gabriel kotliar physics department and center for materials theory rutgers university
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Spectral Density Functional: a first principles approach to the electronic structure of correlated solids. Gabriel Kotliar Physics Department and Center for Materials Theory Rutgers University. 2001 JRCAT-CERC Workshop on Phase Control on Correlated Electron Systems. Outline. - PowerPoint PPT PresentationTRANSCRIPT
Spectral Density Functional: a first principles approach to the electronic
structure of correlated solids
Gabriel Kotliar
Physics Department and
Center for Materials Theory
Rutgers University
2001 JRCAT-CERC Workshop on
Phase Control on Correlated Electron Systems
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Outline Motivation. Some universal
aspects of simple DMFT the Mott transition endpoint in frustrated systems.
Non universal physics requires detailed material modeling. Combining DMFT and band structure a new functional for electronic structure calculations (S. Savrasov and GK)
Results: d electrons Fe and Ni.
(Lichtenstein, Katsenelson and GK, PRL in press)
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Outline
Results: f electrons delta Pu ( S. Savrasov G. K and E. Abrahams,Nature (2001))
Conclusions: further extensions the approach.
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Importance of Mott phenomena
Evolution of the electronic structure between the atomic limit and the band limit. Basic solid state problem. Solved by band theory when the atoms have a closed shell. Mott’s problem: Open shell situation.
The “”in between regime” is ubiquitous central them in strongly correlated systems. Some unorthodox examples
Fe, Ni, Pu.
Solution of this problem and advances in electronic structure theory (LDA +DMFT)
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A time-honored example: Mott transition in V2O3 under pressure
or chemical substitution on V-site
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Phase Diag: Ni Se2-x Sx
G. Czek et. al. J. Mag. Mag. Mat. 3, 58 (1976)
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Mott transition in layered organic conductors Ito et al. (1986) Kanoda (1987) Lefebvre et al. (2001)
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Theoretical Approach to the Mott
endpoint. DMFT.Mean field approach to quantum
many body systems, constructing equivalent impurity models embedded in a bath to be determined self consistently . Use exact numerical techniques (QMC, ED ) as well as semianalytical (IPT) approaches to solve this simplified problem.
Study simple model Hamiltonians (such as the one band model on simple lattices)
Understand the results physically in terms of a Landau theory :certain high temperature aspects are independent of the details of the model and the approximations used.
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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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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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Schematic DMFT phase diagram one band Hubbard model (half filling, semicircular DOS, role of partial frustration) Rozenberg et.al PRL (1995)
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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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Functional Approach
The Landau functional offers a direct connection to the atomic energies
Allows us to study states away from the saddle points,
All the qualitative features of the phase diagram, are simple consequences of the non analytic nature of the functional.
Mott transitions and bifurcations of the functional .
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Insights into the Mott phenomenaThe 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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A time-honored example: Mott transition in V2O3 under pressure
or chemical substitution on V-site
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Evolution of the Spectral Function with Temperature
Anomalous transfer of spectral weight connected to the proximity to an Ising Mott endpoint (Kotliar et.al.PRL 84, 5180 (2000))
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Ising character of Mott endpoint
SingularpartoftheWeissfieldisproportionaltoMax{(p-pc)(T-Tc)}1/inmeanfieldand5in3d
couplestoallphysicalquantitieswhichthenexhibitakinkattheMottendpoint.Resistivity,doubleoccupancy,photoemissionintensity,integratedopticalspectralweight,etc.
Divergenceofthethecompressibility,inparticleholeasymmetricsituations,e.g.FurukawaandImada
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Compressibility
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Mott transition endpoint
Rapid variation has been observed in optical measurements in vanadium oxide and nises mixtures
Experimental questions: width of the critical region. Ising exponents or classical exponents, validity of mean field theory
Building of coherence in other strongly correlated electron systems.
condensation of doubly occupied sites and onset of coherence .
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Insights from DMFT: think in term of spectral functions , the density is not changing!
Resistivity near the metal insulator endpoint ( Rozenberg et.al 1995) exceeds the Mott limit
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Anomalous Resistivity and Mott transition Ni Se2-x Sx
Miyasaka and Tagaki (2000)
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. ARPES measurements on NiS2-xSex
Matsuura et. Al Phys. Rev B 58 (1998) 3690
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Two Roads for first principles calculations of
correlated materials using DMFT.
Correlation functions etc..
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Insights from DMFT Low temperatures several competing phases . Their relative stability depends on chemistry and crystal structure (ordered phases)High temperature behavior around Mott endpoint, more universal regime, captured by simple models treated within DMFT
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LDA+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, substract this out by shifting the heavy level (double counting term)
The U matrix can be estimated from first principles of viewed as parameters
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DMFT +LDA : effective action construction (Fukuda, Valiev and Fernando , Chitra and GK).
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, andf local Greens function by projecting onto the local orbitals.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)]
' ( )* ( , ')( ) ( ')dr dr r G r r i r
' ( )* ( , ')( ) ( ')R Rdr dr r G r r i r
( )R r
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LDA+DMFT
The functional can be built in perturbation theory in the interaction (well defined diagrammatic rules )The functional can also be constructed from the atomic limit.
DFT is useful because e good approximations to the exact density functional DFT(r)] exist, e.g. LDA….
A useful approximation to the exact functional can be constructed, the DMFT +LDA functional.
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LDA+DMFT functional
2
[ , , , ]
log[ / 2 ( , , ')]
( ) ( ) ( ) ( )
1 ( ) ( ')( ) ( ) ' [ ]
2 | ' |
[ ] [ ]
n
lda dmft KS
n KS n
KS n n
i
LDAext xc
DC
V G
Tr i V i r r
V r r dr Tr i G i
r rV r r dr drdr E
r r
G G
w
r
w w
r w w
r rr r
+G S
=- +Ñ - - S -
- S +
+ + +-
F - F
åò
ò ò
( )01[ ] ( 1) , ( )
2i
DC ab
abi
G Un n n T G i ew
w+
F = - = å
Double counting correction
Sum of local 2PI graphs with local U matrix and local G
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Spectral density functionalConnection with atomic limit
1[ ] [ ] [ ] logat atG W Tr G Tr G TrG G-F = D - D - +
1 10 atG G
[ ] atSatW Log e
1 10 0[ ] ( ) ( , ') ( ') ( ) ( ) ( ) ( )at a a abcd a b c d
ab
S G c G c U c c c c
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LDA+DMFT Self-Consistency loop
G0 G
Im puritySolver
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
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Realistic DMFT loop
( )k LMTOt H k E® -LMTO
LL LH
HL HH
H HH
H H
é ùê ú=ê úë û
ki i Ow w®
10 niG i Ow e- = + - D
0 0
0 HH
é ùê úS =ê úSë û
0 0
0 HH
é ùê úD=ê úDë û
0
1 †0 0 ( )( )[ ] ( ) [ ( ) ( )HH n n n n S Gi G G i c i c ia bw w w w-S = + á ñ
110
1( ) ( )
( ) ( ) HH
LMTO HH
n nn k nk
G i ii O H k E i
w ww w
--é ùê ú= +Sê ú- - - Sê úë ûå
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LDA functional
2
[ , , , ]
log[ / 2 ( , , ')]
( ) ( ) ( ) ( )
1 ( ) ( ')( ) ( ) ' [ ]
2 | ' |
[ ] [ ]
n
lda dmft KS
n KS n
KS n n
i
LDAext xc
DC
V G
Tr i V i r r
V r r dr Tr i G i
r rV r r dr drdr E
r r
G G
w
r
w w
r w w
r rr r
+G S
=- +Ñ - - S -
- S +
+ + +-
F - F
åò
ò ò
( )01[ ] ( 1) , ( )
2i
DC ab
abi
G Un n n T G i ew
w+
F = - = å
Double counting correction
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LDA+DMFT References
V. Anisimov, A. Poteryaev, M. Korotin, A. Anokhin and G. Kotliar, J. Phys. Cond. Mat. 35, 7359-7367 (1997).
ALichtensteinandM.KatsenelsonPhys.Rev.B57,6884(1988).
S.SavrasovandG.Kotliar,funcionalformulationforfullselfconsistentimplementation(2001)
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Iron and Nickel: band picture at low T, crossover to real space picture at high T
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Photoemission Spectra and Spin Autocorrelation: Fe(U=2, J=.9ev) (Lichtenstein, Katsenelson,GK prl in press)
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Photoemission and Spin Autocorrelation: Ni (U=3, J=.9 ev)
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Iron and Nickel:mgnetic properties (Lichtenstein, Katsenelson,GK cond-mat 0102297)
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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 3.09 (theory) 3.12 (expt)
Ni 1.50 (theory) 1.62 (expt)
Curie Temperature Tc
Fe 1900 ( theory) 1043(expt) Ni 700 (theory) 631 (expt)
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Fe and Ni
Spin wave stiffness controls the effects of spatial flucuations, it is about twice as large in Ni and in Fe
Classical calculations using measured exchange constants
(Kudrnovski Drachl PRB 2001) Weiss mean field theory gives right Tc for Ni but overestimates
Fe , RPA corrections reduce Tc of Ni by 10% only but reduce Tc of Fe by nearly factor of 2.
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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 Is 35% lower than experimentexperiment
o This is one of the largest discrepancy ever known in DFT based calculations.
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Small amounts of Ga stabilize the phase
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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% lower than Is 35% lower than experimentexperiment
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%
LDA predicts correctly the volume of the phase of Pu, when full potential LMTO (Soderlind and Wills). This is usually taken as an indication that Pu is a weakly correlated system
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Pu: DMFT total energy vs Volume (S. Savrasov )
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Lda vs Exp SpectraD
OS
, st./
[eV
*cel
l]
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Pu Spectra DMFT(Savrasov) EXP (Arko et. Al)
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Conclusion
The character of the localization delocalization in simple( Hubbard) models within DMFT is now fully understood. (Rutgers –ENS), nice qualitative insights.
This has lead to extensions to more realistic models, and a beginning of a first principles approach interpolating between atoms and bands.
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Conclusions 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 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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Conclusions
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.