDADMFT functional , with F
  FHF reduces to LDAU
• Removes inconsistencies of this approach,
• Only in the orbitally ordered Hartree Fock limit,
  the Greens function of the heavy electrons is
  fully coherent
• Gives the local spectra and the total energy
  simultaneously, treating QP and H bands on the
  same footing.

75
LSDADMFT functional
F Sum of local 2PI graphs with local U matrix and
local G
76
DMFT Effective Action point of view.R. Chitra
and G. Kotliar Phys Rev. B.(2000).

• Identify observable, A. Construct an exact
  functional of ltAgta, G a which is stationary at
  the physical value of a.
• Example, density in DFT theory. (Fukuda et. al.)
• When a is local, it gives an exact mapping onto a
  local problem, defines a Weiss field.
• The method is useful when practical and accurate
  approximations to the exact functional exist.
  Example LDA, GGA, in DFT.
• It is useful to introduce a Lagrange multiplier
  l conjugate to a, G a, l .
• It gives as a byproduct a additional lattice
  information.

77
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)

78
LDADMFT Self-Consistency loop
Edc
U
DMFT