![Page 1: Applications: LDA+DMFT scheme - I - uni-hamburg.deDMFT-1.pdf · Outline-I • From Atom to Solids: Multiplets in solids • Functionals: MFT, DFT, SDF • Multiorbital DMFT for General](https://reader031.vdocument.in/reader031/viewer/2022021916/5d490ff888c9931f048bb12e/html5/thumbnails/1.jpg)
Applications: LDA+DMFT scheme - I
A. Lichtenstein University of Hamburg
In collaborations with:A. Poteryaev (Ekaterinburg), M. Rozenberg, S. Biermann, A. Georges (Paris)L. Chioncel (Graz) , I. di Marco, M. Katsnelson (Nijmegen)E. Pavarini (Jülich), O.K. Andersen (Stuttgart)G. Kotliar (Rutgers), S. Savrasov (Devis), A. Rubtsov (Moscow)F. Lechermann, H. Hafermann, T. Wehling, C. Jung, M. Karolak (Hamburg)
http://www.physnet.uni-hamburg.de/hp/alichten/lectures/LDA+DMFT-1.pdf
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Outline-I
• From Atom to Solids: Multiplets in solids
• Functionals: MFT, DFT, SDF
• Multiorbital DMFT for General Lattice
• Matrix version LDA+DMFT Scheme
• Impurity solvers: CT-QMC
• Examples of LDA+DMFT
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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
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- How to incorporate atomic physics in the band structure ?
- How good is a local approximation ?
- What is a best solution for atomic problem in effective medium ?
- What is different from one band Hubbard model?
- How to solve a complicated Quantum multiorbital problem ?
- What is the best Tight-Binding scheme for realistic Many-Body calculation for solids?
From Atom to Solids
![Page 5: Applications: LDA+DMFT scheme - I - uni-hamburg.deDMFT-1.pdf · Outline-I • From Atom to Solids: Multiplets in solids • Functionals: MFT, DFT, SDF • Multiorbital DMFT for General](https://reader031.vdocument.in/reader031/viewer/2022021916/5d490ff888c9931f048bb12e/html5/thumbnails/5.jpg)
Theory of everything: t vs. U
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!
![Page 6: Applications: LDA+DMFT scheme - I - uni-hamburg.deDMFT-1.pdf · Outline-I • From Atom to Solids: Multiplets in solids • Functionals: MFT, DFT, SDF • Multiorbital DMFT for General](https://reader031.vdocument.in/reader031/viewer/2022021916/5d490ff888c9931f048bb12e/html5/thumbnails/6.jpg)
Strong correlations in real systems
Local moments above Tc
Multiplets in solids Hub-I
E
“Real” U?
1 2 3 4 1 2 3 4| |m m m m eeU m m V m m=< >
A.L. and M. Katsnelson, PRB (1998)
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Hubbard-I approximation
![Page 8: Applications: LDA+DMFT scheme - I - uni-hamburg.deDMFT-1.pdf · Outline-I • From Atom to Solids: Multiplets in solids • Functionals: MFT, DFT, SDF • Multiorbital DMFT for General](https://reader031.vdocument.in/reader031/viewer/2022021916/5d490ff888c9931f048bb12e/html5/thumbnails/8.jpg)
Disolving Multiplets in 3d-alkali system
PES: Fe on alkali metalsC. Carbone (Trieste), O. Rader (BESSY), et al DMFT imp.: 5-band in 1-bath
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Functionals: MFT- DFT- DMFT
Weiss Mean-Field Theory (MFT) of classical magnetsKohn Density Functional Theory (DFT) of inhomogeneous electron gas in solids Dynamical Mean-Field Theory (DMFT) of strongly correlated electgron systems
G. Kotliar et. al. (2002), A. Georges (2004)
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Many-body System
The Euclidian Action: x=(r ,τ,σ)
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Functionals: general consideration
The one-electron Green's function
Introduction of the source (constraining field)
Functional derivative:
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Functionals: Legendre Transformation
The Functionals of Green's function
The partition function Z
Self-Energy: Constraining field J=Σthe inverse of the exact Green's function
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Baym-Kadanoff Functional
Exact representation of Φ
Different Functionals:
DFT: G=ρ J=V=Vh+VxcSDF: G=G(iω) J=Σloc(iω)BKF: G=G(k,iω) J=Σ(k,iω)
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DFT-Density Functional TheoryInhomogeneous electron gas in solids ( U=e2/|r-r0| ):
Energy functional with constrained density <n (r) > =ρ (r)
Stationarity in λ insures that:
Construct a functional of ρ (r) only:
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DFT: reference systemNon-interacting electrons in effective potential (t=-∇ 2/2):
Minimization with respect to λ (r):
Coupling constant trick:
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DFT - Functional
where density-density correlations defined as:
Exact relations for density functional:
Local density approximation (LDA):
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Exchange parameters and Functionals
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Exchange interactions from DFTHeisenberg exchagne:
Magnetic torque:
Exchange interactions:
Spin wave spectrum:
M. Katsnelson and A. L., Phys. Rev. 61, 8906 (2000)
Non-collinear magnetism :
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U/tChemical potential
U
t
∑∑ ↓↑+ +=
iiiji
ijij nnUcctH σσ
Hubbard model for correlated electrons
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Dynamical Mean Field Theory
Σ Σ Σ
Σ Σ
Σ Σ Σ
ΣU
( )ττ ′−0G
A.Georges, G.Kotliar, W.Krauth and M.Rozenberg, Rev. Mod. Phys. 68, 13 (1996)G. Kotliar and D. Vollhardt, Physics Today 57, 53 (2004)
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DMFT: Self-Consistent Set of Equations
( ) ( )∑Ω=
BZ
knn ikGiG
r
rωω ,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’
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( ) ∑∑ ↓↑+ +⋅+−=
iiiji
ijijij nnUcctH σσμδ
METAL
INSULATOR
~U
LowerHB
UpperHB
Quasi-particle peak
Hans Bethe
Transition from paramagnetic metal to paramagnetic insulator on the Bethe lattice
DMFT solution for the Hubbard model
Ut
A.Georges, G.Kotliar, W.Krauth and M.Rozenberg, Rev. Mod. Phys. ‘96
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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>
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• Materials-specific (structure, Z, etc.)
• Fast code packages
• Fails for strong correlations
LDA+DMFT
( ) ( ) ( ) ( )[ ]1
0ˆˆˆ1ˆ
−
∑ Σ−−+Ω
=BZ
knnn ikHiIiG
r
rωωμω
LDA Models approaches
• Input parameters unknown
• Computationally expensive
• Systematic many-body scheme
( ) ( ) dcLDA UkHkH ˆˆˆ0 −=
rr
Multi band Quantum Monte Carlo
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Flow diagram for the LDA+DMFT approach:
1,
1,
−− −=∑ locbath GG σσσ
[ ]∑ −− Σ−=κ
σσσ11
,, LDAloc GG
σσσ ∑+= −− 1,
1, locbath GG
Quantum impurity problemBand problem (LDA)
[ ]locGQMC ,: σ
DMFT self-consistency
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Specific features of realistic-DMFT
Matrix form of multiorbital bath Green functions – G0mm’(ω)
General form of the electron-electron interactions – Umm’m’’m’’’
Screened (GW) Coulomb interaction - frequency dependent U(ω)
Accurate Wannier description of one-electron band structure
Complicated solution of LDA+DMFT Quantum-Impurity Model
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Matrix form of Bath Green Function
eg
eg
gt
gt
gt
mm
GmGm
GmGm
GmmmmmmG
00000000000000000000
5
4
23
22
21
54321'
Simple case of d-orbital GF in the cubic lattice:
Genaral case of non-cubic lattice- Matrix form of G0mm’(ω)
Small cluster in DMFT – e.g. double Bethe lattice:
1 2
^11 12
21 22
( ) ( )( ) ( )
⎛ ⎞= ⎜ ⎟⎝ ⎠
G GG
G Gω ωω ω
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Coulomb vertex
The Coulomb interaction matrix for t2g:
033226303225330224
220333223032223301
654321
3
2
1
3
2
1
321321
JUJUUJUJUmJUJUJUUJUmJUJUJUJUUm
UJUJUJUJUmJUUJUJUJUmJUJUUJUJUm
mmmmmmUij
−−−−−−−−−−−−
−−−−−−−−−−−−
↓
↓
↓
↑
↑
↑
↓↓↓↑↑↑
'2)1(/}2/)2()3)[(1({ UJUNNJUJUNNNUUav =−=−−+−−+=
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Multi-band QMC-scheme (Hirsch-Fye)
)}(exp{21)]}(
21[exp{
''1
'''''
mmmmS
mmmmmmmmnnSnnnnU
mm
−=+−Δ− ∑±=
λτ
Discrete HS-transformation (Hirsch, 1983)
Number of Ising fields: N M(2M 1)= −m m ' m m '
1 2 1arccos h exp U _ N ote _ or2 L | U | W
⎡ ⎤ β⎛ ⎞λ = Δτ Δτ = <⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦
Green Functions:
'
1' '
( )
1 1' ' ' '
' ' ''
'
1( , ') ( , ', ) det
( , ', ) ( , ') ( )( ) ( )
1, '1, '
−
− −
= ×
= −
=
+ <⎧= ⎨− >⎩
∑
∑
m m
m m m mS
m m m m m m m
m m m m m m mm
m m
G G S GZ
G S VV S
m mm m
τ
ττ
τ τ τ τ
τ τ τ τ τ δ δ
τ λ τ σ
σ
G
' ''i
ij i j mm m mij mmH t c c U n nσ σσ
+= − +∑ ∑
U
m´
´mm´
τ
m
τ0
β
L Δ τ
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Sign problem and DMFTDeterminant ratio in one-band model:
1
1
ˆdet[ ] 1 [1 ( , )](exp( 2 ) 1) 0ˆdet[ ]new
ii iGR G VGσ τ τ
−
−= = + − − − ≥
Green function in arbitrary Ising fields:' 'ˆ ˆ ˆ ˆ' (1 )( 1) ' [1 (1 )( 1)] [0 1]V V VG G G e G e−= + − − ≈ + − − = ÷G G G
τ
-G
0
1
-1
β− β00 ( ) 1< << <iiGτ β
τ
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DMFT-SCFBethe lattice bath-Green function:
)()( 2nnn iGtii ωμωω −+=-1G
Energy
DO
S
EF
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DMFT-SCFHilbert transform (perovskite lattice):
)()()(
)(
)()(
1nnn
nn
iiGi
iizz
dNzG
ωωω
ωμωεεε
Σ+=
Σ−+=−
=
−
∫
1-G
Energy
DOS
EF
t2g
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DMFT-QMC calculation: Sr2RuO4
QMC for 3-orbitals:
15 auxiliary fields
300000 sweeps
128 imaginary times
T=15 meV
Max-Ent for DOS
LDA+DMFT, A. Liebsch and A.L., PRL(2000)
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Spectral function –ARPES and DMFT
Van Hove=10 meVm*/m=2.1-2.6
ARPES (A. Damascelli, et al PRL2000)LDA+DMFT, A. Liebsch,et al PRL(2000)
SrSr22RuORuO44
LDA
DMFT
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DMFT-SCFLDA+DMFT (orthogonal LMTO-TB):
)(ˆ),(ˆ)(ˆ)(ˆ
),()(
)()(),(1
''
''1
'
αωαω
ωω
ωμωω
α
+
∈∈
∈
−
−
∑∑
∑
=
=
Σ−−+=
UikGUiG
ikGiG
ikHiikG
IBZkn
On
BZknLLnLL
nDMFTLL
LDALLnnLL
h
Energy
DOS
EF
sp
d
ddddpdd
dpppp
dpss
kHΣ+
=Σ+HHH
HHHHHH
s
s
ss
)(ˆ)(ˆ ωr
Correlated d-states:
FBZ-integration
IBZ-integration+symmetrization
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DMFT for a general latticeBath Green function for multiband case:
from the cavity construction:
Gij(0) is the Green function with eliminated 0-site:
Using the Fourier transform:
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DMFT for a general lattice: GF GF in DMFT:
Here:
Taking into account that ∑t(k)=0 we have:
andUsing this formulas we obtained:
Finally for a bath GF we have:
A.L. and M. Katsnelson, PRB (1998)
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LDA+DMFT: Local Dynamics
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. L. and M. Katsnelson PRB, 57, 6884 (1998)
A. Poteryaev, A. L. , and G. Kotliar, PRL 93, 086401 (2004)S. Biermann, A. Poteryaev, A. L., and A. Georges PRL 94, 026404 (2005)
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General Projection formalism for DA+DMFTDELOCALIZED S,P-STATES
CORRELATED D,F-STATES
G. Trimarchi et al. arXiv:0802.4435, JPCM (2008)B. Amadon et al. arXiv:0801.4353 , PRB (2008)
|L>
|G>
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SCF-LDA+DMFT
F. Lechermann, et al, PRB (2007)
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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 et al, PRB, 44, 943 (1991), JPCM 9, 767 (1997)
LDAε
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Full-potential LDA+U: a problem
ELDA+U= LDA + U - DC
= + -
= + - No
OK!
S. Dudarev et. al. PRB 57, 1505 (1998)
Spherical RI-LDA+U
Interchange –possible!A.L. et al. Springer Series in Materials Science. Volume. 54 (2003)
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Static limit: LDA+URotationally invariant LDA+U functional
Local screend Coulomb correlations
LDA-double counting term (nσ =Tr(nmm0σ ) and n=n↑ +n⇓ ):
Occupation matrix for correlated electrons:
![Page 44: Applications: LDA+DMFT scheme - I - uni-hamburg.deDMFT-1.pdf · Outline-I • From Atom to Solids: Multiplets in solids • Functionals: MFT, DFT, SDF • Multiorbital DMFT for General](https://reader031.vdocument.in/reader031/viewer/2022021916/5d490ff888c9931f048bb12e/html5/thumbnails/44.jpg)
Slater parametrization of UMultipole expansion:
Coulomb matrix elements in Ylm basis:
Slater integrals:
Angular part – 3j symbols
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Average interaction: U and JAverage Coulomb parameter:
Average Exchange parameter:
For d-electrons:Coulomb and exchange interactions:
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• eg orbitals
• t2g orbitals
Mn (3+) = 3d4
5x3x
2x eg
t2g
Cubic Crystal field splitting
Spins ||Atomic Hunds rule
––
––
–– ––
––––
d
Orbital degrees of freedom
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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
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Orbital order: KCuF3
hole density of the same symmetry
A.L. 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
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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
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Constrained LDA calculation of U and J
Gunnarsson-1989 supercell with cutting hybridisation
Norman-1995 estimation of screening parameter
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Constrain GW calculations of U
F. Aryasetiawanan et alPRB(2004)
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Wannier - GW and effective U(ω)
T. Miyake and F. AryasetiawanPhys. Rev. B 77, 085122 (2008)
C-GW
GW
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Continuous Time QMC formalism: U(ω)
Partition function and action for fermionic system with pair interactions
Tr( )SZ Te−=
1 2 1 2
1 2 1 2
' ''' ' ' 1 1 2 2' ' 'r r r rr r
r r r r r rS t c c drdr w c c c c drdr dr dr+ + += +∫ ∫ ∫ ∫ ∫ ∫
{ , , }r i sτ=0 i s
dr dβ
τ= ∑∑∫ ∫Splitting of the action into
Gaussian part and interaction 0S S W= +
( )( )2 1 2 2 1
2 1 2 2 1
' ' ' ''0 ' 2 2 '' 'r r r r rr r
r r r r r r rS t w w dr dr c c drdrα += + +∫ ∫ ∫ ∫
( )( )1 2 1 1 2 2
1 2 1 1 2 2
' '' ' ' ' 1 1 2 2' 'r r r r r r
r r r r r rW w c c c c drdr dr drα α+ += − −∫ ∫ ∫ ∫
'rrα -- additional parameters, which are necessary to minimize the sign problem
A. Rubtsov et al Phys Rev B 72, 035122 (2005)
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Continuous Time QMC formalismFormal perturbation-series:
1 1 2 2 1 1 2 20
' ... ' ( , ' ,..., , ' )k k k k kk
Z dr dr dr dr r r r r∞
=
= Ω∑∫ ∫ ∫ ∫
2 1 2 1 21 2
1 2 2 1 2 1 2
' ' ...' '1 1 2 2 0 ' ... '
( 1)( , ' ,..., , ' ) ...!
k k k
k k k
kr r r rr r
k k k r r r r r rr r r r Z w w Dk
−
−
−Ω =
( ) ( )1 2 2 21 1
1 2 1 1 2 2
...' ... ' ' ' ' '...k k k
k k k
r r r rr rr r r r r rD T c c c cα α+ += − −
Since S0 is Gaussian one can apply the Wick theorem
D can be presented as a determinant g0
( ) ( )( ) ( )
2 21 1
1 1 2 2
2 21 1
1 1 2 2
' ' ' ' ''
' ' ' '
...( )
...
k k
k k
k k
k k
r rr rrr r r r rr
r r rr rr r r r
Tc c c c c cg k
T c c c c
α α
α α
+ + +
+ +
− −=
− −The Green function can be
calculated as follows
ratio of determinantsIn practice efficient calculation
of a ratio is possible due to fast-update formulas
A. Rubtsov and A.L., JETP Lett. 80, 61 (2004) www.ct-qmc.ru
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Random walks in the k space
Step k+1Step k-11
1
k
k
w Dk D
+
+
1k
k
k Dw D
−
Acceptance ratio
0 20 40 60
0
Dis
tribu
tion
k
decrease increase
Maximum at 2UNβ
k-1 k+1
Z=… Zk-1 + Zk + Zk+1+ ….
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CT-QMC: fast update k -> k+1
N2 operations
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Metal-Insulator transition: Bethe lattice
0.0 0.5 1.0 1.5 2.0 2.5 3.0-7
-6
-5
-4
-3
-2
-1
0
iωiω
Σ(iω)
0.0 0.5 1.0 1.5 2.0 2.5 3.0-1.8
-1.6
-1.4
-1.2
-1.0
-0.8
-0.6
-0.4
-0.2
0.0
U=2
U=3
U=2
G(iω
)
U=3
-4 -2 0 2 40.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
DO
S
Energy
Density of states for β=64:U=2; U=2.2; U=2.4; U=3
-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5-2
-1
0
1
2
3
4
coexistence of the metallic and insulating solutions: U=2.4, β=64, W=2
iω
G(iω
)
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Dynamical screening in Hubbard model
( ) ( ) ( )U U U Vτ δ τ τ→ = +
10 ( ) ( )G i iω ω μ −= +
2
2 2( )U U VωωΩ
= −+Ω
Compare to Exact solution
0 2 4 6 80.0
0.1
0.2
0.3
0.4
0.5
U=3, V=2, Ω=1, β=8
G(τ
)
τ
0 1 2 3 40.0
0.1
0.2
0.3
0.4
0.5
U=2, V=0.7, Ω=4, β=4
G(τ
)
τ
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Co on Cu: 5d-orbitals CT-QMC calculation
DOS for Co atom in Cu
E. Gorelov et al, to be published
G(τ
) LDA
τ
CT-QMC
U=4, b = 10 (T ~ 1/40 W)
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Advantages of the CT-QMC method
Number of auxiliary spinsin the Hirsch scheme
Short-range interactions Long-range interactions
Local in time interactions Non-local in time interactions
• non-local in time interactions: dynamical Coulomb screening
• non-local in space interactions: multi-band systems, E-DMFT
Auxiliary field (Hirsch) algorithm is time-consuming since it’s necessary to introduce large number of auxiliary fields, while
CT-QMC scheme needs almost the same time as in local case
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Miracle of continuous time QMC
Weak coupling expansion (A. Rubtsov et al)
Strong coupling expansion (P. Werner et al)
Path Integral:
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Comparison of different CT-QMC
Σ Σ Σ
Σ
Σ
Σ
ΣΣ
U
U
G( ’)τ−τ
ττ’
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Double counting in LDA+DMFT
Analytic modelsAround mean fieldFully localized limit
Constraint on particle number
Constraint on self-energyLDA
!
0!
Tr Tr
Tr Tr
GG
GG
=
=
( )
( ) 0 ReTr
00 ReTr !
!
=∞
=
Σ
Σ
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Choice of double counting in LDA+DMFTShift of chemical potential for correlated state
Natural choice :
1( , ) ( ) ( ) [ ] [ ( ) ]LDA dc c cG k i H k Eω ω μ δμ ω δμ→ →
− = + − + − − Σ −
1( , ) ( ) ( ) ( )cLDAG k i H kω ω μ ω→ →
− = + − −ΣTransformations:
11
0 01
10( ) ( )
c c
c cc c
G G
G G
δμ
ω ω δμ
−−
−−
= −
Σ = − = Σ −Condition for (Friedel SR)
0[ ] [ ]Tr G Tr G=0.0 0.5 1.0
-1.0
-0.5
0.0
GG0
G(τ
)
τ
Ni Ferro Eg-up
dc cE δμ=
cδμ
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DC-test for LDA+DMFT: NiO
LDA part includes entire Hilbert space
Coulomb interaction acting on correlatedsubspace only
Double counting
∑ +=k
kkk ccH εLDA
'int ij i j '
ij
1 1 1H U n n2 2 2
σσσ σ
σ
⎛ ⎞⎛ ⎞= − −⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠
∑
∑−=σ
σμi
inH DDC
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NiO – a charge transfer system
LDA band structure (paramagnetic)
Ni-3d orbitals as correlated subspaceO-2d orbitals as uncorrelated subspace
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NiO – double counting
Total particle number (color encoded) as function of chemical potential μ and double counting μDC
(eV)
(eV)
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Peak positions and spectral weights I
Fitting of Green functions to DOS with 3 δ-peaksat εi with spectral weight Zi
( ) ( ) ( )[ ]∑ −Θ−= −
iii neZG i εττ τε
F
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Peak positions and spectral weights II
Energy of peaks in spectral function of Ni-3d and O-2p orbitals as function of double counting. Line thicknesses correspond spectral weight of each peak.
( ) 0 ReTr !=∞Σ
FLLAMF LDA!
0!
Tr Tr
Tr Tr
GG
GG
=
=
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Spectral functions and double counting
Mott insulator
Charge transfer insulator
Almost metallic
eV21DC =μ
eV26DC ≥μ
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Itinerant ferromagnetism
Stoner
T=0
T<Tc
T>Tc
Heisenberg Spin-fluctuation
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Magnetism of metals: LDA+DMFT
A. L., M. Katsnelson and G. Kotliar, PRL 87, 067205 (2001)
0,0 0,2 0,4 0,6 0,8 1,0 1,2 1,4 1,6 1,8 2,0 2,20,0
0,2
0,4
0,6
0,8
1,0
1,2
0,0
0,2
0,4
0,6
0,8
1,0
1,2
χ(T)M(T)
M(T) and χ(T): LDA+DMFT
Ni
Fe
χ-1M
eff2 /3
T c
M(T
)/M(0
)
T/Tc
Global spin flip
Exchange interactions in metalsFinite temperature 3d-metal magnetism
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Ferromagnetism of transiton metals
-8 -6 -4 -2 0 20.0
0.5
1.0
1.5
2.0
2.5
LDA
DMFT
PES
Ni: LDA+DMFT (T=0.9 Tc)
EF
Den
sity
of s
tate
s, e
V-1
Energy, eV
0 2 4 60.0
0.5
1.0
1.5
τ, eV-1
<S(τ)
S(0)
>
Ferromagnetic Ni DMFT vs. LSDA: • 30% band narrowing• 50% spin-splitting reduction• -6 eV satteliteLDA+DMFT with ME
J. Braun, et alPRL (2006)
A. L, M. Katsnelson and G. Kotliar, PRL (2001)
0,0 0,2 0,4 0,6 0,8 1,0 1,2 1,4 1,6 1,8 2,0 2,20,0
0,2
0,4
0,6
0,8
1,0
1,2
0,0
0,2
0,4
0,6
0,8
1,0
1,2
χ(T)M(T)
M(T) and χ(T): LDA+DMFT
Ni
Fe
χ-1M
eff2 /3
T c
M(T
)/M(0
)
T/Tc
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Ferromagnetic Iron: Spectral Function
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Realistic DMFT for Co(111) surfaceFirst model surface DMFT – M.Potthoff (1999)
FPLMTO+DMFT: Igor di Marco PRB (2007)
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Orthorhombic 3dOrthorhombic 3d11 PerovskitesPerovskitesSrSrVVOO33 CaCaVVOO33
LaLaTiTiOO33 YYTiTiOO33
Metal
m*/m=2.7
Metal
m*/m=3.6
Insulator
Gap=0.2eV
Insulator
Gap=1.0eV
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Crystal-field splittings w/in t2g multiplet:(140,200) meV for LaTiO3 ; (200,330) meV for YTiO3
LDA-NMTO results: DOS
all metallicin LDA
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YTiO3
Violet: OxygenOrange: M
Orbital ordering in MTiO3
Occup. LDA DMFT
LaTiO3 0.45 0.88
YTiO3 0.50 0.96
La
Y
E. Pavarini et al. PRL (2004)
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LDALDA
+ + DMFTDMFT
m*/m=2.2 m*/m=3.5
1.00.2
LDALDA++DMFT: DMFT: comparison with comparison with experimentsexperiments
Exp=2.7 Exp=3.6
SrVOSrVO33 CaVOCaVO33
LaTiOLaTiO33 YTiOYTiO33
Exp Exp
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Conclusions
LDA+DMFT is a perfect scheme for realistic description of electronic structure of correlated electron materials
Matrix DMFT formalism can be used for general lattice or cluster compounds