1. Daniel Karlsson, Lund May, 2011, Naples
A TDDFT approach for the 3D Hubbard model in the linear and non-linear regime
Daniel Karlsson, Lund
May, 2011, Naples
Young Researchers’ meeting
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2. 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

3. 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: v1 (2011) (accepted by Chem. Phys.)

4. 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:
Also called SOFT: Site Occupation Function Theory, to distinguish this from the continous case. The same type of argument is possible, and a Levy-search can be made.

5. Reference systems in DFT
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)

6. Time Dependent DFT (TDDFT)
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
Trying to apply the Runge-Gross proof for the continuum to the discrete case does not work. Counter-example in 2D: Sfefanucci and Kurth, two different potentials, same density. Not v representability. Some special situatioons lack uniqueness. But still accurate calculations is possible.

7. TDDFT for the Hubbard model
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
Describe again that this is a lattice model, first term kinetic energy, which does not contain spin flip terms, second on-site interaction, which is purely local. Last, the local potential, which in principle can be spin dependent.
Verdozzi introduced tddft for finite hubbard model, reverse engineered time dependent vxc, showed that infinite 1D reference system is a good ref. system in the ALDA.

8. Overview xc-functionals for 3D systems TDDFT for the Hubbard model
Linear response in the ALDA
Full time evolution: Benchmarking ALDA beyond the linear regime

9. Dynamical Mean Field Theory (DMFT)
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 U
Electron bath
time
The effective model, an anderson impurity model with effective parameters can be solved exactly, and we get the kinetic energy and double occupancy, which gives potential energy. Which gives Total energy
We neglect magnetic phases: spin compensated system. If more details needed, ask me.

10. DMFT-LDA: Exchange-Correlation in 3D
Vxc discontinuous at half-filling density for high interaction, DFT manifestation of the Mott-Hubbard insulator transition
Important: Real metal-insulator transition eqiv. dicontinuity. Lattice is built in inside dmft. Not homogeneous electron gs. Mention potential like this can describe mott insulators
Karlsson, Privitera, Verdozzi, PRL (2011)

11. Overview xc-functionals for 3D systems TDDFT for the Hubbard model
Linear response in the ALDA
Full time evolution: Benchmarking ALDA beyond the linear regime

12. Linear response using ALDA
Object of study: retarded density-density response function for infinite 3D Hubbard model
In TDDFT:

13. Linear response: fxc from DMFT
fxc can become positive at densities close to half-filling

14. Linear response: Reciprocal Space
Quarter-filling: lowers effective interaction
High filling: Positive fxc shifts poles to higher energies
FXCS is always negative in the alda electron gas.
describes the plasmonic behavior. use gw, rpa, the poles of dielectric constants a kuttke zeri excutatuin, number divided by 0. Collective motion, by analogy.

15. Linear response: Double Occupancy
Double occupancy can be negative, in RPA and in ALDA
Intoduce a natural cufoff from the lattice. screening, compared to continous case.
Mention I am not very involved in this, but also there can be unphysical poles in the upper half plane, which also shows in model systems. If interested, I refer to the paper.

16. Overview xc-functionals for 3D systems TDDFT for the Hubbard model
Linear response in the ALDA
Full time evolution: Benchmarking ALDA beyond the linear regime
by the way, I am not going to talk about the linear regime, not things pårimalry involved wiwth,. Howewverm if one uses nay body perturbation theory to inprove the kernel, one is going ti invert the linear response to get fxc, problems due to the fact double poles to chi0. This is very general featuer, in very small cluster calculations, I refer you directly to the paper. Add line: Some shortcomings of tddft, in the linear response, appearsalso in cluster models. maria mention, also refer to lattices.

17. DMFT-TDDFT vs exact in a 125-site cluster
Via symmetry, can be reduced to a 10-site effective cluster
Interaction and perturbation only in the center
Time-dependent Kohn-Sham:
TDDFT time propagation:
Use the ground state vxc from DMFT,
in the ALDA:

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

19. Many-Body Approximations
2nd Born (BA), GW, Tmatrix (TMA)
Includes non-local effects in space and time

20. TDDFT vs Kadanoff-Baym dynamics in 3DU
Strong Gaussian potential
ALDA describes strong interaction better than KBE

21. TDDFT vs Kadanoff-Baym dynamics in 3DU
Step potential
KBE describes non-adiabatic response better than ALDA
Weak
Strong

22. 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
Apart from fundamental development, we intend to apply this to cold atoms in 3D, and see what kind of effects these vxcs and fxcs has on describing cold atoms in 3D. Mott insulators can be described, in 1D but also in 3D. Also a real mott insulator, V2O2, has been described ab-initio by LDA+DMFT.
Verdozzi, Karlsson, Puig, Almbladh, von Barth, arXiv: v1 (accepted by Chem. Phys.)

24. U DMFT-TDDFT vs exact in a 125-site cluster
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
Karlsson, Privitera, Verdozzi, PRL (2010)

25. Comparison between 1D and 3D Vxc
DMFT (3D)
BALDA (1D)
U=8
U=24
1D: Always a discontinuity for
3D: Discontinuity for

26. Mott plateaus in parabolic potential

27. Mott plateaus: time evolution
Parabolic potential

28. Linear response: Real space

29. 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

30. Bloch Oscillations, example in 1D
If constant electric field E applied, electrons oscillate in a periodic potential.
Non-interacting:
oscillations at
Weak interaction U Weak field F Damping
Weak interaction U
strong field F
Beatings with frequency U
Ponomarev, PhD thesis (2008)

31. 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
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)

32. Kadanoff-Baym dynamics
Basic quantity
Dyson Equation in time
Conserving approximations: functional derivative of generating functionals
In equilibrium,
Time propagation on the time square
Total energy, one-particle averages, Excitation energies with  1 particle
Too many equations, simpler, make picture larger. Put other contour also.
- 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)

33. The Kohn-Sham system
Kohn-Sham (1965): Construct fictitious non-interacting system, gives density of interacting system
Also possible in the lattice case.

34. Linear response: F-sum rule
Test successfully performed in reciprocal space

35. Details for linear response
Constant kernel
Structure factor

36. 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

37. References Lima et al, PRL (2003)
Daniel Karlsson, Antonio Privitera, Claudio Verdozzi, Phys. Rev. Lett. 106, (2011)
Claudio Verdozzi, Daniel Karlsson, Marc Puig von Friesen, Carl-Olof Almbladh, Ulf von Barth; arXiv: v1 (accepted by Chemical Physics)
Lima et al, PRL (2003)