Download Issues in Strongly Correlated Electron Physics: A DMFT Perspective

Strongly Correlated
Electron Systems: a DMFT
Perspective
Gabriel Kotliar
Physics Department and
Center for Materials Theory
Rutgers University
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)
k2
k 
2m
e 2 k F (k F l )
h
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
 ~
const
CV ~ T
RH ~ const
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
- Ñ / 2 + VKS (r ) y kj = ekj y kj
2
r (r ) =
å
f (ekj ) | y kj (r ) |2
kj
VKS (r )[r (r )] = Vext (r ) +
ò
dExc
r (r ')
dr '+
[r ]
| r- r'|
dr (r )
•Kohn Sham spectra, proved to be an excelent
starting point for doing perturbatio theory in
screened Coulomb interactions GW.
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LDA+GW: 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 a>>aB. 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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1
T
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
RUTGERS
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 LDA+U 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*/m=1000
(La1-xSrx)MnO3 Colossal Magnetoresistance.
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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
T
S
V
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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 Resistivity:LiV2O4
Takagi et.al. PRL 2000
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Failure of the Standard
Model: Anomalous Spectral
Weight Transfer
Optical Conductivity Schlesinger et.al
(1993)


0
 ( )d
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 LDA+GW 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
LDA+GW
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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 structure
Low 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-xSex
.Matsuura 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:
Miyasaka and
NiSe2-xSx
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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Title:
Anomalous Resistivities when wave
picture does not apply. Doped
Hubbard model
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
H hamiltonian, J electric current , P polarization


 ne2
0  ( )d  iV   P, J   m
Below energy


0
 ( )d 

iV
H eff , J eff , Peff

  Peff , J eff  
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
al. cond-mat/0004455, Phys. Rev. Lett. 85, 5420 (2000)
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S Lefebvre et
•Evolution of the distribution
in k space of the low energy
spectral intensity as the Mott
transition is approached.
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U/D=2, U/D=2.25 (Parcollet et.al.)
Uc=2.35+-.05, Tc/D=1/44
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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
Nc=2 CDMFT
vs Nc=1
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N vs mu in one dimension.
Comparaison of 2+8 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 CDMFT.
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
High Performance Computing Project
Department of Physics and Astronomy
National Science Foundation
- NSF0116068: Acquisition of a
Network Cluster of Advanced
Workstations for First Principles Electronic Structure
Calculations of
Complex Materials
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TOP 500 (ICL-UT)
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TOP 500
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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 ().
Exact functional  [Gii () .
DMFT Approximation to the functional.
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Spectral Density Functional : effective
action construction (Chitra and GK).




Introduce local orbitals, aR(r-R)orbitals, and local
GF
G(R,R)(i ) =  dr ' dr  (r ) *G(r, r ')(i ) a (r ')
R
R
The exact free energy can be expressed as a
functional of the local Greens function and of the
density by introducing sources for (r) and G and
performing a Legendre transformation,
(r),G(R,R)(i)]
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
Nc=2 CDMFT
vs Nc=1
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N vs mu in one dimension.
Compare 2+8 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
-
å
J ij Si S j - h å Si
i, j
Quantum case
 (t

i , j  ,
i
H MF = - heff So
b
ij
 m ij )(ci† c j  c †j ci )  U  ni  ni 
i
b
b
¶
†
ò ò cos (t )[ ¶ t + m- D (t - t ')]cos (t ') +U ò no- no¯
0 0
0
heff
D ( w)
m0 = áS0 ñHMF ( heff )
heff =
å
J ij m j + h
G = - áco†s (iwn )cos (iwn )ñSMF (D )
G (iwn ) =
å
k
j
Phys. Rev. B 45, 6497
1
[D (iwn ) -
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1
- ek ]
G (iwn )[D ]
A. Georges, G. Kotliar (1992)
DMFT: Effective Action point of view.
R. Chitra and G. Kotliar Phys Rev. B.
(2000).
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Identify observable, A. Construct an exact functional of
<A>=a,  [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
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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/D=2, U/D=2.25
(Parcollet et.al.)
Uc=2.35+-.05, Tc/D=1/44
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