Transcript Document

Daniel Karlsson, Lund
May, 2011, Naples
Young Researchers’ meeting
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Why consider model systems?
 Comparisons: can be solved exactly
 Simplicity: clues about strong correlation, non-adiabaticity,
memory effects
 Complex: Can describe cold atoms in optical lattices
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Overview
 TDDFT for the Hubbard model
 xc-functionals for 3D systems
Karlsson, Privitera, Verdozzi, PRL (2011)
 Linear response in the ALDA
 Full time evolution: Benchmarking ALDA beyond the
linear regime
Verdozzi, Karlsson, Puig, Almbladh, von Barth,
arXiv:1103.2291v1 (2011) (accepted by Chem. Phys.)
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Density Functional Theory (DFT)
 Continous case: Hohenberg-Kohn (1964)
 Mapping between densities and potentials:
n(x ) )
v(x )
 Lattice case: Gunnarsson, Schönhammer (1986)
Gunnarsson, Schönhammer , Noack (1995)
 Mapping between site occupancy and potentials:
n (x i ) )
v(x i )
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Reference systems in DFT
 Energy is split into several terms:
R
E [n] = T0 [n] + E H [n] +
 Continuous case:
vex t n + E x c [n]
Ex c [n]
from homogeneous electron gas
 Lattice case: Depends on coordination number!
 1D:
Ex c [n]
from homogeneous 1D Hubbard chain
 3D simple cubic: Homogeneous 3D simple cubic
 Other dimensionalities: Different Exc:s (surface different than
bulk)
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Time Dependent DFT (TDDFT)
 Continuous case: Runge-Gross theorem (1984)
n(x ; t ) ) v(x ; t )
 Lattice case: Theorem from continous case does not go
through
 Mapping in 1D due to TDCDFT
 Open problem for D>1
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TDDFT for the Hubbard model
X
H = ¡ t
hR R 0 i ¾
T^
+
a R ¾ a R 0 ¾+
X
X
U R n^ R " n
^R# +
R
w R ¾( t ) n
^R¾
R ¾
^
U
Hubbard model
V^ex t
Lattice TDDFT
 Verdozzi (2008): ALDA for 1D finite Hubbard cluster
 Exc from 1D homogeneous Hubbard model: Exactly
solvable by Bethe Ansatz
vxAcL D A [n](R i ; t ) = vxgsc (n(R i ; t ))
 No exact solution for 3D
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Overview
 TDDFT for the Hubbard model
 xc-functionals for 3D systems
 Linear response in the ALDA
 Full time evolution: Benchmarking ALDA beyond the
linear regime
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Dynamical Mean Field Theory (DMFT)
• Hubbard model remapped
into impurity model
• Infinite number of nearest
neighbors: exact mapping
• Impurity model: Interacting
impurity + reservoir of noninteracting electrons with
effective parameters
•Non-perturbative in the
interaction U: strong
correlations possible
Electron bath
U
time
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DMFT-LDA: Exchange-Correlation in 3D
Exc = ED M
F T
¡ T0 ¡ E H
 Vxc discontinuous at half-filling density for high interaction,
DFT manifestation of the Mott-Hubbard insulator transition
Karlsson, Privitera, Verdozzi, PRL (2011)
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Overview
 TDDFT for the Hubbard model
 xc-functionals for 3D systems
 Linear response in the ALDA
 Full time evolution: Benchmarking ALDA beyond the
linear regime
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Linear response using ALDA
 Object of study: retarded density-density response
function for infinite 3D Hubbard model
 ( R ; t ) = ¡ i µ( t ) h[ n
~R ( t ) ; n
~ 0 ( 0) ]i g s
Â0 (q; ! )
 In TDDFT: Â(q; ! ) =
1 ¡ (U + f x c )Â0 (q; ! )
Â0 (q ; ! ) =
Z
2
( 2¼)
3
3
d k
nF (²k ) ¡ nF (²k +
²k ¡ ²k +
q
q
)
+ ! + i´
² k = ¡ 2t(coskx + cosky + coskz )
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Linear response: fxc from DMFT
f
xc
n
 fxc can become positive at densities close to half-filling
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Linear response: Reciprocal Space
U = 8
n = 0:85
q =
( ¼; 0 ; 0 )
q =
( ¼; ¼; 0 )
q =
( ¼; ¼; ¼)
¡ = Â( q ; ! )
n = 0 :5
U = 8
n = 0:5
U = 8
n = 0:5
 Quarter-filling: f x c > 0 lowers effective interaction
 High filling: Positive fxc shifts poles to higher energies
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Linear response: Double Occupancy
1
¡
¼
Z
1
= mÂ(R = 0; R 0 = 0; ! )d! = hni ¡ hni 2 + 2hn " n # i
0
P a r a m et er s
hn " n # i
U = 8 ; n = 0 :5
0 .0 3 6 3
0 .0 6 2
- 0 .0 1 0
- 0 .0 0 7
U = 2 4 ; n = 0 :5
0 .0 1 6 1
0 .0 6 2
- 0 .0 4 7
- 0 .0 3 5
U = 8 ; n = 0 :8 5
0 .1 1 4
0 .1 7 8
0 .0 7 2
0 .0 5 9
D M F T
hn " n # i
0
hn " n # i
R P A
hn " n # i
A L D A
T a b l e 1 : D o u b l e o ccu p a n ci es o b t a i n ed f r o m D M F T , co m p a r ed a g a i n st r esu l t s
f r o m l i n ea r r esp o n se.
 Double occupancy can be negative, in RPA and in ALDA
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Overview
 TDDFT for the Hubbard model
 xc-functionals for 3D systems
 Linear response in the ALDA
 Full time evolution: Benchmarking ALDA beyond the
linear regime
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DMFT-TDDFT vs exact in a 125-site cluster
U
U
Interaction and perturbation
only in the center
TDDFT time propagation:
Via symmetry, can be
reduced to a 10-site
effective cluster
Time-dependent Kohn-Sham:
³
´
d' i ( t )
^
T 0 + v^ ef f ( t ) ' i ( t ) = i ¹h d t
• Use the ground state vxc from DMFT,
in the ALDA:
vx c ( R i ; t ) !
v xDcM
F T
(n(R i ; t))
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Kadanoff-Baym dynamics
• Basic quantity: Single-particle Greens function:
• Dyson Equation in time
• Two times: Iterative time propagation on the time square to obtain nonequilibrium Green’s function
Kadanoff and Baym (1962); Keldysh, JETP (1965); Danielewicz, AoP (1984)
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Many-Body Approximations
2nd Born (BA), GW, Tmatrix (TMA)
• Includes non-local effects in space and time
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TDDFT vs Kadanoff-Baym dynamics in 3D
U
Strong Gaussian potential
U = 8
U = 24
 ALDA describes strong interaction better than KBE
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TDDFT vs Kadanoff-Baym dynamics in 3D
U
Step potential
Weak
U = 8
U = 8
Strong
 KBE describes non-adiabatic response better than ALDA
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Conclusions
• Linear response for 3D homogeneous Hubbard model:
• fxc can become positive at high densities
• Double occupancy can become negative in RPA and in ALDA
• Time evolution for finite 3D clusters:
• High interaction: Manageable by ALDA
• Non-adiabatic perturbations: non-local effects needed, non-
equilibrium Green’s functions can help
Verdozzi, Karlsson, Puig, Almbladh, von Barth,
arXiv:1103.2291v1 (accepted by Chem. Phys.)
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DMFT-TDDFT vs exact in a 125-site cluster
Ne=40
U
Ne=40
U=8
U=8
U=24
U=24
Ne=70
U=8
n0
U=24
n0
Vex t
•
ALDA performance good in a)-e) but worsens considerably in panel f)
Why?
a-e): exact vKS local. f): vKS non local; ALDA-DMFT misses non-locality
Karlsson, Privitera, Verdozzi, PRL (2010)
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Comparison between 1D and 3D Vxc
DMFT (3D)
BALDA (1D)
U=8
DMFT (3D)
BALDA (1D)
U=24
 1D: Always a discontinuity for U > 0 +
 3D: Discontinuity for
U > 13
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Mott plateaus in parabolic potential
 Parabolic potential
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Mott plateaus: time evolution
 Parabolic potential
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Linear response: Real space
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DMFT: Self-consistent scheme
Schematics of DMFT
• Auxiliary AIM solved via
Lanczos diagonalisation
with N=8 (converged)
degrees of freedom
• Local occupancies/
potential energy from AIM
• Kinetic Energy from the
lattice Green’s function
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Bloch Oscillations, example in 1D
If constant electric field E applied, electrons oscillate in a
periodic potential.
Non-interacting:
x =
2
E
cos E t oscillations at ! = E
Weak interaction U
Weak field F
Damping
Weak interaction U
strong field F
Beatings with frequency U
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Ponomarev, PhD thesis (2008)
Bloch oscillations in the 3D Hubbard model
•Semiclassically
F
x =
2
E
cos E t
• Center-of-mass:
• Interactions: Different phenomena: depending
on E and U, e.g. beatings and damped behavior

• Cluster: 33 x 5 x 5
• Correct beating behavior observed, splitting
• No clear signature of damping: non-adiabatic potentials needed
Karlsson, Privitera, Verdozzi PRL (2011)
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Kadanoff-Baym dynamics
• Basic quantity
Total energy, one-particle averages, Excitation energies with  1 particle
• Dyson Equation in time
• Conserving approximations:
functional derivative of generating functionals
• In equilibrium,
• Time propagation on the time square
- Iteration of Dyson’s equation until convergence
- For building blocks of S: predictor corrector algorithm
Kadanoff and Baym (1962); Keldysh, JETP (1965); Danielewicz, AoP (1984)
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The Kohn-Sham system
 Kohn-Sham (1965): Construct fictitious non-
interacting system, gives density of interacting system
 Also possible in the lattice case.
³
´
T^ + v^ ef
P
occ
i= 1
f
'
i
= ²i '
i
j' i (x )j 2 = n(x )
v^ef f = v^ex t + v^H + v^x c
v^x c =
±E x c
±n
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Linear response: F-sum rule
Z
2
M (q) = ¡
Im(Â0 (q; ! ))! d! =
¼
Z
2
y
3
¡
d
khc
k ck i (²(k + q) + ²(k ¡ q) ¡ 2²(k )):
3
(2¼)
 Test successfully performed in reciprocal space
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Details for linear response
0
0
f x c (R ; R ; t; t ) =
f xAcL D A (R ; R 0; t; t 0) =
±v x c ( R ;t )
±n R 0 ( t 0)
f
xc
=
±v x c
±n
dv x c ( n R ( t ) )
dn R 0 ( t 0)
Constant kernel
f xAcL D A (R ; R 0; t; t 0) = vx0c (n R (t))±R R 0±(t ¡ t 0)
f
AL D A
xc
(q; ! ) = f
xc
 = Â0 + Â0 (U + f x c )Â
1
S(! ) =
2¼
Z
1
hn(t)ni 0 ei ! t dt
¡ 1
Structure factor
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TDDFT: Drawbacks and advantages
 Time Dependent Density Functional Theory (TDDFT):
 Advantages:
Accurate
 Can treat large systems
 Drawbacks:
 Describing strong correlation


Non-adiabatic effects
 Dynamical Mean Field Theory (DMFT):

Non-perturbative, can describe strong correlation
 Kadanoff-Baym Dynamics (KBE):
 Able to treat memory effects
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References
Daniel Karlsson, Antonio Privitera, Claudio Verdozzi, Phys. Rev. Lett. 106, 116401 (2011)
Claudio Verdozzi, Daniel Karlsson, Marc Puig von Friesen, Carl-Olof Almbladh, Ulf
von Barth; arXiv:1103.2291v1 (accepted by Chemical Physics)
Lima et al, PRL (2003)
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