Electronic Structure of Strongly Correlated Materials:Insights from Dynamical Mean Field Theory (DMFT).

1
Electronic Structure of Strongly Correlated
MaterialsInsights from Dynamical Mean Field
Theory (DMFT).

• Gabriel Kotliar
• Physics Department and
• Center for Materials Theory
• Rutgers UniversityCenter for Materials Theory
  Rutgers University
• CPTH Ecole Polytechnique Palaiseau, and CPTH
  CEA Saclay , France

REUNIÓN NACIONAL DE FÍSICA DEL ESTADO SÓLIDO.
GEFES IV Alicante Spain. February 1-3 (2006)
upport NSF DMR . Blaise Pascal Chair
Fondation de lEcole Normale.
2
Electrons in a Solidthe Standard Model
Band Theory electrons as waves. Landau Fermi
Liquid Theory. At low energies the electrons
behave as non interacting quasiparticles.
Rigid bands , optical transitions ,
thermodynamics, transport

• Quantitative Tools. Density Functional Theory
  GW Perturbation Theory.

3
LDAGW semiconducting gaps. Reviews J. Wilkins,
M. VanSchilfgaarde
Success story Density Functional linear
response Review Baroni et.al, Rev. Mod. Phys,
73, 515, 2001
4
Correlated Electron Materials

• Are not well described by either the itinerant
  or the localized framework .
• Compounds with partially filled f and d shells.
  Need new starting point for their description.
  Non perturbative problem. New reference frame
  for computing their physical properties.
• Have consistently produce spectacular big
  effects thru the years. High temperature
  superconductivity, huge resistivity changes
  across the MIT, colossal magneto-resistance, huge
  volume collapses, large masses in heavy Fermions,
  ..

5
Breakdown of the Standard Model Large Metallic
Resistivities
6
Transfer of optical spectral weight non local in
frequency Schlesinger et. al. (1994), Vander
Marel (2005) Takagi (2003 ) Neff depends on T
7
Localization vs Delocalization Strong Correlation
Problem

• Many interesting compounds do not fit within the
  Standard Model.
• Tend to have elements with partially filled d
  and f shells. Competition between kinetic and
  Coulomb interactions.
• Breakdown of the wave picture. Need to
  incorporate a real space perspective (Mott).
• Non perturbative problem.
• Require a framework that combines both atomic
  physics and band theory. DMFT.

8
DMFT Cavity Construction. A. Georges and G.
Kotliar PRB 45, 6479 (1992). First happy marriage
of atomic and band physics.
Reviews A. Georges G. Kotliar W. Krauth and M.
Rozenberg RMP68 , 13, 1996 Gabriel Kotliar and
Dieter Vollhardt Physics Today 57,(2004)
9
Mean-Field Classical vs Quantum
Classical case
Quantum case
A. Georges, G. Kotliar (1992)
Phys. Rev. B 45, 6497
10
Cluster Extensions of Single Site DMFT
Many Techniques for solving the impurity model
QMC, (Fye-Hirsch), NCA, ED(Krauth Caffarel),
IPT, For a review see Kotliar et. Al to
appear in RMP (2006)
11
For reviews of cluster methods see Georges
et.al. RMP (1996) Maier et.al RMP (2005), Kotliar
et.al cond-mat 0511085. to appear in RMP (2006)
Kyung et.al cond-mat 0511085
Parametrizes the physics in terms of a few
functions .
D , Weiss Field
Alternative (T. Stanescu and G. K. ) periodize
the cumulants rather than the self energies.
12
Testing CDMFT (G.. Kotliar,S. Savrasov, G.
Palsson and G. Biroli, Phys. Rev. Lett. 87,
186401 (2001) ) with two sites in the Hubbard
model in one dimension V. Kancharla C. Bolech and
GK PRB 67, 075110 (2003)M.Capone M.Civelli V
Kancharla C.Castellani and GK PR B 69,195105
(2004)
U/t4.
13
Mott transition in V2O3 under pressure or
chemical substitution on V-site. How does the
electron go from localized to itinerant.
14
Pressure Driven Mott transition
How does the electron go from the localized to
the itinerant limit ?
15
M. Rozenberg et. al. Phys. Rev. Lett. 75, 105
(1995)
T/W
Phase diagram of a Hubbard model with partial
frustration at integer filling. Thinking about
the Mott transition in single site DMFT. High
temperature universality
16
V2O3Anomalous transfer of spectral weight
M. Rozenberg G. Kotliar H. Kajueter G Tahomas
D. Rapkikne J Honig and P Metcalf Phys. Rev.
Lett. 75, 105 (1995)
17
Anomalous transfer of optical spectral weight,
NiSeS. Miyasaka and Takagi 2000
18
Anomalous Resistivity and Mott transition Ni
Se2-x Sx
Crossover from Fermi liquid to bad metal to
semiconductor to paramagnetic insulator.
M. Rozenberg G. Kotliar H. Kajueter G Tahomas D.
Rapkikne J Honig and P Metcalf Phys. Rev. Lett.
75, 105 (1995)
19
Single site DMFT and kappa organics
20
Ising critical endpoint! In V2O3 P. Limelette
et.al. Science 302, 89 (2003)
21
ARPES measurements on NiS2-xSexMatsuura et. Al
Phys. Rev B 58 (1998) 3690. Doniaach and Watanabe
Phys. Rev. B 57, 3829 (1998) Mo et al., Phys.
Rev.Lett. 90, 186403 (2003).
.
22
Conclusions.

• Three peak structure, quasiparticles and Hubbard
  bands.
• Non local transfer of spectral weight.
• Large metallic resistivities.
• The Mott transition is driven by transfer of
  spectral weight from low to high energy as we
  approach the localized phase.
• Coherent and incoherence crossover. Real and
  momentum space.
• Theory and experiments begin to agree on a broad
  picture.

23
Cuprate superconductors and the Hubbard Model .
PW Anderson 1987
24
RVB physics and Cuprate Superconductors

• P.W. Anderson. Connection between high Tc and
  Mott physics. Science 235, 1196 (1987)
• Connection between the anomalous normal state of
  a doped Mott insulator and high Tc.
• Slave boson approach. ltbgt
  coherence order parameter. k, D singlet formation
  order parameters.Baskaran Zhou Anderson ,
  Ruckenstein et.al (1987) .

Other states flux phase or sid ( G. Kotliar
(1988) Affleck and Marston (1988) have point
zeros.
25
RVB phase diagram of the Cuprate Superconductors.
Superexchange.

• The approach to the Mott insulator renormalizes
  the kinetic energy Trvb increases.
• The proximity to the Mott insulator reduce the
  charge stiffness , TBE goes to zero.
• Superconducting dome. Pseudogap evolves
  continously into the superconducting state.

G. Kotliar and J. Liu Phys.Rev. B 38,5412 (1988)
Related approach using wave functionsT. M. Rice
group. Zhang et. al. Supercond Scie Tech 1, 36
(1998, Gross Joynt and Rice (1986) M. Randeria
N. Trivedi , A. Paramenkanti PRL 87, 217002
(2001)
26
Problems with the approach.

• Neel order. How to continue a Neel insulating
  state ? Need to treat properly finite T.
• Temperature dependence of the penetration depth
  Wen and Lee , Ioffe and Millis .
  TheoryrsTx-Ta x2 , Exp rT x-T a.
• Mean field is too uniform on the Fermi surface,
  in contradiction with ARPES.
• No quantitative computations in the regime
  where there is a coherent-incoherent crossover
  which compare well with experiments. e.g. Ioffe
  Kotliar 1990

The development of CDMFT solves may solve many of
these problems.!!
27
Photoemission spectra near the antinodal
direction in a Bi2212 underdoped sample.
Campuzano et.al
EDC along different parts of the zone, from Zhou
et.al.
28
M. Rozenberg et. al. Phys. Rev. Lett. 75, 105
(1995)
T/W
Phase diagram of a Hubbard model with partial
frustration at integer filling. Thinking about
the Mott transition in single site DMFT. High
temperature universality
29
The development of CDMFT solves may solve many of
the problems of the early slave bosons RVB
theory .!! Theoretical approach study the
plaquette CDMFT equations.

• Ignore inhomogeneities and phase separation.
• Follow separately each mean field state.
• Focus on the physics results from the proximity
  to a Mott insulating state and to which extent it
  accounts for the experimental observations.

30
Competition of AF and SC M. Capone M. Civelli and
GK (2006)
31
Superconductivity in the Hubbard model role of
the Mott transition and influence of the
super-exchange. ( M. Capone et.al. V. Kancharla
et. al. CDMFTED, 4 8 sites t0) .
Pd
32
Order Parameter and Superconducting Gap do not
always scale! ED study in the SC state Capone
Civelli Parcollet and GK (2006)
33
Evolution of DOS with doping U8t. Capone et.al.
Superconductivity is driven by transfer of
spectral weight , slave boson b2 !
34
Anomalous Self Energy. (from Capone et.al 2006.)
Notice the remarkable increase with decreasing
doping! True superconducting pairing!! U8t
Significant Difference with Migdal-Eliashberg.
35
Follow the normal state with doping. Civelli
et.al. PRL 95, 106402 (2005) Spectral Function
A(k,??0) -1/p G(k, ? ?0) vs k U16 t, t-.3
K.M. Shen et.al. 2004
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. Different for electron doped!
2X2 CDMFT
36
Interpretation in terms of lines of zeros and
lines of poles of G T.D. Stanescu and G.K
cond-matt 0508302
37
Conclusion

• CDMFT delivers the spectra.
• Path between d-wave and insulator. Dynamical RVB!
• Lines of zeros. Connection with other work. of
  A. Tsvelik and collaborators. (Perturbation
  theory in chains , see however Biermann et.al).
  T.Stanescu, fully self consistent scheme.
• Weak coupling RG (T. M. Rice and collaborators).
  Truncation of the Fermi surface.
• CDMFT presents it as a strong coupling
  instability that begins FAR FROM FERMI SURFACE.

38
Realistic Descriptions of Materials and a First
Principles Approach to Strongly Correlated
Electron Systems.

• Incorporate realistic band structure and orbital
  degeneracy.
• Incorporate the coupling of the lattice degrees
  of freedom to the electronic degrees of freedom.
• Predict properties of matter without empirical
  information.

39
LDADMFT V. Anisimov, A. Poteryaev, M. Korotin,
A. Anokhin and G. Kotliar, J. Phys. Cond. Mat.
35, 7359 (1997).

• Realistic band structure and orbital degeneracy.
  Describes the excitation spectra of many strongly
  correlated solids. .

Spectral Density Functionals. Chitra and Kotliar
PRB 2001 Savrasov et. al. Nature (2001) Savrasov
and Kotliar PRB (2005)

• Determine the self energy , the density and the
  structure of the solid by extremizing a
  functional of these quantities. Coupling of
  electronic degrees of freedom to structural
  degrees of freedom.

40
Mott Transition in Actinides
The f electrons in Plutonium are close to a
localization-delocalization transition
(Johansson, 1974) . Modern understanding of the
phenomena with DMFT (Savrasov and Kotliar
2002-2003)
Mott Transition
after G. Lander, Science (2003).
after J. Lashley et.al, cond-mat (2005).
This regime is not well described by traditional
techniques of electronic structure techniques and
require new methods which take into account the
itinerant and the localized character of the
electron on the same footing.
41
DMFT Phonons in fcc d-Pu
( Dai, Savrasov, Kotliar,Ledbetter, Migliori,
Abrahams, Science, 9 May 2003)
(experiments from Wong et.al, Science, 22 August
2003)
42
Summary

• Review the standard model of solids.
• Introduced some of the problems posed by strongly
  correlated electron materials.
• Dynamical Mean Field Theory (DMFT). New reference
  frame to think about the physics of these
  materials and compute its properties.
• The Mott Transition in 3d frustrated transition
  metal oxides and in high temperature
  superconductors.
• Future Directions. The field of correlated
  electrons is at the brink of a revolution.(C)
  DMFT Rapid development of conceptual tools and
  computational abilities. Theoretical Spectroscopy
  in the making.
• Prelude to theoretical material design using
  strongly correlated elemenets.
• Focus on the deviations between CDMFT and
  experiments to elucidate the role of long
  wavelength non Gaussian fluctuations.

43

• Gracias por invitarme y por vuestra atencion!

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Functional formulation to achieve more realistic
calculations For a review see Kotliar et.al. to
appear in RMP..

• Spectral Density Functional
• LDADMFT

Final Goal

Savrasov Kotliar and Abrahams Nature 410,793
(2001).
V. Anisimov, A. Poteryaev, M. Korotin, A. Anokhin
and G. Kotliar, J. Phys. Cond. Mat. 35, 7359
(1997) generalizing LDA U
46
Approaching the Mott transition CDMFT Picture

• Fermi Surface Breakup. Qualitative effect,
  momentum space differentiation. Formation of hot
  cold regions is an unavoidable consequence of
  the approach to the Mott insulating state!
• D wave gapping of the single particle spectra as
  the Mott transition is approached. Real and
  Imaginary part of the self energies grow
  approaching half filling. Unlike weak coupling!
• Similar scenario was encountered in previous
  study of the kappa organics. O Parcollet G.
  Biroli and G. Kotliar PRL, 92, 226402. (2004)
  . Both real and imaginary parts of the self
  energy get larger. Strong Coupling instability.

47
Two paths for the calculation of electronic
structure of materials
Crystal structure Atomic positions
Model Hamiltonian
Correlation Functions Total Energies etc.
Hubbard Model
48
ARPES spectra for La2-xSrxCuO4 at doping x
0.063, 0.09, 0.22. From Zhou et al
49
Functional formulation. Chitra and Kotliar Phys.
Rev. B 62, 12715 (2000) and Phys. Rev.B (2001)
. 
Introduce Notion of Local Greens functions, Wloc,
Gloc GGlocGnonloc .
Ex. IrgtR, rgt GlocG(R r, R r) dR,R
Sum of 2PI graphs
One can also view as an approximation to an
exact Spectral Density Functional of Gloc and
Wloc.
50
Order in Perturbation Theory

• n1

Order in PT
n2
Basis set size.
l1
DMFT
r1

l2
r site CDMFT
r2
GW first vertex correction
GW
llmax
Range of the clusters
51

• Spectral Density Functional
• LDADMFT

Savrasov Kotliar and Abrahams Nature 410,793
(2001).
V. Anisimov, A. Poteryaev, M. Korotin, A. Anokhin
and G. Kotliar, J. Phys. Cond. Mat. 35, 7359
(1997)
52
Conclusions sp systems.

• Not well described by single site DMFT. But very
  well described by first principles cdmft with
  relatively small clusters. 2 or 3 coordination
  spheres
• Weakly correlated materials. Use cheap impurity
  solvers.
• Fast, self consistent way of getting first
  principles electronic structure without LDA. Good
  trends for semiconducting gaps and band withds.

53
Earlier approximations as limiting casessinlge
site DMFT for models A. Georges G. Kotliar PRB
(1992)

• Spectral Density Functional
• LDADMFT

Savrasov Kotliar and Abrahams Nature 410,793
(2001).
V. Anisimov, A. Poteryaev, M. Korotin, A. Anokhin
and G. Kotliar, J. Phys. Cond. Mat. 35, 7359
(1997)
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Spectral shapes. Large Doping Stanescu and GK
cond-mat 0508302
56
Small Doping. T. Stanescu and GK cond-matt 0508302
57
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58
Interpretation in terms of lines of zeros and
lines of poles of G T.D. Stanescu and G.K
cond-matt 0508302
59
Lines of Zeros and Spectral Shapes. Stanescu and
GK cond-matt 0508302
60
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61
Two paths for 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.
62

• The combination of realistic band theory and many
  body physics, is a very broad subject.
• Having a practical and tractable non
  perturbative method for solving many body
  Hamiltonians, the next step is to bring more
  realistic descriptions of the materials Orbital
  degeneracy and realistic band structure.
• LDADMFT V. Anisimov, A. Poteryaev, M. Korotin,
  A. Anokhin and G. Kotliar, J. Phys. Cond. Mat.
  35, 7359 (1997).
• The light, sp (or spd) electrons are extended,
  well described by LDA .The heavy, d (or f)
  electrons are localized treat by DMFT.
• Functional formulation similar to DFT Total
  Energy as functional of local Green function
  Spectral Density Functional Theory(Chitra,
  Kotliar, PRB 2001, Savrasov, Kotliar, Abrahams,
  Nature 2001).

63

• Focus on the local spectral function A(w)
  (and of the local screened Coulomb interaction
  W(w) ) of the solid.
• Write a functional of the local spectral function
  such that its stationary point, give the energy
  of the solid.
• No explicit expression for the exact functional
  exists, but good approximations are available.
  LDADMFT.
• The spectral function is computed by solving a
  local impurity model in a medium .Which is a new
  reference system to think about correlated
  electrons.
• Add non local perturbative corrections
  GWDMFT.
• Explosion of papers, refining the techniques and
  applying it to many different materials.

64
Actinies , role of Pu in the periodic table
65
Pu phases A. Lawson Los Alamos Science 26,
(2000)
LDA underestimates the volume of fcc Pu by
30. Within LDA fcc Pu has a negative shear
modulus. LSDA predicts d Pu to be magnetic with
a 5 ub moment. Experimentally it is not.
Treating f electrons as core overestimates the
volume by 30
66
Pu is not MAGNETIC, alpha and delta have
comparable susceptibility and specifi heat.
67
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, ,S5/2
  Ms5 mB . Mtot0
• More realistic calculations, (GGAU),itineracy,
  crystal fields G7 G8, ML-3.9 Mtot1.1. S. Y.
  Savrasov and G. Kotliar, Phys. Rev. Lett., 84,
  3670 (2000)
• This moment is quenched or screened by spd
  electrons, and other f electrons. (e.g. alpha
  Ce).
• Contrast Am(5f)6

68
Total Energy as a function of volume for Pu W
(ev) vs (a.u. 27.2 ev)
(Savrasov, Kotliar, Abrahams, Nature ( 2001) Non
magnetic correlated state of fcc Pu.
Zein Savrasov and Kotliar (2004)
69
Double well structure and d Pu

• Qualitative explanation of negative thermal
  expansion G. Kotliar J.Low Temp. Physvol.126,
  1009 27. (2002)See also A . Lawson et.al.Phil.
  Mag. B 82, 1837

70
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.

71
.
CDMFT study of cuprates

• AFunctional of the cluster Greens function.
  Allows the investigation of the normal state
  underlying the superconducting state, by forcing
  a symmetric Weiss function, we can follow the
  normal state near the Mott transition.
• Earlier studies use QMC (Katsnelson and
  Lichtenstein, (1998) M Hettler et. T. Maier
  et. al. (2000) . ) used QMC as an impurity
  solver and DCA as cluster scheme. (Limits U to
  less than 8t )
• Use exact diag ( Krauth Caffarel 1995 ) as a
  solver to reach larger Us
• and smaller Temperature and CDMFT as the
  mean field scheme.
• Recently (K. Haule and GK ) the region near the
  superconducting normal state transition
  temperature near optimal doping was studied
  using NCA DCA .
• DYNAMICAL GENERALIZATION OF SLAVE BOSON ANZATS
• w-S(k,w)m w/b2 -(Db2 t) (cos kx cos ky)/b2
  l
• b--------gt b(k), D -----? D(w), l -----?
  l (k )
• Extends the functional form of self energy to
  finite T and higher frequency.

72
Phonon freq (THz) vs q in delta Pu X. Dai et. al.
Science vol 300, 953, 2003
73
Inelastic X Ray. Phonon energy 10 mev, photon
energy 10 Kev.
E Ei - Ef Q ki - kf
74
DMFT Phonons in fcc d-Pu
( Dai, Savrasov, Kotliar,Ledbetter, Migliori,
Abrahams, Science, 9 May 2003)
(experiments from Wong et.al, Science, 22 August
2003)
75
J. Tobin et. al. PHYSICAL REVIEW B 68, 155109
,2003
76
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.

77

• Pu strongly correlated element, at the brink of a
  Mott instability.
• Realistic implementations of DMFT total
  energy, photoemission spectra and phonon
  dispersions of delta Pu.
• Clues to understanding other Pu anomalies.

78
Outline

• Introduction to strongly correlated electrons.
• Introduction to Dynamical Mean Field Theory
  (DMFT)
• The Mott transition problem. Theory and
  experiments.
• More realistic calculations. Pu the Mott
  transition across the actinide series.
• Conclusions . Current developments and future
  directions.

79
Mott transition into an open (right) and closed
(left) shell systems. AmAt room pressure a
localised 5f6 systemj5/2. S -L 3 J 0
apply pressure ?
S
S
.g T
Log2J1
???
Uc
S0
U
U
g 1/(Uc-U)
80
Americium under pressure
Experimental Equation of State
(after Heathman et.al, PRL 2000)
Mott Transition?
Soft
Hard

• Density functional based electronic structure
  calculations
• Non magnetic LDA/GGA predicts volume 50 off.
• Magnetic GGA corrects most of error in volume but
  gives m6mB
• (Soderlind et.al., PRB 2000).
• Experimentally, Am has non magnetic f6 ground
  state with J0 (7F0)

81
Mott transition in open (right) and closed (left)
shell systems.
S
S
g T
Tc
Log2J1
???
Uc
J0
U
U
g 1/(Uc-U)
82
Am under pressure J.C. GriveauJ. Rebizant G.
Lander G. Kotliar PRL (2005)
83
Collaborators References

• Reviews A. Georges G. Kotliar W. Krauth and M.
  Rozenberg RMP68 , 13, (1996).
• Reviews G. Kotliar S. Savrasov K. Haule V.
  Oudovenko O. Parcollet and C. Marianetti.
  Submitted to RMP (2005).
• Gabriel Kotliar and Dieter Vollhardt Physics
  Today 57,(2004)

84
Conclusion

• DMFT. Electronic Structure Method under
  development. Local Approach. Cluster extensions.
• Quantitative results , connection between
  electronic structure, scales and bonding.
• Qualitative understanding by linking real
  materials to impurity models. Concepts to think
  about correlated materials.
• Closely tied to experiments. System specific.
  Many materials to be studied, realistic matrix
  elements for each spectroscopy. Optics.
• Role of loclity.
• Material design using strongly correlated
  systems.

85
Anomalous Resistivity
PRL 91,061401 (2003)
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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?
• LDADMFT 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.

89
Epsilon Plutonium.
90
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 T0. But it has a much larger phonon
  entropy than delta.
• At the phase transition the volume shrinks but
  the phonon entropy increases.
• Estimates of the phase transition following
  Drumont and G. Ackland et. al. PRB.65, 184104
  (2002) (and neglecting electronic entropy).
  TC 600 K.

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92
Pu in the periodic table
actinides
93
Small amounts of Ga stabilize the d phase (A.
Lawson LANL)
94
Total Energy as a function of volume for Pu W
(ev) vs (a.u. 27.2 ev)
(Savrasov, Kotliar, Abrahams, Nature ( 2001) Non
magnetic correlated state of fcc Pu.
Zein Savrasov and Kotliar (2004)
95
DMFT Phonons in fcc d-Pu
( Dai, Savrasov, Kotliar,Ledbetter, Migliori,
Abrahams, Science, 9 May 2003)
(experiments from Wong et.al, Science, 22 August
2003)
96
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. AmAt
room pressure a localised 5f6 systemj5/2. S
-L 3 J 0 apply pressure ?
S
S
.g T
Log2J1
???
Uc
S0
U
U
g 1/(Uc-U)
97
J. C. Griveau et. al. (2004)
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99
H.Q. Yuan et. al. CeCu2(Si2-x Gex). Am under
pressure Griveau et. al.
Superconductivity due to valence fluctuations ?
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101
Evolution of the Spectral Function with
Temperature
Anomalous transfer of spectral weight connected
to the proximity to the Ising Mott endpoint
(Kotliar Lange nd Rozenberg Phys. Rev. Lett. 84,
5180 (2000)
102
Epilogue, the search for a quasiparticle peak and
its demise, photoemission, transport.
Confirmation of the DMFT predictions

• ARPES measurements on NiS2-xSexMatsuura et. Al
  Phys. Rev B 58 (1998) 3690. Doniaach and Watanabe
  Phys. Rev. B 57, 3829 (1998)
• S.-K. Mo et al., Phys Rev. Lett. 90, 186403
  (2003).
• Limelette et. al. Science G. Kotliar
  Science.

103
ARPES measurements on NiS2-xSexMatsuura et. Al
Phys. Rev B 58 (1998) 3690. Doniaach and Watanabe
Phys. Rev. B 57, 3829 (1998)
.
104
One Particle Local Spectral Function and Angle
Integrated Photoemission
e

• Probability of removing an electron and
  transfering energy wEi-Ef,
• f(w) A(w) M2
• Probability of absorbing an electron and
  transfering energy wEi-Ef,
• (1-f(w)) A(w) M2
• Theory. Compute one particle greens function and
  use spectral function.

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105
Dynamical Mean Field Theory

• Focus on the local spectral function A(w) of the
  solid.
• Write a functional of the local spectral function
  such that its stationary point, give the energy
  of the solid.
• No explicit expression for the exact functional
  exists, but good approximations are available.
• The spectral function is computed by solving a
  local impurity model. Which is a new reference
  system to think about correlated electrons.
• Ref A. Georges G. Kotliar W. Krauth M.
  Rozenberg. Rev Mod Phys 68,1 (1996) .
  Generalizations to realistic electronic
  structure. (G. Kotliar and S. Savrasov in )

106
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.
107
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108
V. Kancharla C. Bolech and GK PRB 67, 075110
(2003)M.CaponeM.Civelli V Kancharla
C.Castellani and GK P. R B 69,195105 (2004)
Testing CDMFT (G.. Kotliar,S. Savrasov, G.
Palsson and G. Biroli, Phys. Rev. Lett. 87,
186401 (2001) ) with two sites in the Hubbard
model in one dimension.
U/t4.
109
Site? Cell. Cellular DMFT. C-DMFT. G..
Kotliar,S. Savrasov, G. Palsson and G. Biroli,
Phys. Rev. Lett. 87, 186401 (2001)
tˆ(K) hopping expressed in the superlattice
notations.

• Other cluster extensions (DCA Jarrell
  Krishnamurthy, M Hettler et. al. Phys. Rev. B 58,
  7475 (1998)Katsnelson and Lichtenstein periodized
  scheme, Nested Cluster Schemes , causality
  issues, O. Parcollet, G. Biroli and GK cond-matt
  0307587 .

110
Searching for a quasiparticle peak
111
Schematic DMFT phase diagram Hubbard model
(partial frustration). Evidence for QP peak in
V2O3 from optics.
M. Rozenberg G. Kotliar H. Kajueter G Thomas D.
Rapkine J Honig and P Metcalf Phys. Rev. Lett.
75, 105 (1995)
112
ARPES measurements on NiS2-xSexMatsuura et. Al
Phys. Rev B 58 (1998) 3690. Doniaach and Watanabe
Phys. Rev. B 57, 3829 (1998)
.
113
QP in V2O3 was recently found Mo et.al
114
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115
k organics

• ET BEDT-TTFBisethylene dithio
  tetrathiafulvalene
• K (ET)2 X
• Increasing pressure -----? increasing t
  ?------------
• X0 X1 X2
  X3
• (Cu)2CN)3 Cu(NCN)2 Cl Cu(NCN2)2Br Cu(NCS)2
• Spin liquid Mott transition

116
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117
Large and ultrafast optical nonlinearities
Sr2CuO3 (T Ogasawara et.a Phys. Rev. Lett. 85,
2204 (2000) )
118
More examples

• LiCoO2
• Used in batteries, laptops, cell phones

119
Large thermoelectric power in a metal with a
large number of carriers NaCo2O4
120
Vanadium Oxide Transport under pressure.
Limelette etal
121
Mean-Field Classical vs Quantum
Classical case
Quantum case
A. Georges, G. Kotliar (1992)
Phys. Rev. B 45, 6497
122
Expt. Wong et. al.
123
V. Kancharla C. Bolech and GK PRB 67, 075110
(2003)M.CaponeM.Civelli V Kancharla
C.Castellani and GK P. R B 69,195105 (2004)
Testing CDMFT (G.. Kotliar,S. Savrasov, G.
Palsson and G. Biroli, Phys. Rev. Lett. 87,
186401 (2001) ) with two sites in the Hubbard
model in one dimension.
U/t4.
124
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.
125
Failure of the standard model Anomalous
ResistivityLiV2O4
Takagi et.al. PRL 2000
126

1
2
4
3
A. Georges and G. Kotliar PRB 45, 6479 (1992).
G. Kotliar,S. Savrasov, G. Palsson and G.
Biroli, PRL 87, 186401 (2001) .
127
Mott Transition in Actinides
The f electrons in Plutonium are close to a
localization-delocalization transition
(Johansson, 1974) .
Mott Transition
after G. Lander, Science (2003).
after J. Lashley et.al, cond-mat (2005).
This regime is not well described by traditional
techniques of electronic structure techniques and
require new methods which take into account the
itinerant and the localized character of the
electron on the same footing.
128
Resistivity in Americium
(after Griveau et.al, PRL 2005)
Resistivity behavior
Superconductivity

• Under pressure, resistivity of Am raises almost
  an order of magnitude and
• reaches its value of 500 mWcm
• Superconductivity in Am is observed with Tc
  0.5K

129
Photoemission in Am, Pu, Sm
Atomic multiplet structure emerges from measured
photoemission spectra in Am (5f6), Sm(4f6) -
Signature for f electrons localization.
after J. R. Naegele, Phys. Rev. Lett. (1984).
130
Am Equation of State LDADMFT Predictions
Self-consistent evaluations of total energies
with LDADMFT using matrix Hubbard I method.
Accounting for full atomic multiplet structure
using Slater integrals F(0)4.5 eV, F(2)8 eV,
F(4)5.4 eV, F(6)4 eV
New algorithms allow studies of complex
structures.
Theoretical P(V) using LDADMFT
Predictions for Am I

• LDADMFT predictions
• Non magnetic f6 ground state with J0 (7F0)
• Equilibrium Volume
• Vtheory/Vexp0.93
• Bulk Modulus Btheory47 GPa
• Experimentally B40-45 GPa

Predictions for Am II
Predictions for Am III
Predictions for Am IV
131
Photoemission Spectrum from 7F0 Americium
LDADMFT Density of States
Matrix Hubbard I Method
F(0)4.5 eV F(2)8.0 eV F(4)5.4 eV F(6)4.0 eV
Experimental Photoemission Spectrum (after J.
Naegele et.al, PRL 1984)
132
Atomic Multiplets in Americium
LDADMFT Density of States
Matrix Hubbard I Method
F(0)4.5 eV F(2)8.0 eV F(4)5.4 eV F(6)4.0 eV
Exact Diag. for atomic shell F(0)4.5 eV
F(2)8.0 eV F(4)5.4 eV F(6)4.0 eV
133
Alpha and delta Pu
134
Failure of the StandardModel Anomalous Spectral
Weight Transfer
Optical Conductivity o of FeSi for T20,40, 200
and 250 K from Schlesinger et.al (1993)
Neff depends on T
135
DMFT Impurity cavity construction
136
A. C. Lawson et. al. LA UR 04-6008F(T,V)Fphonons
Finvar
137
Invar model A. C. Lawson et. al. LA UR 04-6008
138
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139
Small amounts of Ga stabilize the d phase (A.
Lawson LANL)
140
Breakdown of standard model

• Large metallic resistivities exceeding the Mott
  limit.
• Breakdown of the rigid band picture.
• Anomalous transfer of spectral weight in
  photoemission and optics.
• LDAGW loses its predictive power.
• Need new reference frame, to think about and
  compute the properties of correlated materials.
• Need new starting point to do perturbation
  theory.

141
Limit of large lattice coordination
Metzner Vollhardt, 89
Muller-Hartmann 89
142
K. Haule , Pu- photoemission with DMFT using
vertex corrected NCA.
143
The electron in a solid particle picture.

• Array of hydrogen 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
144
M. Rozenberg et. al. Phys. Rev. Lett. 75, 105
(1995)
T/W
Phase diagram of a Hubbard model with partial
frustration at integer filling. Thinking about
the Mott transition in single site DMFT. High
temperature universality
145
Optical transfer of spectral weight , kappa
organics. Eldridge, J., Kornelsen, K.,Wang,
H.,Williams, J., Crouch, A., and Watkins, D.,
Sol. State. Comm., 79, 583 (1991).
146
RESTRICTED SUM RULES
Below energy
Low energy sum rule can have T and doping
dependence . For nearest neighbor it gives the
kinetic energy.
147
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148
Treatement needs refinement

• The kinetic energy of the Hubbard model contains
  both the kinetic energy of the holes, and the
  superexchange energy of the spins.
• Physically they are very different.
• Experimentally only measures the kinetic energy
  of the holes.

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151
Hubbard model

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

Mott transition as a function of doping, pressure
temperature etc.
152
Single site DMFT cavity construction A.
Georges, G. Kotliar, PRB, (1992)
Weiss field
Semicircular density of states. Behte lattice.
153
Photoemission and the Theory of Electronic
Structure
Local Spectral Function
Limiting case itinerant electrons
Limiting case localized electrons Hubbard bands

154
Cellular DMFT studies of the doped Mott insulator
the plaquette as a reference frame. Dynamical
RVB
Collaborators M. Civelli, K. Haule, M. Capone,
O. Parcollet, T. D. Stanescu, (Rutgers) V.
Kancharla (RutgersSherbrook) A. M Tremblay, D.
Senechal B. Kyung (Sherbrooke) .
155
Evolution of the Spectral Function with
Temperature
Anomalous transfer of spectral weight connected
to the proximity to the Ising Mott endpoint
(Kotliar Lange nd Rozenberg Phys. Rev. Lett. 84,
5180 (2000)
156
One Particle Spectral Function and Angle
Integrated Photoemission
e

• Probability of removing an electron and
  transfering energy wEi-Ef, and momentum k
• f(w) A(w, K) M2
• Probability of absorbing an electron and
  transfering energy wEi-Ef, and momentum k
• (1-f(w)) A(w K ) M2
• Theory. Compute one particle greens function and
  use spectral function.

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