Strongly Correlated Electron Systems: a DMFT Perspective

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Strongly Correlated Electron Systems a DMFT
Perspective

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

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• REVIEW OF SOLID STATE THEORY.
• Chapter 1. The Standard Model of Solids.

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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
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Standard Model of Solids
RIGID BAND PICTURE. Optical response, transitions
between bands.
Quantitative tools DFT, LDA, GGA, total
energies,good starting point for spectra, GW,and
transport

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Density functional and Kohn Sham reference
system

• Kohn Sham spectra, proved to be an excelent
  starting point for doing perturbatio theory in
  screened Coulomb interactions GW.

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LDAGW semiconducting gaps
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• Solid State Physics
• Chapter 2 . Mott insulators.

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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

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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.

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• Solid State Physics
• Chapter 3, strongly correlated electrons.
• Status unfinished.

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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 GW or LDAU or Hartree Fock work
  well.

• Need approach which interpolates correctly
  between atoms and bands. Treats QP bands and
  Hubbard bands. New reference point, to replace
  the Kohn Sham system.

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Failure of the standard model

• DMFT is a new reference frame to approach
  strongly correlated phenomena, and describes
  naturally , NON RIGID BAND picture, highly
  resistive states, treats quasiparticle
  excitations and Hubbard bands on the same
  footing..

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Correlated Materials do big things

• Mott transition.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.

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Pressure Driven Mott transition
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V2O3 resistivity
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Cuprate Superconductors
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• Correlated Electron Materials are based on
  different physical principles outside the
  standard model, exciting perspectives for
  technological applications (e.g. high Tc).

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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).
• Large and ultrafast optical nonlinearities
  Sr2CuO3 (T Ogasawara et.a Phys. Rev. Lett. 85,
  2204 (2000) )
• Huge volume collapses, Ce, Pu.

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Large thermoelectric power in a metal with a
large number of carriers NaCo2O4
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Large and ultrafast optical nonlinearities
Sr2CuO3 (T Ogasawara et.a Phys. Rev. Lett. 85,
2204 (2000) )
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More examples

• LiCoO2
• Used in batteries, laptops, cell phones

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Breakdown of standard model

• Many Qualitative Failures
• Large metallic resistivities exceeding the Mott
  limit. Anderson, Emery and Kivelson
• Breakdown of the rigid band picture.
• Anomalous transfer of spectral weight in
  photoemission and optics. G. Sawatzki

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Failure of the standard model Anomalous
ResistivityLiV2O4
Takagi et.al. PRL 2000
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Failure of the StandardModel Anomalous Spectral
Weight Transfer
Optical Conductivity Schlesinger et.al (1993)
Neff depends on T
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Breakdown of the standard model Anomalous
transfer of optical weight D. Van der Marel
group
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Breakdown of the Standard Model

• The LDAGW program fails badly,
• Qualitatively incorrect predictions.
• Incorrect phase diagrams.
• Physical Reason The one electron spectra,
  contains both Hubbard Bands and Quasiparticle
  featurs.

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Basic Difficulties

• Lack of a small parameter. Kinetic energy is
  comparable to Coulomb energies.
• Relevant degrees of freedom change their form in
  different energy scales, challenge for
  traditional RG methods.
• WANTED a simple picture of the physical
  phenomena, and the physics underlying a given
  material.
• WANTED a computational tool to replace LDAGW

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• Breakthru Development of Dynamical Mean Field
  Theory.

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Dynamical Mean Field Theory

• Basic idea reduce the quantum many body problem
  to a one site or a cluster of sites, in a medium
  of non interacting electrons obeying a self
  consistency condition.
• Basic idea instead of using functionals of the
  density, use more sensitive functionals of the
  one electron spectral function. density of
  states for adding or removing particles in a
  solid, measured in photoemission

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The Mott transition

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

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Mott transition in V2O3 under pressure or
chemical substitution on V-site
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Pressure Driven Mott transition
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Insight

• Phase diagram in the T, U plane of a frustrated
  ((the magnetic order is supressed)) correlated
  system at integer filling.
• At high temperatures, the phase diagram is
  generic, insensitive to microscopid details.
• At low temperatures, every detail matters.

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Schematic DMFT phase diagram one band Hubbard
model (half filling, semicircular DOS, partial
frustration) Rozenberg et.al PRL (1995)
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Pressure Driven Mott transition
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Insight, in the strongly correlated region the
one particle density of states has a three peak
structureLow energy Quasiparticle Peak plus
Hubbard bands.
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DMFT has bridged the gap between band theory and
atomic physics.

• Delocalized picture, it should resemble the
  density of states, (perhaps with some additional
  shifts and satellites).
• Localized picture. Two peaks at the ionization
• and affinity energy of the atom.

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One electron spectra near the Mott transition.
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Insights from DMFT

• The Mott transition is driven by transfer of
  spectral weight from low to high energy as we
  approach the localized phase
• Control parameters doping, temperature,pressure

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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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ARPES measurements on NiS2-xSexMatsuura et. Al
Phys. Rev B 58 (1998) 3690. Doniaach and Watanabe
Phys. Rev. B 57, 3829 (1998)
.
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QP in V2O3 was recently found Mo et.al
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Anomalous metallic resistivities

• In the in between region anomalous
• resistivities are the rule rather than the
  exception.

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Failure of the Standard Model NiSe2-xSx
Miyasaka and Takagi (2000)
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Anomalous Resistivity and Mott transition
(Rozenberg et. Al. ) Ni Se2-x Sx
Insights from DMFT think in term of spectral
functions (branch cuts) instead of well defined
QP (poles )
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More recent work, organics, Limelette et. al.
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Anomalous Resistivities when wave picture does
not apply. Doped Hubbard model
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Qualitative single site DMFT predictions Optics

• Spectra of the strongly correlated metallic
  regime contains both quasiparticle-like and
  Hubbard band-like features.
• Mott transition is drive by transfer of spectral
  weight. Consequences for optics.

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Anomalous transfer of spectral weight heavy
fermions Rozenberg Kajueter Kotliar (1996)
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Anomalous Spectral Weight Transfer Optics
Below energy
ApreciableT dependence found.
Schlesinger et.al (FeSi) PRL 71 ,1748 , (1993) B
Bucher et.al. Ce2Bi4Pt3PRL 72, 522 (1994),
Rozenberg et.al. PRB 54, 8452, (1996).
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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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Ising critical endpoint! In V2O3 P. Limelette
et.al. Science Vol 302 (2003).
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Conclusion.

• An electronic model accounts for all the
  qualitative features of the finite temperature of
  a frustrated system at integer occupancy.
• The observation of the spinodal lines and the
  wide classical critical region where mean field
  holds indicate the coupling to the lattice is
  quantitatively very important.

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• Formations of structures in k space.
• Cluster dynamical mean field study.
• Parcollet Biroli and Kotliar Cond-Matt 0308577

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Mott transition in layered organic conductors
S Lefebvre et al. cond-mat/0004455, Phys. Rev.
Lett. 85, 5420 (2000)
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• Evolution of the distribution in k space of the
  low energy spectral intensity as the Mott
  transition is approached.

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U/D2, U/D2.25 (Parcollet et.al.)
Uc2.35-.05, Tc/D1/44
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Conjecture

• Formation of hot regions is a more general
  phenomena due to the proximity to the Mott point.
• Plaquette reference system is good enough to
  contain the essential features of momentum space
  differentiation. Application to cuprates.

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Lattice and cluster self energies
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Mechanism for hot spot formation
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Deviations from single site DMFT
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System specific application Pu
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Pu in the periodic table
actinides
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Electronic Physics of Pu
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DFT studies.

• Underestimates the volume by 35
• Predicts Pu to be magnetic.
• Largest quantitative failure of DFT-LDA-GA
• Fails to predict a stable delta phase. (negative
  shear)

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Alpha and delta Pu
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Pu DMFT total energy vs Volume (Savrasov
Kotliar and Abrahams 2001)
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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 until
  recently.

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Phonon freq (THz) vs q in delta Pu X. Dai et. al.
Science vol 300, 953, 2003
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Expts Wong et. al. Science 301. 1078 (2003)
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• Plutonium is just one correlated element, there
  are many many more strongly correlated COMPOUNDS
  which can be studies with this method.
• Worldwide activity.

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• Test the idea that the crucial physics of
  strongly correlated materials can be captured
  from a local reference set.
• Test worst case scenario, one dimension.
  Kancharla and Bolech Capone and Civelli.

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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
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N vs mu in one dimension.Comparaison of 28 vs
exact Bethe Anzats, M. Capone and M.Civelli
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What do we want from materials theory?

• New concepts , qualitative ideas
• Understanding, explanation of existent
  experiments, and predictions of new ones.
• Quantitative capabilities with predictive
• power.
• Notoriously difficult to achieve in strongly
  correlated materials. DMFT is delivering on both
  counts.

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Outlook

• Local approach to strongly correlated electrons
  offers a new starting point or reference frame to
  describe new physics.
• Breakdown of rigid band picture.
• Many extensions, make the approach suitable for
  getting insights and quantitative results in many
  correlated materials.
• RESEARCH OPPORTUNITY.

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Networks.

• Psik f electrons. European Research and training
  network, with nodes in Denmark, UK, France,
  Germany, and Holland and NJ.
• Computational Material Science Network.
• CMSN, with nodes at Brookhaven, UC Davis,
  ORNL, NJIT, Rutgers, NRL, Cornell,

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Students and Postdocs

• Marcelo Rozenberg. Development of DMFT.
• Goetz Moeller. Theory of the Mott transition.
• Henrik Kajueter. Development of techniques for
  solving DMFT equations.
• Indranil Paul. Thermal Transport in Correlated
  Materials.
• Sergej Pankov. Extensions of DMFT and studies of
  disordered system and electron phonon
  interactions.

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Students and Postdocs

• Vlad Dobrosavlevic. Studies of disordered systems
  with DMFT. Metal Insulator Transition in two
  dimensions.
• Ping Sun. Combinations of EDMFT and GW. Studies
  of Heavy fermion critical points.
• Sahana Murthy. Study of the Mott transition in
  americium.

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Students and Postdocs

• Sergej Savrasov DMFT study of the volume
  collapse and photoemission in plutonium.
• Viktor Udovenko Thermoelectric power of
  correlated materials. Optical studies of
  correlated materials.
• Christjan Haule New techniques for solving the
  DMFT equations. Optical studies of Cerium.

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Postdocs Students.

• Harald Jeschke, development of DMFT solvers, and
  molecular dynamics using DMFT.
• Qimiao Si. Non Fermi liquid states using DMFT
  and its extensions.
• Ekke Lange. Magneto-transport studies of
  correlated materials. Landau theory of the Mott
  transition.
• Michael Sindel Andrea Perali and Marcello
  Civelli hot spots in cuprates.

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Students and Postdocs

• Chris Marianetti, DMFT studies of materials for
  battery applications.
• Olivier Parcollet. and Giulio Biroli Extensions
  of DMFT to clusters. High temperature
  superconductitivity and organic materials..
• Venky Kancharla and Carlos Bolech, development of
  DMFT-DMRG for clusters. Applications to charge
  density wave materials

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Students and Postdoc

• Marcello Civelli and Massimo Capone. High
  temperature superconductivity using C-DMFT.
• Antonina Toropova CrO2, a high temperature half
  metallic systems.
• Tudor Stanescu. Recent improvement of DMFT
• Xi Dai. Phonons in plutonium.

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Acknowledgements Development of DMFT
Collaborators V. Anisimov,G. Biroli, R.
Chitra, V. Dobrosavlevic, X. Dai, D. Fisher,
A. Georges, H. Kajueter, K. Haujle, W.Krauth, E.
Lange, A. Lichtenstein, G. Moeller, Y. Motome,
O. Parcollet , 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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Materials Science

• New concepts.
• Techniques. Analytical. Quantum Field Theory and
  the Renormalization Group.
• Numerical. New algoritms. Hardware.

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High Performance Computing http//beowulf.rutgers
.edu
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Epsilon Plutonium.
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Anomalous transfer of spectral weight heavy
fermions
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Anomalous transfer of spectral weight
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Anomalous transfer of spectral weigth heavy
fermions
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Example DMFT for lattice model (e.g. single band
Hubbard).Muller Hartman 89, Chitra and Kotliar 99.

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

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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.

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Mott transition and superexchange
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How good is the LOCAL approximation?
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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
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N vs mu in one dimension.Compare 28 vs exact
Bethe Anzats, M. Capone and M.Civelli
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Strongly correlated systems are usually treated
with model Hamiltonians
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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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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.
• DMFT, build functionals of the LOCAL spectral
  function.
• Exact functionals exist. We also have good
  approximations!
• Extension to an ab initio method.

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Realistic applications of DMFT References

• 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 and Abrahams
  funcional formulation for full self consistent
  Nature \bf 410, 793(2001).
• Reviews Held et.al. , Psi-k Newsletter \\bf
  56 (April 2003), p. 65 Lichtenstein Katsnelson
  and and Kotliar cond-mat/0211076

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Double Occupancy vs U

• CDMFT Parcollet, Biroli GK

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Compare with single site results Rozenberg Chitra
Kotliar PRL 2002
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Mott transition in cluster
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Evolution of the Spectral FunctionU/D2, U/D2.25
(Parcollet et.al.)
Uc2.35-.05, Tc/D1/44