Introduction to Dynamical Mean Field Theory (DMFT) and its Applications to the Electronic Structure of Correlated Materials

1
Introduction to Dynamical Mean Field Theory
(DMFT) and its Applications to the Electronic
Structure of Correlated Materials
G.Kotliar Physics Department Center for
Materials Theory Rutgers University.
Collaborators K. Haule (Rutgers), C. Marianetti
(Rutgers ) S. Savrasov (UC Davis)

• Zacatecas Mexico PASSI School . Montauk June
  (2006).

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References

• Electronic structure calculations with dynamical
  mean-field theory G. Kotliar, S. Savrasov, K.
  Haule, V. Oudovenko, O. Parcollet, and
  C. Marianetti, Rev. of Mod. Phys. 78, 000865
  (2006).Dynamical Mean Field Theory of Strongly
  Correlation Fermion Systems and the Limit of
  Infinite Dimensions  A. Georges, G.
  Kotliar, W. Krauth, and M. Rozenberg, Rev. of
  Mod. Phys. 68, 13-125 (1996).
• Electronic Structure of Strongly Correlated
  Materials Insights from Dynamical Mean Field
  Theory Gabriel Kotliar and Dieter Vollhardt,
  Physics Today 57, 53 (2004).

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• What is a strongly correlated material ?

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Electrons in a Solidthe Standard Model
Band Theory electrons as waves. Landau Fermi
Liquid Theory.
n band index, e.g. s, p, d,,f
Rigid bands , optical transitions ,
thermodynamics, transport

• Quantitative Tools. Density Functional Theory
• Kohn Sham (1964)

Static Mean Field Theory.
Kohn Sham Eigenvalues and Eigensates Excellent
starting point for perturbation theory in the
screened interactions (Hedin 1965)
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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)
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Kohn Sham reference system
Excellent starting point for computation of
spectra in perturbation theory in screened
Coulomb interaction GW.
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GW approximation (Hedin )
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Kohn Sham Eigenvalues and Eigensates Excellent
starting point for perturbation theory in the
screened interactions (Hedin 1965)
Self Energy
VanShilfgaarde (2005)
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Strong Correlation Problemwhere the standard
model fails

• Fermi Liquid Theory works but parameters
  cant be computed in perturbation theory.
• Fermi Liquid Theory does NOT work . Need new
  concepts to replace of rigid bands !
• 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.

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10
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 (fully localized or fully
  itinerant).
• Situation realized by applying a control
  parameters, e.g. pressure. Metal to Insulator
  Transition.
• Some materials have several species of
  electrons, some localized (f s ds ) some
  itinerant (sp, spd) . OSMT. Heavy Fermions.
• Introducing carries (electrons or holes) to a
  Mott insulator. Doping Driven Mott
  transition.

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• Why is it worthwhile to study correlated electron
  materials ?

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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.

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Strongly correlated systems

• Copper Oxides. High Temperature
  Superconductivity.
• Cobaltates Anomalous thermoelectricity.
• Manganites . Colossal magnetoresistance.
• Heavy Fermions. Huge quasiparticle masses.
• 2d Electron gases. Metal to insulator
  transitions.
• Lanthanides, Transition Metal Oxides,
  Multiferroics..

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Basic Questions

• How does the electron go from being localized to
  itinerant.
• How do the physical properties evolve.
• How to bridge between the microscopic information
  (atomic positions) and experimental measurements.
• New concepts, new techniques

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• How do we probe SCES experimentally ?

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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.

n
n
e
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Spectral Function Photoemission and correlations
e

• Probability of removing an electron and
  transfering energy wEi-Ef, and momentum k
• f(w) A(w, K) M2

1. Weak Correlation
2. Strong Correlation

n
n
Angle integrated spectral function
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Strong Correlation Problemwhere the standard
model fails

• Fermi Liquid Theory works but parameters
  cant be computed in perturbation theory.
• Fermi Liquid Theory does NOT work . Need new
  concepts to replace of rigid bands !
• 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.

4
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• How do we approach the problem of
• strongly correlated electron stystems ?

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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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Strongly correlated systems are usually treated
with model Hamiltonians

• Tight binding form. Eliminate the irrelevant
  high energy degrees of freedom

Add effective Coulomb interaction terms.
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One Band Hubbard model

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

Mott transition as a function of doping, pressure
temperature etc.
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• How do we reduce the many body problem to
  something tractable ?

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DMFT Cavity Construction. Happy marriage of
atomic and band physics.
Extremize a functional of the local spectra.
Local self energy.
Reviews A. Georges G. Kotliar W. Krauth and M.
Rozenberg RMP68 , 13, 1996 Gabriel Kotliar and
Dieter Vollhardt Physics Today 57,(2004). G.
Kotliar S. Savrasov K. Haule V. Oudovenko O.
Parcollet and C. Marianetti Rev. Mod. Phys. 78,
865 (2006) . G. Kotliar and D . Vollhardt
Physics 53 Today (2004)
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Mean-Field Classical vs Quantum
Classical case
Quantum case
A. Georges, G. Kotliar (1992)
Phys. Rev. B 45, 6497
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Single site DMFT Impurity cavity construction
A. Georges, G. Kotliar, PRB, (1992)
Weiss field
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Extension to clusters. Cellular DMFT. C-DMFT. G.
Kotliar,S.Y. Savrasov, G. Palsson and G. Biroli,
Phys. Rev. Lett. 87, 186401 (2001)
tˆ(K) is the hopping expressed in the
superlattice notations.

• Other cluster extensions (DCA, nested cluster
  schemes, PCMDFT ), causality issues, O.
  Parcollet, G. Biroli and GK cond-matt 0307587
  (2003)

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What is the structure of the DMFT problem ?

• Embedding and truncation

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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)

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• How do we generalize this construction realistic
  systems ?

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More general DMFT loop
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Dynamical Mean Field Theory. Cavity Construction.
A. Georges and G. Kotliar PRB 45, 6479 (1992).
A(w)
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A(w)
A. Georges, G. Kotliar (1992)
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Dynamical Mean Field Theory

• Weiss field is a function. Multiple scales in
  strongly correlated materials.
• Exact in the limit of large coordination (Metzner
  and Vollhardt 89) , kinetic and interaction
  energy compete on equal footing.
• Immediate extension to real materials

DFTDMFT
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Evolution of the DOS. Theory and experiments
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DMFT Qualitative Phase diagram of a frustrated
Hubbard model at integer filling
T/W
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Interaction with Experiments. Photoemission Three
peak strucure. V2O3Anomalous transfer of
spectral weight
M. Rozenberg G. Kotliar H. Kajueter G Thomas
D. Rapkine J Honig and P Metcalf Phys. Rev. Lett.
75, 105 (1995)
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Photoemission measurements and Theory
V2O3 Mo, Denlinger, Kim, Park, Allen, Sekiyama,
Yamasaki, Kadono, Suga, Saitoh, Muro, Metcalf,
Keller, Held, Eyert, Anisimov, Vollhardt PRL .
(2003)
.
NiSxSe1-xMatsuura Watanabe Kim Doniach Shen Thio
Bennett (1998)
Poteryaev et.al. (to be published)
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How do we solve the impurity model

• ?

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Methods of solution some examples

• Iterative perturbation theory. A Georges and G
  Kotliar PRB 45, 6479 (1992). H Kajueter and G.
  Kotliar PRL (1996). Interpolative schemes
  (Oudovenko et.al.)
• Exact diag schemes Rozenberg et. al. PRL 72, 2761
  (1994)Krauth and Caffarel. PRL 72, 1545 (1994)
• Projective method G Moeller et. al. PRL 74 2082
  (1995).
• NRG R. Bulla PRL 83, 136 (1999)

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• QMC M. Jarrell, PRL 69 (1992) 168, Rozenberg
  Zhang Kotliar PRL 69, 1236 (1992) ,A Georges and
  W Krauth PRL 69, 1240 (1992) M. Rozenberg PRB 55,
  4855 (1987).
• NCA Prushke et. al. (1993) . SUNCA K. Haule
  (2003).
• Analytic approaches, slave bosons.
• Analytic treatment near special points.

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How good is DMFT ?
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Single site DMFT is exact in the Limit of large
lattice coordination
Metzner Vollhardt, 89
Muller-Hartmann 89
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C-DMFT test in one dimension. (Bolech, Kancharla
GK PRB 2003)
Gap vs U, Exact solution Lieb and Wu, Ovshinikov
Nc2 CDMFT vs Nc1
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N vs mu in one dimensional Hubbard model
.Compare 2 site cluster (in exact diag with
Nb8) vs exact Bethe Anzats, M. Capone C.
Castellani M.Civelli and GK (2003)
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• How do we incorporate the Long Range Coulomb
  Interactions

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DMFT Impurity cavity construction
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How do we merge band theory and DMFT ?

• How do we extract total energies ?

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Spectral density functional theory
(G. Kotliar et.al., RMP 2006).
observable of interest is the "local Green's
functions (spectral function)
Currently feasible approximations LDADMFT
Variation gives st. eq.
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Exact QMC impurity solver, expansion in terms
of hybridization
K.H. Phys. Rev. B 75, 155113 (2007)
P. Werner, Phys. Rev. Lett. 97, 076405 (2006)
General impurity problem
Diagrammatic expansion in terms of hybridization
D Metropolis sampling over the diagrams

• Exact method samples all diagrams!
• Allows correct treatment of multiplets

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• What are the characteristics of the spectra of a
  correlated system , in the simplest model ?

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Pressure Driven Mott transition
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Mott transition in one band model. Review
Georges et.al. RMP 96
T/W
Phase diagram of a Hubbard model with partial
frustration at integer filling. Rozenberg et.
al. PRL 1995 Evolution of the Local Spectra as a
function of U,and T. Mott transition driven by
transfer of spectral weight Zhang Rozenberg
Kotliar PRL (1993)..
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Spectral Evolution at T0 half filling full
frustration
X.Zhang M. Rozenberg G. Kotliar (PRL 1993)
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Parallel development Fujimori et.al
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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 nd Rozenberg Phys. Rev. Lett. 84,
5180 (2000)
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Order in Perturbation Theory

• n1

Order in PT
n2
Basis set size.
l1
DMFT
r1

l2
r site CDMFT
r2
GW
llmax
GW first vertex correction
Range of the clusters
68

• How do we go directly from structure to
  physical observables ?

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Can the various approaches (DMFT, DFT, DFTU be
unified )?
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Spectral density functional. Effective action
construction.e.g Fukuda et.al
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In practice we need good approximations to the
exchange correlation, in DFT LDA. In spectral
density functional theory, DMFT. Review
Kotliar et.al. Rev. Mod. Phys. 78, 865 (2006)
Kohn Sham equations
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Different methods differ by the choice of
variable a used.

• DFT
• Spin and Density FT

Spectral Density Functional R. Chitra and G.K
Phys. Rev. B 62, 12715 (2000). S. Savrasov and
G.K PRB (2005)
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C DMFT extend the notion of localityto several
unit cells
DFTDMFT

• U (and form of dc) are input parameters.
• EDMFT a Gloc Wloc Cluster Greens Function
  and Screened interaction, No input
  parameters.
• Recently impelemented and tested for sp systems.
  Si C .
• N. Zein et.al.PRL 96, (2006) 226403 Zein and
  Antropov PRL 89,126402
• Review Kotliar et.al. Rev. Mod. Phys. 78, 865
  (2006)

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• What is the ultimate theory , without any
  external parameters ?

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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 Spetral Density Functional of Gloc and
Wloc.
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Mean-Field Classical vs Quantum
Classical case
Quantum case
Hard!!!
Easy!!!
QMC J. Hirsch R. Fye (1986) NCA T. Pruschke
and N. Grewe (1989) PT Yoshida and Yamada
(1970) NRG Wilson (1980)
A. Georges, G. Kotliar (1992)
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EDMFT loop G. Kotliar and S. Savrasov in New
Theoretical Approaches to Strongly Correlated G
Systems, A. M. Tsvelik Ed. 2001 Kluwer Academic
Publishers. 259-301 . cond-mat/0208241 S. Y.
Savrasov, G. Kotliar, Phys. Rev. B 69, 245101
(2004)

• Full implementation in the context of a a one
  orbital model. P Sun and G. Kotliar Phys. Rev. B
  66, 85120 (2002).
• After finishing the loop treat the graphs
  involving Gnonloc Wnonloc in perturbation theory.
  P.Sun and GK PRL (2004). Related work, Biermann
  Aersetiwan and Georges PRL 90,086402 (2003) .
•  

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Conclusion

• DMFT, method under very active development. But
  there is now a clear formulation (and to large
  extent implementation) as a fully self
  consistent, controlled many body approach to
  solids.
• It gives good quantitative results for total
  energies, phonon and photoemission spectra, and
  transport of materials. Many examplesall over
  the periodic table.
• Helpful in developing intuition and
  qualitative insights in correlated electron
  materials.
• With advances in implementation, we will be able
  to focus on deviations from (cluster) dynamical
  mean field theory.

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The Mott transition problem

• Universal and non universal aspects.
• Frustration and the success of DMFT. In the
  phases without long range order, DMFT is valid if
  T gt Jeff. Need frustration to supress it. When T
  lt Jeff LRO sets in. If Tneel is to high it
  oblitarates the Mott phenomena.
• t vs U fundamental competition and secondary
  instabilities.

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V2O3 under pressure or
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Schematic DMFT phase diagram of a partially
frustrated integered filled Hubbard model.
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S.-K. Mo et al., Phys. Rev. Lett. 90, 186403
(2003)..
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Schematic DMFT phase Implications for transport.
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Material Properties total energy and phonon
spectra