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THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
Insights into real materials : DMFT at work. From theoretical solid state physics to
materials science.
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
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. Inspired by the LDA+U approach (Anisimov, Andersen and Zaanen)
Anisimov Poteryaev Korotin Anhokin and Kotliar (1997). Lichtenstein and Katsenelson (1998)
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
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, substract this out by shifting the heavy level (double counting term)
The U matrix can be estimated from first principles or viewed as parameters
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
Spectral Density Functional : 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, 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)]
' ( )* ( , ')( ) ( ')R Rdr dr r G r r i r
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
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.
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
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
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
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
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
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.
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
Case study in f electrons, Mott transition in the actinide series. B. Johanssen 1974 Smith and Kmetko Phase Diagram 1984.
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
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.
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
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, using full potential LMTO (Soderlind and Wills). This is usually taken as an indication that Pu is a weakly correlated system.
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
Pu: DMFT total energy vs Volume (Savrasov Kotliar and Abrahams Nature 2001)
THE STATE UNIVERSITY OF NEW JERSEY
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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 much larger than the ordered moment.
Magnetic anisotropy
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
Iron and Nickel: crossover to a real space picture at high T (Lichtenstein, Katsnelson and GK PRL 2001)
THE STATE UNIVERSITY OF NEW JERSEY
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Iron and Nickel:magnetic properties (Lichtenstein, Katsenelson,GK PRL 01)
0 3( )q
Meff
T Tc
0 3( )q
Meff
T Tc
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
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)
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
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
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
Ni low T puzzles
Magnetic anisotropy puzzle. LDA predicts the incorrect easy axis(100) for Nickel .(instead of the correct one (111)
LDA Fermi surface has features which are not seen in DeHaas Van Alphen ( Lonzarich)
Use LDA+ U to tackle these refined issues, ( compare parameters with DMFT results )
I. Yang S. Savrasov and G. Kotliar PRL2001
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
Fe and Ni 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
Mean field calculations using measured exchange constants(Kudrnovski Drachl PRB 2001) right Tc for Ni but overestimates Fe , RPA corrections reduce Tc of Ni by 10% and Tc of Fe by 50%.
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
DMFT: References
Collaborators: V. Anisimov, R. Chitra, V. Dobrosavlevic, D. Fisher, A. Georges, H. Kajueter, W.Krauth, E. Lange, G. Moeller, Y. Motome, G. Palsson, M. Rozenberg, S. Savrasov, Q. Si, V. Udovenko, X.Y. Zhang
Other work: A. Brandt, W. Nolting, R. Bulla, M. Jarrell, D. Logan, J. Freericks, T. Prushke, W. Metzner, F. Gebhardt, A. Lichtenstein, M. Fleck D. Vollhardt ……………….
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
Reviews of DMFT
Prushke T. Jarrell M. and Freericks J. Adv. Phys. 44,187 (1995)
A. Georges, G. Kotliar, W. Krauth and M. Rozenberg Rev. Mod. Phys. 68,13 (1996)]
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
Challenges The photoemission in cuprates has a strong
momentum dependence Strong Magnetic Correlations (no orbital
degeneracy) Single Site DMFT does not capture these
effects
2
1~ 0ij i j
j
J S S dd
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
Cuprates: Photoemission
–
Transfer of Spectral Weight
with a) temperature and b) doping
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
DMFT Spin Orbital Ordered StatesLonger range interactions Coulomb, interactions, Random Exchange (Sachdev and Ye, Parcollet and Georges, Kajueter and Kotliar, Si and Smith, Chitra and Kotliar,)Short range magnetic correlations. Cluster Schemes. (Ingersent and Schiller, Georges and Kotliar, cluster expansion in real space, momentum space cluster DCA Jarrell et.al., C-DMFT Kotliar et. al ).
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
DMFTFormulation as an electronic structure method (Chitra and Kotliar)Density vs Local Spectral FunctionExtensions to treat strong spatial inhomogeneities. Anderson Localization (Dobrosavlevic and Kotliar),Surfaces (Nolting),Stripes (Fleck Lichtenstein and Oles)Practical Implementation (Anisimov and Kotliar, Savrasov, Katsenelson and Lichtenstein)
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
Cuprates: Photoemission
–
Transfer of Spectral Weight
with a) temperature and b) doping
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
Anomalous Resistivities: DopedHubbard Model (Prushke and Jarrell 1993)
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THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
Anomalous Resistivities:Doped Hubbard ModelG. Palsson 1998
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IPT
NCA
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
Failure of the “Standard Model”: Cuprates
Anomalous Resistivity
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
Mott transition in layered organic conductors S Lefebvre et al. cond-mat/0004455
THE STATE UNIVERSITY OF NEW JERSEY
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Standard Model
Typical Mott values of the resistivity 200 Ohm-cm
Residual instabilites SDW, CDW, SC
Odd # electrons -> metal
Even # electrons -> insulator
Theoretical foundation: Sommerfeld, Bloch and Landau
Computational tools DFT in LDA
Transport Properties, Boltzman equation , low temperature dependence of transport coefficients2 ( )
MottF Fe k k l
h
Mott
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
Failure of the “Standard Model”: Cuprates
Anomalous Resistivity
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
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,
NRG,ED….)•Analytical Methods
G0 G
Im puritySo lver
S .C .C .
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
DMFTFormulation as an electronic structure method (Chitra and Kotliar)Density vs Local Spectral FunctionExtensions to treat strong spatial inhomogeneities. Anderson Localization (Dobrosavlevic and Kotliar),Surfaces (Nolting),Stripes (Fleck Lichtenstein and Oles)Practical Implementation (Anisimov and Kotliar, Savrasov, Katsenelson and Lichtenstein)
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
DMFT Spin Orbital Ordered StatesLonger range interactions Coulomb, interactions, Random Exchange (Sachdev and Ye, Parcollet and Georges, Kajueter and Kotliar, Si and Smith, Chitra and Kotliar,)Short range magnetic correlations. Cluster Schemes. (Ingersent and Schiller, Georges and Kotliar, cluster expansion in real space, momentum space cluster DCA Jarrell et.al., C-DMFT Kotliar et. al ).
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
Photoemission Spectra and Spin Autocorrelation: Fe (U=2, J=.9ev,T/Tc=.8) (Lichtenstein, Katsenelson,GK prl 2001)
THE STATE UNIVERSITY OF NEW JERSEY
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Photoemission and T/Tc=.8 Spin Autocorrelation: Ni (U=3, J=.9 ev)
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Strongly Correlated Electrons
Competing Interaction
Low T, Several Phases Close in Energy
Complex Phase Diagrams
Extreme Sensitivity to Changes in External Parameters
Need for Quantitative Methods
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
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
0( )d
Neff depends on T
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
Hubbard model
† †
, ,
( )( )ij ij i j j i i ii j i
t c c c c U n n
U/t
Doping d or chemical potential
Frustration (t’/t)
T temperatureMott transition as a function of doping, pressure temperature etc.
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
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)
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
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)
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
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.
THE STATE UNIVERSITY OF NEW JERSEY
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A time-honored example: Mott transition in V2O3 under pressure or chemical substitution on V-site
THE STATE UNIVERSITY OF NEW JERSEY
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Ising character of the transfer of spectral weight
Ising –like dependence of the photo-emission intensity and the optical spectral weight near the Mott transition endpoint
THE STATE UNIVERSITY OF NEW JERSEY
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X.Zhang M. Rozenberg G. Kotliar (PRL 1993)
Spectral Evolution at T=0 half filling full frustration