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LDA+U and beyondA. Lichtenstein
University of Hamburg
In collaborations with:
V. Anisimov (Ekaterinburg)M. Katsnelson (Nijmegen)
Outline
• Complexity of Transition Metal Systems
• LDA+U: spin-charge-orbital ordering
• LDA+DMFT: dynamical effects
• Conclusions
From Atom to Solids
Theory: interactions vs. hopping
Coulomb inraatomic interaction
Multiband Hubbard model (<im|jm0 >=δ ijδ mm0 )
Matrix elements of electron-electron interactions:
Exact diagonalization of atom: tij=0 gives multiplets!Solution with hoppings tij≠0 in solids is unknown!
Strong correlations in real f-systems
Multiplets in solids: Hubbard-I1 2 3 4 1 2 3 4| |m m m m eeU m m V m m=< >
A.L. and M.Katsnelson, PRB (1998)
Control parameters• Bandwidth (U/W)• Band filling• Dimensionality
Degrees of freedom• Charge / Spin• Orbital • Lattice
3d - 4fopen shells
materials
U<<WCharge fluct.
U>>WSpin fluct.
• Kondo• Mott-Hubbard• Heavy Fermions• High-Tc SC• Spin-charge order• Colossal MR
Nd2-xCexCuO4 La2-xSrxCuO4
0.3 0.2 0.10
100
200
300
SC
AFTem
pera
ture
(K)
Dopant Concentration x0.0 0.1 0.2 0.3
SC
AF
Pseudogap
'Normal'Metal
La1-xCaxMnO3
Dopant Concentration x
CMR
FM
I II IIIb IVb Vb VIb VIIb VIIIb Ib IIb III IV V VI VII 0H HeLi Be B C N O F NeNa Mg Al Si P S Cl Ar
Rb Sr Y Zr Nb Mo Tc Ru Rh Pd Ag Cd In Sn Sb Te I XeCs Ba La* Hf Ta W Re Os Ir Pt Au Hg Tl Pb Bi Po At RnFr Ra Ac** Rf Db Sg Bh Hs Mt
Lanthanides *Actinides ** Th Pa U Np Pu Am Cm Bk Cf Es Fm Md No Lr
K Ca Sc Ti V Cr Mn Fe Co Ni Cu Zn Ga Ge As Se Br Kr
Ce Pr Nd Pm Sm Eu Gd Tb Dy Ho Er Tm Yb Lu
Strongly Correlated Electron Systems
DFT: Computational Material Science
CERAN-plate
LDA-modeling:LiAlSiO4
A.L. & R. Jones
Strong correlations in real system
Local moments above Tc
Multiplets in solids
E
“Real” U?
1 2 3 4 1 2 3 4| |m m m m eeU m m V m m=< >
Correlation driven MIT
U/W
photoemission spectra (DOS)A. Fujimori et al.
Charge transfer TMO insulators
Zaanen-Sawatzky-Allen(ZSA) phase diagram
Mott-Hubbard
Charge-Transfer
Eg
Eg~U
~ Δ
(WM+WL
)/2 Δ
Insulator
U
MW
NiOFeO
LaMnO3
V2O
3
TiO
V 2O5p-
met
al
d-metal
CuO
EFN
(E)
EW
U
Δ
dn-1
pL
n+1d
• eg orbitals
• t2g orbitals
Mn (3+) = 3d4
5x3x
2x eg
t2g
3d-ion in cubic crystal field
d
Orbital degrees of freedom
Model Hamiltonians
Hubbard and Anderson models unknown parameters many-body explicit Coulomb correlations
Density Functional Theory
LDA, GGAab-initioone-electron averaged Coulomb interaction
LDA++
Coulomb correlations problem
combined LDA+U and LDA+DMFT approaches(GW, TD-DFT are alternative ways)
LDA+U: static mean-filed approximation
LDA+U functional:
One-electron energies: )n21(U
nE
iLDAii −+ε=
∂∂=ε
Occupied states: 2U1n LDAii −ε=ε⇒=
LDA i j d dij
U UE E n n - n (n -1)2 2
= + ∑
Empty states:2U0n LDAii +ε=ε⇒=
Mott-Hubbard
gap
d
LDAn
U∂ε∂≡
V. Anisimov, PRB, 44, 943 (1991)
LDAε
Rotationally invariant LDA+ULDA+U functional
Local screend Coulomb correlations
LDA-double counting term (nσ =Tr(nmm0σ ) and n=n↑ +n⇓ ):
Occupation matrix for correlated electrons:
A. I. Lichtenstein, J. Zaanen, and V. I. Anisimov, PRB 52, R5467 (1995)
Slater parametrization of UMultipole expansion:
Coulomb matrix elements in Ylm basis:
Slater integrals:
Angular part – 3j symbols
Average interaction: U and JAverage Coulomb parameter:
Average Exchange parameter:
For d-electrons:Coulomb and exchange interactions:
Constrained LDA calculation of U and J
Gunnarsson-1989 supercell with cutting hybridisation
Norman-1995 estimation of screening parameter
Full-potential LDA+U: a problem
ELDA+U= LDA + U - DC
= + -
= + - No
OK!
S. Dudarev et. A. PRB 57, 1505 (1998)
Spherical RI-LDA+U
Interchange –possible!
Exchange interaction couplings
Calculation of J from LDA+U results:↓
′↑
′′′′′′′′′′′′′′ −≡χ= ∑ imm
imm
imm
jmm
ijmmmm
}m{
immij VVIIIJ
A.Lichtenstein et al, Phys. Rev.B 52, R5467 (1995)
mjlk'n
'ilmk'n
mjlnk
ilmnk
'knn k'nnk
k'nnkijmmmm cccc
ff ′′′
↓↓∗′′
↑∗↑
↓↑
↑↓′′′′′′ ∑ ε−ε
−=χ
ji
2
ijij
jiij
EJSSJEθ∂θ∂
∂== ∑
rr
Heisenberg Hamiltonian parameters:
LDA+U eigenvaluesand eigenfunctions: ∑ >=Ψε σσσ
ilm
ilmnknknk ilm|c;
Exchange interactions from LDA++Heisenberg exchagne:
Magnetic torque:
Exchange interactions:
Spin wave spectrum:
M. Katsnelson and A. Lichtenstein, Phys. Rev. 61, 8906 (2000)
Non-collinear magnetism :
Electronic structure of TMO: LDA+U
0
4
8
12MnO
Den
sity
of S
tate
s (s
tate
s/eV
form
ula
unit)
LSDA
0
4
8U= 5eV
0
4
8U= 9eV
-12 -8 -4 0 4Energy (eV)
0
4
8U= 13eV
NiO
LSDA
U= 5eV
U= 9eV
-12 -8 -4 0 4 8Energy (eV)
U= 13eV0
100
200
300
400
w(q
), m
eV
G Z F G L
U =13LDA5
791113exp
DOS
Spin-waveSpectrum
NiOI. Solovyev
MnO NiO
O2p3d 3d
Orbital order: KCuF3
hole density of the same symmetry
A.Lichtenstein et al, Phys. Rev.B 52, R5467 (1995);
In KCuF3 Cu+2 ion has d9 configuration
with a single hole in eg doubly degenerate subshell.
Experimental crystal structure
antiferro-orbital order
LDA+U calculations for undistortedperovskite structure
Cooperative Jahn-Teller distortions in KCuF3
Quadrupolar distortion in KCuF3
Superexchange interaction
J(K) Jc JabTheory -240 +6Exp. -202 +3
KCuF3
LSDA gave cubic perovskite crystal structure stable in respect to Jahn-Teller distortion of CuF6 octahedra
Only LDA+U produces total energy minimum for distorted structure
Mechanism: OO – Kugel-Khomskii
Spin and Orbital moments in CoO
LDA+U+SO+non-collinear
L
L
S
I. Solovyev, A. L, and K. Terakura, PRL 80, 5758 (1998)
CoPt: LDA+U calculation of MAE
A. Shik, O. Mryasov, PRB (2003)
LDA+U : Forces and Orbital OrderingAFM FMLaTiOLaTiO33 YTiOYTiO33
S. Okatov, et. al.Europhys. Lett. (2004)
LDA and charge order problem
;nU)nn(;nU)nn(;dndU 00
LSDA200
LSDA1 δ+ε=δ+εδ−ε=δ−ε
ε=
Charge disproportionation in LSDA is unstable due to self-interaction problem
in LDA+U self-interaction is explicitly canceled
)n21(U))nn(
21(U)nn(
)n21(U))nn(
21(U)nn(
0000LSDA22
0000LSDA11
−−ε=δ+−+δ+ε=ε
−−ε=δ−−+δ−ε=ε
Charge order in Fe3O4
half of the octahedral positions is occupied by Fe+3 and other half by Fe+2.
V.Anisimov et al, Phys. Rev.B 54, 4387 (1996)
Fe3O4 has spinelcrystal structure
one Fe+3 ion in tetrahedral position (A)
two Fe+2.5 ions in octahedral positions (B)
Below TV=122K a charge ordering happens Verwey transition
Simultaneous metal-insulator transition:
LDA+U: charge ordering in Fe3O4
Charge and orbital order in experimental low-temperature monoclinic crystal structure Fe3O4
ΔQ=0.1 eΔQm=0.7 e
I.Leonov et al, PRL93,146404 (2004)
CaV2O5 and MgV2O5 CaV3O7 CaV4O9
V.Anisimov et al, Phys.Rev.Lett. 83, 1387 (1999)
Exchange interactions in layered vanadates
n=3: CaV3O7 has unusual long-range spin ordern=4: CaV4O9 is a frustrated (plaquets) system with a spin gap value 107K n=2: CaV2O5 is a set of weakly coupled dimerswith a large spin gap 616 K isostructural MgV2O5 has very small spin gap value < 10K
CaVnO2n+1
QMC solution of Heisenberg model
M. Troyer et. al.: Comparison of the calculated and measured susceptibility
LDA+U in fully-localized limit (LDA+U-FLL)
LDA+U in around mean-field limit (LDA+U-AMF)
( ) ( )( )↓↓↓↑↓↑↓↑↓↑↓↑ −−+−+= nnnnUnnnnnnUnnU mmmmmm '''
“mean-field” = LDA LDA+U
[ ]431
4321
21 24314231 ,,,,21
γγγγγγ
γγ δγγγγγγγγδ nUUnE AFM −= ∑
∑−=+
=−=l
lmmmn
lnnnn σσσ
γγσ
γγ δδγγ ,12
1 , 21
1
2121
Around mean-field limit of LDA+U+SO
m ,γ = σ Spin-orbitals
General LDA+U formulation22
2( )2 2 2
−⎡ ⎤= − − +⎢ ⎥⎣ ⎦∑ ∑U
JnUn U JH S n P nσσ σ σ σ
σ σ
AMF: 1/(2 1), 0 SIC: 0, 1/ 2
S l PS P= + == =
FLL is the right “DFT” mean field for localized systems, nmσ= 1 or 0
AMF is the right “DFT” mean field for for uniform occupancy, nmσ= <nσ>
2 2/(2 1) mm
n l n nσ σ σ+ ≤ ≤∑Generalization: (2l+1)Sσ+Pσ=1
A. Petukhov, et. al., PRB 67, 153106 (2003).
Electronic structure of δ-Pu in AMF LDA+U
Pu f-band configuration in AFM LDA+U is close to f6: f5/2 states are filled and 8 states f7/2 are empty
A. Shick et al. Europhys. Lett. 69, 588 (2005)
From Atom to Solid
E
N(E)
EF
QPLHB UHB
E
N(E)
EF
Atomic physics Bands effects (LDA)
LDA+DMFT
E
N(E)
EF dndn+ 1| SL>
Dynamical Mean Field Theory
Σ Σ Σ
Σ Σ
Σ Σ Σ
ΣU
( )ττ ′−0G
W. Metzner and D. Vollhardt PRL(1989)A. Georges and G. Kotliar PRB(1992)Europhysics Prize (2006)
DMFT: Self-Consistent Set of Equations
( ) ∑→
⎟⎠⎞
⎜⎝⎛
Ω=
→BZ
k
nn ikGiG ωω ,ˆ1ˆ
( ) ( ) ( )nnn iiGi ωωω Σ+= −− ˆˆˆ 110G
QMC ED
DMRG IPTFLEX
( ) ( ) ( )nnnnew iGii ωωω 110
ˆˆˆ −− −=Σ G
Quantum Impurity Solver
Σ Σ Σ
Σ
Σ
Σ
ΣΣ
U
U
G( ’)τ−τ
ττ’
Local Dynamics: LDA+DMFT
LDA+UStatic mean-field approximationEnergy-independent potential
|minlVinlm|V̂mm
mm σ′<>σ= ∑σ′
σ′
LDA+DMFTDynamic mean-field approximation
Energy-dependent self-energy operator
|minl)(inlm|)(ˆmm
mm σ′<εΣ>σ=εΣ ∑σ′
σ′
Applications:Insulators with long-range
spin-,orbital- and charge order
Applications:Paramagnetic, paraorbitalstrongly correlated metals
short range spin and orbital order
Cluster LDA+DMFT approximation
V. Anisimov, et al. J. Phys. CM 9, 7359 (1997)A. Lichtenstein, et. al. PRB, 57, 6884 (1998)
A. Poteryaev, A. Lichtenstein, and G. Kotliar, PRL 93, 086401 (2004)S. Biermann, A. Poteryaev, A. I. Lichtenstein, and A. Georges
Phys. Rev. Lett. 94, 026404 (2005)
• Materials-specific (structure, Z, etc.)
• Fast code packages
• Fails for strong correlations
LDA+DMFT
( ) ( ) ( )1
0ˆˆˆˆ
−→
∑ ⎥⎦
⎤⎢⎣
⎡ Σ−⎟⎠⎞
⎜⎝⎛−+=
BZ
knnn ikHIiiG ωμωω
LDA Models approaches
• Input parameters unknown
• Computationally expensive
• Systematic many-body scheme
dcLDA EkHkH −⎟⎠⎞
⎜⎝⎛=⎟
⎠⎞
⎜⎝⎛ →→ ˆˆ
0
Multi band Quantum Monte Carlo
Multiorbital CT-QMC: general U-vertex
-4 -2 0 2 40.0
0.2
0.4
0.6
W=2U=2J=0.2
DO
S (1
/eV)
Energy (eV)
5 orbitals, full U-vertex
τ
G(τ) Uijkl
Udiag
E. Gorelov, et. al. to be published A. Rubtsov and A.L., JETP Lett. (2004)
Spectral function –ARPES and DMFT
Van Hove=10 meVm*/m=2.1-2.6
ARPES (A. Damascelli, et al PRL2000)LDA+DMFT
SrSr22RuORuO44
before renormalization
after renormalization
Conclusions
LDA+U is an accurate scheme for realistic description of electronic structure, spin, orbital and charge ordering in complex transition metal systems
LDA+DMFT method is useful for dynamical, short-range non-local Coulomb correlations effects in solids: metal-insulator transition