Electronic Structure of Correlated Materials : a DMFT Perspective
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Electronic Structure of Correlated Materials : a DMFT Perspective
Gabriel Kotliar
Physics Department andCenter for Materials Theory
Rutgers Universityand KITP
Institute for Theoretical PhysicsUCSB Santa Barbara
Brookhaven National Laboratory
September 12th 2002
Supported by the NSF DMR 0096462
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Outline
The Mott transition problem and electronic structure.
Dynamical Mean Field Theory Model Hamiltonian Studies of the Mott
transition. Universal aspects. System specific studies of materials.
LDA+DMFT. Some case studies. Outlook
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Weakly correlated electrons:band theory. Simple conceptual picture of the ground
state, excitation spectra, transport properties of many systems (simple metals, semiconductors,….).
A methods for performing quantitative calculations. (Density functional theory, in various approximations).
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Momentum Space (Sommerfeld)
Standard model of solids (Bloch, Landau) Periodic potential, waves form bands , k in Brillouin zone . Interactions renormalize away.
2 ( )F Fe k k l
h
The electron in a solid: wave picture
Maximum metallic resistivity 200 ohm cm
2
2k
k
m
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Standard Model of Solids Qualitative predictions: low temperature dependence of
thermodynamics and transport.
Optical response, transition between the bands. Qualitative predictions: filled bands give rise to
insulting behavior. Compounds with odd number of electrons are metals.
Quantitative tools: Density Functional Theory with approximations suggested by the Kohn Sham formulation, (LDA GGA) is a successful computational tool for the total energy. Good starting point for perturbative calculation of spectra,eg. GW. Kinetic equations yield transport coefficients.
~H constR~ const S T ~VC T
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Kohn Sham reference system
2 / 2 ( ) KS kj kj kjV r y e y- Ñ + =
( ')( )[ ( )] ( ) ' [ ]
| ' | ( )
LDAxc
KS ext
ErV r r V r dr
r r r
drr r
dr= + +
-ò
2( ) ( ) | ( ) |kj
kj kjr f rr e y=å
Excellent starting point for computation of spectra in perturbation theory in screened Coulomb interaction GW.
Success story : Density Functional Linear Success story : Density Functional Linear ResponseResponse
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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The electron in a solid: particle picture.Ba
Array of hydrogen atoms is insulating if a>>aB.
Mott: correlations localize the electron
e_ e_ e_ e_
Superexchange
Ba
Think in real space , solid collection of atoms
High T : local moments, Low T spin-orbital order
1
T
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Mott : Correlations localize the electron
Low densities, electron behaves as a particle,use atomic physics, 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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Photoemission spectroscopy.
Measures density of states for (BIS)adding and (PES) removing electrons
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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 LDA+U 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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Strong correlation anomalies
Metals with resistivities which exceed the Mott Ioffe Reggel limit.
Transfer of spectral weight which is non local in frequency.
Dramatic failure of DFT based approximations in predicting physical properties.
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Correlated Materials do big things
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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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) )
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Mott transition in V2O3 under pressure or chemical substitution on V-site
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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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Failure of the Standard Model: NiSe2-xSx
Miyasaka and Takagi (2000)
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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
†( )( ) ( )
MFL o n o n HG c i c iw 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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Limit of large lattice coordination
1~ d ij nearest neighborsijt
d
† 1~i jc c
d
†
,
1 1~ ~ (1)ij i j
j
t c c d Od d
~O(1)i i
Un n
Metzner Vollhardt, 89
1( , )
( )k
G k ii i
Muller-Hartmann 89
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1
10
1( ) ( )
( )n nn k nk
G i ii t i
w ww m w
-
-é ùê ú= +Sê ú- + - Sê úë ûå
DMFT Impurity cavity construction
1
10
1( ) ( )
V ( )n nk nk
D i ii
w ww
-
-é ùê ú= +Pê ú- Pê úë ûå
0
1 † 10 0 ( )( )[ ] ( ) [ ( ) ( ) ]n n n n S Gi G G i c i c ia bw w w w- -S = + á ñ
†
0 0
( ) ( , ') ( ') ( , ') o o o o o oc Go c n n U n nb b
s st t t t d t t ¯ ¯+òò
† †
, ,
( )( )ij ij i j j i i ii j i
t c c c c U n n
()
1 100 0 0( )[ ] ( ) [ ( ) ( ) ]n n n n Si G D i n i n iw w w w- -P = + á ñ
,ij i j
i j
V n n
( , ')Do t t+
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E-DMFT references
H. Kajueter and G. Kotliar (unpublished and Kajuter’s Ph.D thesis).
Q. Si and Smith PRL [analysis of quantum critical points]
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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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Insights from DMFT Low temperature Ordered phases . Stability depends on chemistry and crystal structureHigh temperature behavior around Mott endpoint, more universal regime, captured by simple models treated within DMFT. Role of magnetic frustration.
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Schematic DMFT phase diagram Hubbard model (partial frustration)
M. Rozenberg G. Kotliar H. Kajueter G Thomas D. Rapkikne J Honig and P Metcalf Phys. Rev. Lett. 75, 105 (1995)
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Kuwamoto Honig and Appell PRB (1980)M. Rozenberg G. Kotliar H. Kajueter G Thomas D. Rapkikne J Honig and P Metcalf Phys. Rev. Lett. 75, 105 (1995)
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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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Phase Diag: Ni Se2-x Sx
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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 and 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. Doniach and Watanabe Phys. Rev. B 57, 3829 (1998)
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Mott transition in V2O3 under pressure or chemical substitution on V-site
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Anomalous transfer of optical spectral weight V2O3
:M Rozenberg G. Kotliar and H. Kajuter Phys. Rev. B 54, 8452 (1996).
M. Rozenberg G. Kotliar H. Kajueter G Tahomas D. Rapkikne J Honig and P Metcalf Phys. Rev. Lett. 75, 105 (1995)
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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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. 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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Anomalous transfer of optical spectral weight, NiSeS. [Miyasaka and Takagi]
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Anomalous Resistivity and Mott transition 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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Qualitative phase diagram in the U, T , plane (two band Kotliar Murthy Rozenberg PRL (2002).
Coexistence regions between localized and delocalized spectral functions.
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QMC calculationof n vs (Kotliar Murthy Rozenberg PRL 2002, 2 band, U=3.0)
diverges at generic Mott endpoints
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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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Insights from DMFT Mott transition as a bifurcation of an
effective action
Important role of the incoherent part of the spectral function at finite temperature
Physics is governed by the transfer of spectral weight from the coherent to the incoherent part of the spectra. Real and momentum space.
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Two roads for ab-initio calculation of electronic structure of strongly correlated materials
Correlation Functions Total Energies etc.
Model Hamiltonian
Crystal structure +Atomic positions
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Realistic Calculationsof the Electronic Structure of Correlated materials
Combinining DMFT with state of the art electronic structure methods to construct a first principles framework to describe complex materials.
Anisimov Poteryaev Korotin Anhokin and Kotliar J. Phys. Cond. Mat. 35, 7359 (1997)
Savrasov Kotliar and Abrahams Nature 410, 793 (2001))
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Combining LDA and DMFT
The light, SP (or SPD) electrons are extended, well described by LDA
The heavy, D (or F) electrons are localized,treat by DMFT.
LDA already contains an average interaction of the heavy electrons, subtract this out by shifting the heavy level (double counting term)
The U matrix can be estimated from first principles or viewed as parameters
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Materials……
Pu Fe, Ni, La1-x Srx TiO3 NiO …………….
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Case study in f electrons, Mott transition in the actinide series. B. Johanssen 1974 Smith and Kmetko Phase Diagram 1984.
Problems with density functional treatements of PuPu
•DFT in the LDA or GGA is a well established tool for the calculation of ground state properties.•Many studies (APW Freeman, Koelling 1972, ASA and FP-LMTO, Soderlind et. al 1990, Kollar et.al 1997, Boettger et.al 1998, Wills et.al. 1999) show•an equilibrium volume of the an equilibrium volume of the phasephaseIs 35% lower than experimentIs 35% lower than experiment•This is the largest discrepancy ever known in DFT based calculations.•LSDA predicts magnetic long range order which is not observed experimentally (Solovyev et.al.)•If one treats the f electrons as part of the core LDA overestimates the volume by 30%•Weak correlation picture for alpha phase.
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Pu DMFT total energy vs Volume (Savrasov Kotliar and Abrahams Nature 410, 793 (2001)
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Lda vs Exp Spectra (Joyce et.al.)
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Pu Spectra DMFT(Savrasov) EXP (Joyce , Arko et.al)
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Pu
Alpha and delta are strongly correlated,
The coexistence in the toy model, give rise to
two distinct phases in the realistic calculation.
In progress: phonon spectrum, epsilon phase……
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LaxSr1-xO3
Adding holes to a Mott insulator in three dimensions.
Canonical example of a Brinkman Rice system.
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(Tokura et. Al. 1993)A doped Mott insulator:LaxSr1-xO3
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DMFT calculation U near the Mott transition, Rozenberg et.al 94
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Hall Coefficient, electron like.
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La1-xSrxTiO3 photoemission
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Evolution of spectra with doping U=4
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Optical conductivity
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Realistic Computation of Optical Properties : La1-xSrxTiO3
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Case study Fe and Ni
Archetypical itinerant ferromagnets LSDA predicts correct low T moment Band picture holds at low T Main puzzle: at high temperatures has a
Curie Weiss law with a moment larger than the ordered moment.
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Iron and Nickel: crossover to a real space picture at high T (Lichtenstein, Katsnelson and Kotliar Phys Rev. Lett 87, 67205 , 2001)
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Iron and Nickel:magnetic properties (Lichtenstein, Katsenelson,GK PRL 01)
0 3( )q
Meff
T Tc
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Ni and Fe: theory vs exp eff high T moment
Fe 3.1 (theory) 3.12 (expt)
Ni 1.5 (theory) 1.62 (expt)
Curie Temperature Tc
Fe 1900 ( theory) 1043(expt) Ni 700 (theory) 631 (expt)
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Fe and Ni Consistent picture of Fe (more localized) and Ni
(more correlated) Satellite in minority band at 6 ev, 30 % reduction
of bandwidth, exchange splitting reduction .3 ev Spin wave stiffness controls the effects of spatial
flucuations, it is about twice as large in Ni and in Fe
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Photoemission Spectra and Spin Autocorrelation: Fe (U=2, J=.9ev,T/Tc=.8) (Lichtenstein, Katsenelson,Kotliar Phys Rev. Lett 87, 67205 , 2001)
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Photoemission and T/Tc=.8 Spin Autocorrelation: Ni (U=3, J=.9 ev)
Failures of lda for ground state properties. Failures of lda for ground state properties.
NiO dielectric constant. LSDA:35.7 Exp:5.7
Lattice dynamics cannot be predicted:
• Optical G-phonon in MnO within LSDA: 3.04 THz, Experimentally: 7.86 THz (Massidda, et.al, PRL 1999)
• Bulk modulus for metallic Plutonium is one order of magnitude too large within LDA (214 GPa vs. 30 GPa) Also elastic constants are off.(Bouchet, et.al, J.Phys.C, 2001)
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Functional approach allows computation of linear response.(S. Savrasov and GK 2002
Apply to NiO, canonical Mott insulator.
U=8 ev, J=.9ev
Simple Impurity solver Hubbard 1.
Results for NiO: PhononsResults for NiO: Phonons
Solid circles – theory, open circles – exp. (Roy et.al, 1976)
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NiO U=8ev, J=1ev, Savrasov and GK (2002)
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Summary
Introduction to strongly correlated electrons Dynamical Mean Field Theory Model Hamiltonian Studies. Universal
aspects insights from DMFT System specific studies: LDA+DMFT Outlook
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Very Partial list of application of realistic DMFT to materials QP bands in ruthenides: A. Liebsch et al (PRL 2000) phase of Pu: S. Savrasov et al (Nature 2001) MIT in V2O3: K. Held et al (PRL 2001) Magnetism of Fe, Ni: A. Lichtenstein et al PRL (2001) transition in Ce: K. Held et al (PRL 2000); M. Zolfl et al
PRL (2000). 3d doped Mott insulator La1-xSrxTiO3 (Anisimov et.al
1997, Nekrasov et.al. 1999, Udovenko et.al 2002) ………………..
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Acknowledgements: Development of DMFT
Collaborators: V. Anisimov, R. Chitra, V. Dobrosavlevic, D. Fisher, A. Georges, H. Kajueter, W.Krauth, E. Lange, A. Lichtenstein, G. Moeller, Y. Motome, G. Palsson, M. Rozenberg, S. Savrasov, Q. Si, V. Udovenko, X.Y. Zhang
Support: National Science Foundation.
Work on Pu: Departament of Energy and LANL.
Work on Fe and Ni: Office of Naval Research
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Challenges
Short Range Magnetic Correlations without magnetic order.
Single Site DMFT does not capture these effects
2
1~ 0ij i j
j
J S S dd
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Pu: Anomalous thermal expansion (J. Smith LANL)
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Problems with LDA
o DFT in the LDA or GGA is a well established tool for the calculation of ground state properties.
o Many studies (Freeman, Koelling 1972)APW methods
o ASA and FP-LMTO Soderlind et. Al 1990, Kollar et.al 1997, Boettger et.al 1998, Wills et.al. 1999) give
o an equilibrium volume of the an equilibrium volume of the phasephaseIs 35% Is 35% lower than experimentlower than experiment
o This is the largest discrepancy ever known in DFT based calculations.
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Problems with LDA
LSDA predicts magnetic long range order which is not observed experimentally (Solovyev et.al.)
If one treats the f electrons as part of the core LDA overestimates the volume by 30%
Notice however that LDA predicts correctly the volume of the phase of Pu, when full potential LMTO (Soderlind Eriksson and Wills). This is usually taken as an indication that Pu is a weakly correlated system
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Conventional viewpoint Alpha Pu is a simple metal, it can be
described with LDA + correction. In contrast delta Pu is strongly correlated.
Constrained LDA approach (Erickson, Wills, Balatzki, Becker). In Alpha Pu, all the 5f electrons are treated as band like, while in Delta Pu, 4 5f electrons are band-like while one 5f electron is deloclized.
Same situation in LDA + U (Savrasov andGK Bouchet et. al. [Bouchet’s talk]) .Delta Pu has U=4,Alpha Pu has U =0.
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Problems with the conventional viewpoint of Pu
The specific heat of delta Pu, is only twice as big as that of alpha Pu.
The susceptibility of alpha Pu is in fact larger than that of delta Pu.
The resistivity of alpha Pu is comparable to that of delta Pu.
Only the structural and elastic properties are completely different.
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Pu Specific Heat
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Anomalous ResistivityJ. Smith LANL
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MAGNETIC SUSCEPTIBILITY
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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 the model hamiltonian phase diagram once electronic structure is about to vary.
This result resolves one of the basic paradoxes in the physics of Pu.
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Pu: DMFT total energy vs Volume
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Lda vs Exp Spectra
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Pu Spectra DMFT(Savrasov) EXP (Arko et. Al)
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PU: ALPHA AND DELTA
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Case study Fe and Ni
Archetypical itinerant ferromagnets LSDA predicts correct low T moment Band picture holds at low T But at high temperatures, they resemble
more a collection of atoms with reduced moment.
The crossover from low T to high T requires DMFT.
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Iron and Nickel: crossover to a real space picture at high T (Lichtenstein, Katsnelson and GK)
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However not everything in low T phase is OK as far as LDA goes..
Magnetic anisotropy puzzle. LDA predicts the incorrect easy axis for Nickel .(instead of 111)
LDA Fermi surface has features which are not seen in DeHaas Van Alphen ( Lonzarich)
Use LDA+ U to tackle these refined issues, (WE cannot be resolved with DMFT, compare parameters with Lichtenstein’s )
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Some Earlier Work:
Kondorskii and E Straube Sov Phys. JETP 36, 188 (1973)
G. H Dallderop P J Kelly M Schuurmans Phys. Rev. B 41, 11919 (1990)
Trygg, Johansson Eriksson and Wills Phys. Rev. Lett. 75 2871 (1995) Schneider M Erickson and Jansen J. Appl Phys. 81 3869 (1997)
I Solovyev, Lichenstein Terakura Phys. Rev. Lett 80, 5758 (LDA+U +SO Coupling)…….
Present work : Imseok Yang, S Savrasov and GK
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Origin of Magnetic Anisotropy Spin orbit coupling L.S L is a variable which is sensitive to
correlations, a reminder of the atomic physics Crystal fields quench L, interactions enhance
it, T2g levels carry moment, eg levels do not
any redistribution of these no matter how small will affect L.
Both J and U matter !
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Magnetic anisotropy of Fe and Ni LDA+ U
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Conclusion
The character of the localization delocalization in simple( Hubbard) models within DMFT is now fully understood, nice qualitative insights.
This has lead to extensions to more realistic models, and a beginning of a first principles approach interpolating between atoms and band, encouraging results for simple elements
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1
10
1( ) ( )
( )n nn k nk
G i ii t i
w ww m w
-
-é ùê ú= +Sê ú- + - Sê úë ûå
DMFT Review: A. Georges, G. Kotliar, W. Krauth and M. Rozenberg Rev. Mod. Phys. 68,13 (1996)]
†
0 0 0
[ ] ( )[ ( , ')] ( ')o o o oS Go c Go c n nb b b
s st t t t ¯= +òò ò
† †
, ,
( )( )ij ij i j j i i ii j i
t c c c c U n n
0
†( )( ) ( ) ( )L n o n o n S GG i c i c iw w w=- á ñ
10 ( ) ( )n n nG i i iw w m w- = + - D
0
1 † 10 0 ( )( )[ ] ( ) [ ( ) ( ) ]n n n n S Gi G G i c i c ia bw w w w- -S = + á ñ
Weiss field
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Resistivities
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Resistivities
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LiVO4
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Thermodynamics LiVO4
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Resistivity saturation
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Anomalous Resistivities:Doped Hubbard ModelG. Palsson 1998
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IPTNCA
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Anomalous Resistivities: DopedHubbard Model (QMC)
Prushke and Jarrell 1993
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Outlook Systematic improvements, short range correlations. Take a cluster of sites, include the effect of the rest
in a G0 (renormalization of the quadratic part of the effective action). What to take for G0:
DCA (M. Jarrell et.al) , CDMFT ( Savrasov Palsson and GK )
include the effects of the electrons to renormalize the quartic part of the action (spin spin , charge charge correlations) E. DMFT (Kajueter and GK, Si et.al)
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Outlook
Extensions of DMFT implemented on model systems, carry over to more realistic framework. Better determination of Tcs…………
First principles approach: determination of the Hubbard parameters, and the double counting corrections long range coulomb interactions E-DMFT
Improvement in the treatement of multiplet effects in the impurity solvers, phonon entropies, ………
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Ni moment
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Fe moment
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Magnetic anisotropy vs U , J=.95 Ni
1 3
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Magnetic anisotropy Fe J=.8
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Conventional viewpoint Alpha Pu is a simple metal, it can be
described with LDA + correction. In contrast delta Pu is strongly correlated.
Constrained LDA approach (Erickson, Wills, Balatzki, Becker). In Alpha Pu, all the 5f electrons are treated as band like, while in Delta Pu, 4 5f electrons are band-like while one 5f electron is deloclized.
Same situation in LDA + U (Savrasov andGK Bouchet et. al. [Bouchet’s talk]) .Delta Pu has U=4,Alpha Pu has U =0.
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Delocalization-Localization across the actinide series
o f electrons in Th Pr U Np are itinerant . From Am on they are localized. Pu is at the boundary.
o Pu has a simple cubic fcc structure,the phase which is easily stabilized over a wide region in the T,p phase diagram.
o The phase is non magnetic.o Many LDA , GGA studies ( Soderlind et. Al 1990,
Kollar et.al 1997, Boettger et.al 1998, Wills et.al. 1999) give an equilibrium volume of the an equilibrium volume of the phasephaseIs 35% lower than experimentIs 35% lower than experiment
o This is one of the largest discrepancy ever known in DFT based calculations.
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Pu: Anomalous thermal expansion (J. Smith LANL)
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Problems with LDA
o DFT in the LDA or GGA is a well established tool for the calculation of ground state properties.
o Many studies (Freeman, Koelling 1972)APW methods
o ASA and FP-LMTO Soderlind et. Al 1990, Kollar et.al 1997, Boettger et.al 1998, Wills et.al. 1999) give
o an equilibrium volume of the an equilibrium volume of the phasephaseIs 35% Is 35% lower than experimentlower than experiment
o This is the largest discrepancy ever known in DFT based calculations.
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Problems with LDA
LSDA predicts magnetic long range order which is not observed experimentally (Solovyev et.al.)
If one treats the f electrons as part of the core LDA overestimates the volume by 30%
Notice however that LDA predicts correctly the volume of the phase of Pu, when full potential LMTO (Soderlind Eriksson and Wills). This is usually taken as an indication that Pu is a weakly correlated system
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Conventional viewpoint Alpha Pu is a simple metal, it can be
described with LDA + correction. In contrast delta Pu is strongly correlated.
Constrained LDA approach (Erickson, Wills, Balatzki, Becker). In Alpha Pu, all the 5f electrons are treated as band like, while in Delta Pu, 4 5f electrons are band-like while one 5f electron is deloclized.
Same situation in LDA + U (Savrasov andGK Bouchet et. al. [Bouchet’s talk]) .Delta Pu has U=4,Alpha Pu has U =0.
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Problems with the conventional viewpoint of Pu
The specific heat of delta Pu, is only twice as big as that of alpha Pu.
The susceptibility of alpha Pu is in fact larger than that of delta Pu.
The resistivity of alpha Pu is comparable to that of delta Pu.
Only the structural and elastic properties are completely different.
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Pu Specific Heat
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Anomalous ResistivityJ. Smith LANL
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MAGNETIC SUSCEPTIBILITY
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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 the model hamiltonian phase diagram once electronic structure is about to vary.
This result resolves one of the basic paradoxes in the physics of Pu.
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LDA+DMFT functional2 *log[ / 2 ( ) ( )]
( ) ( ) ( ) ( )
1 ( ) ( ')( ) ( ) ' [ ]
2 | ' |
[ ]
R R
n
n KS
KS n n
i
LDAext xc
DC
R
Tr i V r r
V r r dr Tr i G i
r rV r r dr drdr E
r r
G
a b ba
w
w c c
r w w
r rr r
- +Ñ - - S -
- S +
+ + +-
F - F
åò
ò òå
Sum of local 2PI graphs with local U matrix and local G
1[ ] ( 1)
2DC G Un nF = - ( )0( ) iab
abi
n T G i ew
w+
= å
KS ab [ ( ) G V ( ) ]LDA DMFT a br r
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Comments on LDA+DMFT• Static limit of the LDA+DMFT functional , with = HF
reduces to LDA+U• Removes inconsistencies and shortcomings of this
approach. DMFT retain correlations effects in the absence of orbital ordering.
• Only in the orbitally ordered Hartree Fock limit, the Greens function of the heavy electrons is fully coherent
• Gives the local spectra and the total energy simultaneously, treating QP and H bands on the same footing.
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Anomalous Resistivities: DopedHubbard Model (QMC)
Prushke and Jarrell 1993
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Anomalous Resistivities:Doped Hubbard ModelG. Palsson 1998
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DMFT: Methods of Solution
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LDA functional
2log[ / 2 ] ( ) ( )
1 ( ) ( ')( ) ( ) ' [ ]
2 | ' |
n KS KS
LDAext xc
Tr i V V r r dr
r rV r r dr drdr E
r r
w r
r rr r
- +Ñ - -
+ +-
ò
ò ò
[ ( )]LDA r
[ ( ), ( )]LDA KSr V r
Conjugate field, VKS(r)
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Minimize LDA functional
[ ]( )( ) ( ) '
| ' | ( )
LDAxc
KS ext
ErV r V r dr
r r r
d rrdr
= + +-ò
0*2
( ) { )[ / 2 ]
( ) ( ) n
n
ikj kj kj
n KSkj
r f tri V
r r ew
w
r e yw
y +=
+Ñ -=å å
Kohn Sham eigenvalues, auxiliary quantities.
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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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V2O3 resistivity
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LDA+DMFT Self-Consistency loop
G0 G
Im puritySo lver
S .C .C .
0( ) ( , , ) i
i
r T G r r i e w
w
r w+
= å
2| ( ) | ( )k xc k LMTOV H ka ac r c- Ñ + =
DMFT
U
E
0( , , )HHi
HH
i
n T G r r i e w
w
w+
= å
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1
10
1( ) ( )
( )n nn k nk
G i ii t i
w ww m w
-
-é ùê ú= +Sê ú- + - Sê úë ûå
DMFT Impurity cavity construction: A. Georges, G. Kotliar, PRB, (1992)]
†
0 0 0
[ ] ( )[ ( , ')] ( ')o o o oS Go c Go c U n nb b b
s st t t t ¯= +òò ò
† †
, ,
( )( )ij ij i j j i i ii j i
t c c c c U n n
0
†( )( ) ( ) ( )L n o n o n S GG i c i c iw w w=- á ñ
10 ( ) ( )n n nG i i iw w m w- = + - D
0
1 † 10 0 ( )( )[ ] ( ) [ ( ) ( ) ]n n n n S Gi G G i c i c ia bw w w w- -S = + á ñ
Weiss field
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Landau Functional
† †,
2
2
[ , ] ( ) ( ) ( )†
† † † †
0
†
Mettalic Order Para
( )[ ] [ ]
mete
[ ]
[ , ] [ [ ] ]
( )( )
r: ( )
( ) 2 ( )[ ]( )
loc
LG imp
L f f f i i f i
imp
loc f
imp
iF T F
t
F Log df dfe
dL f f f e f Uf f f f d
d
F iT f i f i TG i
i
i
2
2
Spin Model An
[ ] [[ ]2 ]
alogy:
2LG
t
hF h Log ch h
J
G. Kotliar EPJB (1999)
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Ni and Fe: theory vs exp ( T=.9 Tc)/ ordered moment
Fe 1.5 ( theory) 1.55 (expt) Ni .3 (theory) .35 (expt)
eff high T moment
Fe 3.1 (theory) 3.12 (expt) Ni 1.5 (theory) 1.62 (expt)
Curie Temperature Tc
Fe 1900 ( theory) 1043(expt) Ni 700 (theory) 631 (expt)
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Problems with LDA
o DFT in the LDA or GGA is a well established tool for the calculation of ground state properties.
o Many studies (Freeman, Koelling 1972)APW methods
o ASA and FP-LMTO Soderlind et. Al 1990, Kollar et.al 1997, Boettger et.al 1998, Wills et.al. 1999) give
o an equilibrium volume of the an equilibrium volume of the phasephaseIs 35% Is 35% lower than experimentlower than experiment
o This is the largest discrepancy ever known in DFT based calculations.
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Spectral Density Functional The exact functional can be built in perturbation
theory in the interaction (well defined diagrammatic rules )The functional can also be constructed expanding around the the atomic limit. No explicit expression exists.
DFT is useful because good approximations to the exact density functional DFT(r)] exist, e.g. LDA, GGA
A useful approximation to the exact functional can be constructed, the DMFT +LDA functional. Savrasov Kotliar and Abrahams Nature 410, 793 (2001))
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LDA functional
2log[ / 2 ] ( ) ( )
1 ( ) ( ')( ) ( ) ' [ ]
2 | ' |
n KS KS
LDAext xc
Tr i V V r r dr
r rV r r dr drdr E
r r
w r
r rr r
- +Ñ - -
+ +-
ò
ò ò
[ ( )]LDA r
[ ( ), ( )]LDA KSr V r
Conjugate field, VKS(r)
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Failure of the StandardModel: Anomalous Spectral Weight TransferOptical Conductivity o of FeSi for T=,20,20,250 200 and 250 K from Schlesinger et.al (1993)
0( )d
Neff depends on T
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Degenerate Hubbard model
† †
, ,
( )( )ij ij i j j i i ii j i
t c c c c U n n
U/t
Doping or chemical potential
Frustration (t’/t)
T temperatureMott transition as a function of doping, pressure temperature etc.
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Outlook
The Strong Correlation Problem:How to deal with a multiplicity of competing low temperature phases and infrared trajectories which diverge in the IR
Strategy: advancing our understanding scale by scale
Generalized cluster methods to capture longer range magnetic correlations
New structures in k space. Cellular DMFT
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Small amounts of Ga stabilize the phase
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Insights from DMFT: think in term of spectral functions (branch cuts) instead of well defined QP (poles )
Resistivity near the metal insulator endpoint ( Rozenberg et.al 1995) exceeds the Mott limit
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res85520219st9v9t9es
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LDA functional
2log[ / 2 ] ( ) ( )
1 ( ) ( ')( ) ( ) ' [ ]
2 | ' |
n KS KS
LDAext xc
Tr i V V r r dr
r rV r r dr drdr E
r r
w r
r rr r
- +Ñ - -
+ +-
ò
ò ò
[ ( )]LDA r
[ ( ), ( )]LDA KSr V r
Conjugate field, VKS(r)
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Hubbard model
† †
, ,
( )( )ij ij i j j i i ii j i
t c c c c U n n
U/t
Doping or chemical potential
Frustration (t’/t)
T temperatureMott transition as a function of doping, pressure temperature etc.
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Dynamical Mean Field Theory, cavity constructionA. Georges G. Kotliar 92
†
0 0 0
( )[ ( ')] ( ')eff o o o oc c U n nSb b b
s st m t t tt ¯
¶= + - D - +
¶òò ò ( )wD
†( )( ) ( )
MFo n o n SG c i c iw w D=- á ñ 1( )
1( )
( )[ ][ ]
nk
n kn
G ii
G i
ww e
w
=D - -
D
å
† †
, ,
( )( )ij ij i j j i i ii j i
t c c c c U n n
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1
10
1( ) ( )
( )n nn k nk
G i ii t i
w ww m w
-
-é ùê ú= +Sê ú- + - Sê úë ûå
DMFT Impurity cavity construction: A. Georges, G. Kotliar, PRB, (1992)]
†
0 0 0
[ ] ( )[ ( , ')] ( ')o o o oS Go c Go c U n nb b b
s st t t t ¯= +òò ò
† †
, ,
( )( )ij ij i j j i i ii j i
t c c c c U n n
0
†( )( ) ( ) ( )L n o n o n S GG i c i c iw w w=- á ñ
10 ( ) ( )n n nG i i iw w m w- = + - D
0
1 † 10 0 0 0 ( )( )[ ] ( ) [ ( ) ( ) ]n n n n S Gi G G i c i c iw w w w- -S = + á ñ
Weiss field
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Solving the DMFT equations
G 0 G
I m p u r i t yS o l v e r
S . C .C .
•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)]
G0 G
Im puritySo lver
S .C .C .
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Single site DMFT, functional formulation. Construct a functional of the observable ,local Greens function. Express in terms of Weiss field, (semicircularDOS) [G. Kotliar EBJB 99]
† †,
2
2
[ , ] ( ) ( ) ( )†
( )[ ] [ ]
[ ]loc
imp
L f f f i i f i
imp
iF T F
t
F Log df dfe
2
Ising analgoy
[ ] [ [2 ]]2LG
hF h Log ch h
J
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Compressibilty divergence : One band case (Kotliar Murthy and Rozenberg 2002)
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X.Zhang M. Rozenberg G. Kotliar (PRL 1993)
Spectral Evolution at T=0 half filling full frustration
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Spectral Density Functional : effective action construction (Fukuda, Valiev and Fernando , Chitra and Kotliar).
DFT, consider the exact free energy as a functional of an external potential. Express the free energy as a functional of the density by Legendre transformation. DFT(r)]
Introduce local orbitals, R(r-R)orbitals, and local GFG(R,R)(i ) = The exact free energy can be expressed as a functional
of the local Greens function and of the density by introducing (r),G(R,R)(i)]
A useful approximation to the exact functional can be constructed, the DMFT +LDA functional. Savrasov Kotliar and Abrahams Nature 410, 793 (2001))
' ( )* ( , ')( ) ( ')R Rdr dr r G r r i r
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Optical spectral weight
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LDA+DMFT Self-Consistency loop
G0 G
Im puritySo lver
S .C .C .
0( ) ( , , ) i
i
r T G r r i e w
w
r w+
= å
2| ( ) | ( )k xc k LMTOV H ka ac r c- Ñ + =
DMFT
U
E
0( , , )HHi
HH
i
n T G r r i e w
w
w+
= å
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Parallel development: Fujimori et.al
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C-DMFT
C:DMFT The lattice self energy is inferred from the cluster self energy.
0 0cG G ab¾¾® c
abS ¾¾®Sij ijt tab¾¾®
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