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Daniel Karlsson, Lund

May, 2011, Naples

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Young Researchers’ meeting 1

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

Linear response in the ALDA

Full time evolution: Benchmarking ALDA beyond the linear regime

Karlsson, Privitera, Verdozzi, PRL (2011)

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:

Lattice case: Gunnarsson, Schönhammer (1986)

Gunnarsson, Schönhammer , Noack (1995) Mapping between site occupancy and potentials:

n(x) ) v(x)

n ( x i ) ) v ( x i )

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Energy is split into several terms:

Continuous case: from homogeneous electron gas

Lattice case: Depends on coordination number!

1D: from homogeneous 1D Hubbard chain

3D simple cubic: Homogeneous 3D simple cubic

Other dimensionalities: Different Exc:s (surface different than bulk)

E [n] = T0 [n] + EH [n] +Rvextn + Exc[n]

Exc[n]

Exc[n]

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Reference systems in DFT

Continuous case: Runge-Gross theorem (1984)

Lattice case: Theorem from continous case does not go through

Mapping in 1D due to TDCDFT

Open problem for D>1

Time Dependent DFT (TDDFT)

n(x; t) ) v(x; t)

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Lattice TDDFT

Verdozzi (2008): ALDA for 1D finite Hubbard cluster

Exc from 1D homogeneous Hubbard model: Exactly solvable by Bethe Ansatz

No exact solution for 3D


Hubbard model

vALDAxc [n](Ri; t) = vgsxc(n(Ri; t))

T̂ Û V̂ext

H = ¡ tX

hR R 0 i¾

a+

R ¾aR 0¾+

X

R

U R n̂ R " n̂ R # +

X

R ¾

w R ¾ ( t) n̂ R ¾

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Overview





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Dynamical Mean Field Theory (DMFT)

U Electron bath

time

• Hubbard model remapped into impurity model

• Infinite number of nearest neighbors: exact mapping

• Impurity model: Interacting impurity + reservoir of non-interacting electrons with effective parameters

•Non-perturbative in the interaction U: strong correlations possible

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Vxc discontinuous at half-filling density for high interaction, DFT manifestation of the Mott-Hubbard insulator transition

DMFT-LDA: Exchange-Correlation in 3D


Exc = EDM F T ¡ T0 ¡ EH

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Overview





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Linear response using ALDA Object of study: retarded density-density response

function for infinite 3D Hubbard model

In TDDFT:

Â (R ; t) = ¡ iµ (t)h [ ~nR (t) ; ~n 0 (0 )]ig s

Â(q; !) =Â0(q; !)

1¡ (U + fxc)Â0(q; !)

Â 0 (q ; ! ) =2

(2¼ )3

Zd3kn F (²k ) ¡ n F (²k+ q )

²k ¡ ²k+ q + ! + i´

²k =¡2t(coskx +cosky +coskz)12

Linear response: fxc from DMFT

fxc can become positive at densities close to half-filling

fxc

n

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Linear response: Reciprocal Space

Quarter-filling: lowers effective interaction

High filling: Positive fxc shifts poles to higher energies

q = ( ¼ ; 0 ; 0 )

q = ( ¼ ; ¼ ; 0 )

q = ( ¼ ; ¼ ; ¼ )

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n = 0 :5

n = 0 :8 5

n = 0 :5

n = 0 :5

¡=Â(q;!)

U = 8

U = 8

U = 8

f x c > 0

Linear response: Double Occupancy

Double occupancy can be negative, in RPA and in ALDA

P a ram e te r s hn " n # iD M F T hn " n # i 0 hn " n # iR P A hn " n # iA LD AU = 8 ; n = 0 :5 0 .0 3 6 3 0 .0 6 2 -0 .0 1 0 -0 .0 0 7

U = 24 ; 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

Ta b le 1 : D o u b le o c c u p a n c ie s o b ta in ed from D M FT , c om p a re d a g a in st r e su lt s

from lin e a r r e sp o n se .

¡ 1¼

Z 1

0

=mÂ(R = 0;R0 = 0; !)d! = hni ¡ hni2 + 2hn"n#i

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Overview





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• Use the ground state vxc from DMFT, in the ALDA:

TDDFT time propagation:

DMFT-TDDFT vs exact in a 125-site cluster

Interaction and perturbation only in the center

U

U

Via symmetry, can be reduced to a 10-site effective cluster

vxc (R i ; t) ! vDM F Txc (n (R i ; t))

³T̂0 + v̂ef f (t)

´' i(t) = i¹h

d' i (t)

dt

Time-dependent Kohn-Sham:

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• Basic quantity: Single-particle Greens function: • Dyson Equation in time • Two times: Iterative time propagation on the time square to obtain non-equilibrium Green’s function

Kadanoff-Baym dynamics

Kadanoff and Baym (1962); Keldysh, JETP (1965); Danielewicz, AoP (1984) 18

Many-Body Approximations

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2nd Born (BA), GW, Tmatrix (TMA)

• Includes non-local effects in space and time

ALDA describes strong interaction better than KBE

TDDFT vs Kadanoff-Baym dynamics in 3D

U

Strong Gaussian potential

U = 8

U = 24

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TDDFT vs Kadanoff-Baym dynamics in 3D

U

KBE describes non-adiabatic response better than ALDA

U = 8

U = 8

Step potential

Weak

Strong

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• 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

Conclusions

Verdozzi, Karlsson, Puig, Almbladh, von Barth,

arXiv:1103.2291v1 (accepted by Chem. Phys.) 22

DMFT-TDDFT vs exact in a 125-site cluster

U=8 U=8 U=8

U=24 U=24 U=24

Ne=40 Ne=40 Ne=70

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

n0

n0

Vext


U

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Comparison between 1D and 3D Vxc

1D: Always a discontinuity for

3D: Discontinuity for

DMFT (3D)

BALDA (1D)

U=8

U=24

DMFT (3D)

BALDA (1D)

U > 0+

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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Weak interaction U

Weak field F

Damping

Weak interaction U

strong field F

Beatings with frequency U

Ponomarev, PhD thesis (2008)

Bloch Oscillations, example in 1D

Non-interacting: x = 2E

cos E t oscillations at ! = E

If constant electric field E applied, electrons oscillate in a periodic potential.

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Bloch oscillations in the 3D Hubbard model

•Semiclassically

• Center-of-mass:

• Interactions: Different phenomena: depending

on E and U, e.g. beatings and damped behavior

• Correct beating behavior observed, splitting • No clear signature of damping: non-adiabatic potentials needed

F

x =2

Ecos E t

• Cluster: 33 x 5 x 5

Karlsson, Privitera, Verdozzi PRL (2011) 31

• Basic quantity

• Dyson Equation in time

• Conserving approximations: functional derivative of generating functionals

• In equilibrium,

• Time propagation on the time square

Kadanoff-Baym dynamics

Kadanoff and Baym (1962); Keldysh, JETP (1965); Danielewicz, AoP (1984)

Total energy, one-particle averages, Excitation energies with 1 particle

- Iteration of Dyson’s equation until convergence

- For building blocks of S: predictor corrector algorithm

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Kohn-Sham (1965): Construct fictitious non-interacting system, gives density of interacting system

Also possible in the lattice case.

The Kohn-Sham system

³T̂ + v̂ef f

´' i = ²i' i

Pocci=1

j'i(x)j2 = n(x)

v̂eff = v̂ext + v̂H + v̂xc

v̂xc =±Exc±n

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Linear response: F-sum rule

Test successfully performed in reciprocal space

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M(q) = ¡ 2¼

ZIm(Â0(q; !))!d! =

¡ 2(2¼)3

Zd3khcykcki(²(k+ q) + ²(k¡ q) ¡ 2²(k)):

Â = Â0 +Â0(U + fxc)Â

fxc(R;R0; t; t0) =

±vxc(R;t)

±nR0(t0)

fALDAxc (R;R0; t; t0) =

dvxc(nR(t))

dnR0 (t0)

fALDAxc (R;R0; t; t0) = v0xc(nR(t))±RR0±(t¡ t0) f ALDAxc (q ; ! ) = fxc

fxc =± vxc

±n

S(!) =1

2¼

Z 1

¡1hn(t)ni0ei!tdt Structure factor

Constant kernel

Details for linear response

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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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