strongly correlated electron systems: a dmft perspective

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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Momentum Space (Sommerfeld)

Standard model of solids Periodic potential, waves form bands , k in Brillouin zone

2 ( )F Fe k k l

h

The electron in a solid: wave picture

Maximum metallic resistivity 200 ohm cm

2

2k

k

m

Landau: Interactions renormalize away

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Standard Model of Solids

~ const ~H constR~VC T

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

2 / 2 ( ) KS kj kj kjV r y e y- Ñ + =

( ')( )[ ( )] ( ) ' [ ]

| ' | ( )xc

KS ext

ErV r r V r dr

r r r

drr r

dr= + +

-ò

2( ) ( ) | ( ) |kj

kj kjr f rr e y=å

[ ]

•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_

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

1

T •Superexchange

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Mott : Correlations localize the electronLow 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….)

• Quantitative calculations of Hubbard bands and exchange constants, LDA+ U, Hartree Fock. Atomic Physics.

•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

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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 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

TS

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 StandardModel: 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 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 DMFTThe Mott transition is driven by transfer of spectral weight from low to high energy as we approach the localized phaseControl 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: 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

0( ) ,eff effd P J

iV

Schlesinger et.al (FeSi) PRL 71 ,1748 , (1993) B Bucher et.al. Ce2Bi4Pt3PRL 72, 522 (1994), Rozenberg et.al. PRB 54, 8452, (1996).

2

0( ) ,

ned P J

iV m

ApreciableT dependence found.

, ,H hamiltonian J electric current P polarization

, ,eff eff effH J PBelow energy

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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/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 predictivepower.

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 ProjectDepartment 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                   

High Performance Computing

http://beowulf.rutgers.edu

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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.

[ , ] log[ ] ( ) ( ) [ ]DMFT DMFTij ii iin n niG Tr i t Tr i G i Gw w w-G S =- - S - S +Få

[ ] Sum of 2PI graphs with local UDMFT atom ii

i

GF = Få

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

Introduce local orbitals, R(r-R)orbitals, and local GF G(R,R)(i ) =

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.

' ( )* ( , ')( ) ( ')R Rdr dr r G r r i r

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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 Quantum case

Phys. Rev. B 45, 6497 A. Georges, G. Kotliar (1992)

†

0 0 0

( )[ ( ')] ( ')o o o oc c U n nb b b

s st m t t tt ¯

¶+ - D - +

¶òò ò

( )wD

†( )( ) ( )

MFo n o n SG c i c is sw w D=- á ñ

1( )

1( )

( )[ ][ ]

nk

n kn

G ii

G i

ww e

w

=D - -

D

å

,ij i j i

i j i

J S S h S- -å å

MF eff oH h S=-

effh

0 0 ( )MF effH hm S=á ñ

eff ij jj

h J m h= +å

† †

, ,

( )( )ij ij i j j i i ii j i

t c c c c U n n

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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 <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 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