Dipartimento di Fisica dell'Universita' di Roma “Tor...

Andrea Marini

Solids beyond the single-particle approximation: an introduction to Ab-Initio

Many-Body perturbation theory

Dipartimento di Fisica dell'Universita' di Roma “Tor Vergata”. Roma, Italy.http://people.roma2.infn.it/~marini/

Lille January 25, 2005

[G. Onida, L. Reining and A. Rubio, Rev. Mod. Phys. 74, 601 (2002)]DFT and beyond ...

H.A. Bethe

E.E. Salpeter

The Bethe Salpeter equation (BSE):an equation for bound-state problems

[E.E. Salpeter and H.A. Bethe, Phys. Rev. 84, 1232 (1951)]

Part 1: Static and dynamic excitons in solids: the Bethe-Salpeter equation

Part 2: Bound excitons: BSE versus Time-dependent Density-Functional

Theory.

Part 0: Particles and quasiparticles: a brief introduction.

DFT-LDA band structure of Cu [A. Marini et al PRB, 64, 195125 (2001)]

Part 0: Particles

quasiparticles

Discrepancies both large and k-point/band dependent

''It is intriguing to find such pronounced deviations from band theory'' [V.N. Strocov et al.,

PRL 81, 4943 (1998)]

[G. Onida, L. Reining and A. Rubio, Rev. Mod. Phys. 74, 601 (2002)]DFT and beyond ...

Dyson Equation:

[F. Aryasetiawan, Rep. Prog. Phys.  61, 237­312 (1998)]

Quasiparticles: the GW approximationPart 0: Particles

quasiparticles

0th order = Kohn-Sham hamiltonian (LDA approximation) Quasiparticle wavefunction

The GW ''soup''

DFT states

The pseudopotential approach

=Kohn-Sham states=Plasmons/Electron-hole states

The self-energy physics

GW quasiparticle band structure of Copper

A. Marini et al PRL, 88, 016403 (2002)

From "Quasiparticle calculations in solids", W.G. Aulbur, L. Jönsson and J.W. Wilkins, Solid State Physics 54 1 (2000), also available in preprint form at http://www.physics.ohio-

state.edu/~wilkins/vita/publications.html#reviews

EnQP≃n

KSn∣nKS−vxc∣n

GW quasiparticle gaps of insulators

light

Polarisation function: a two-body problem

[G. Onida, L. Reining and A. Rubio, Rev. Mod. Phys. 74, 601 (2002)]DFT and beyond ...

BSE: Static excitons in metals and semiconductors

Part 1: the BSE[Onida, Reining and Rubio, Rev. Mod. Phys. 74, 601 (2002)]

S(tatic)BSE

Static approximation

for W(t)

The BSE is not solvable in the space of single-time Green's functions

=Z<1

Quasiexcitons DBSE

Dyson

Quasiexciton Green's function

AM and R. Del Sole, PRL 91, 176402 (2003)

The BSE: a O(NxN) problem

S(tatic)BSE

[G. Onida, L. Reining and A. Rubio, Rev. Mod. Phys. 74, 601 (2002)]DFT and beyond ...

xc-functional>vxc>fxc .

Sham Schluter, PRL 51, 1888; Tokatly Pankratov PRL 86 2078; Tokatly Stubner Pankratov PRB

65 113107.

... Many-Body perturbation theory

Onida, Reining and Rubio, Rev. Mod. Phys. 74, 601 (2002)

EXX functional>vx>fx .

M. Stadele et al., PRL 79 2089; Kim Gorling, PRL 89,

096402;PRB 66 035114.

<ij| fxc |kl>=<ij| W|kl>

Sottile, Olevano, Reining, PRL 91, 056402.

Long-range model for fxc: Reining, Olevano, Rubio,

Onida PRL 88, 066404.

Time Dependent DFT versus ...

xc-functionals a la' TDDFT, Ulf von Barth.

Many-Body approach to the Exchange-Correlation Kernel of TDDFT

Hp. (1)

Hp. (2)

The fxc series does not converge unless the BS kernel diagonal is embodied in the IP polarization function, i.e.

Using this idea the higher order corrections to the first-order fxc are

not only well defined but numerically stable at all orders.

BSETDDFT Experiment

QP-RPA

`

Bound excitons in TDDFTAM, R. Del Sole, A. Rubio PRL 91, 256402 (2003)

Causal Kernel !

Resonant Kernel !

Low dimensional systems (1D): polyacetylene

 Compared to an isolated chain [M. Rohlfing and S.G. Louie PRL 82 1959

(1999)] the exciton binding energy is reduced in the

molecular crystal by an order of magnitude [P. Pusching and C.A.

Draxl PRL 89 056405 (2002)]

The present TDDFT kernel, instead,

successfully reproduces the BSE results, in good

agreement with the experiment.

In LDA (ALDA) the xc potential lacks of a term counteracting

the applied electric field[S.J.A. Van

Gisbergen PRL 83 694 (1999)]. Hence ALDA overestimates the chain

polarizability.

0D/2D systems: polyacetylene and Silicon surface

Compared to the approximations for the

TDDFT/CDFT kernel present on the “market” the MB-

based fxc works equally well for 0-1-2-3D.

[D. Varsano and O. Pulci, private communications]

Is TDDFT ''fast'' compared to the BSE ?

=

[# of frequencies] ×100 x10000×W 10000×10000× '10000×100

When the only optical spectra is calculated TDDFT is as time consuming as BSE...

...but when the full dielectric matrix is needed TDDFT is more favorable than the BSE

Ab-Initio Many-Body perturbation theory is ...

...reliable...... accurate ...

...helpful...

AM, R. Del Sole, A. Rubio PRL 91, 256402 (2003)

AM R. Del Sole PRL 91, 176402 (2003)

Onida, Reining and Rubio, Rev. Mod. Phys. 74, 601 (2002)

F. Aryasetiawan, Rep. Prog. Phys. 61, 237-312 (1998)

A. Marini et al PRL, 88, 016403 (2002)

AcknowledgmentsDaniele VarsanoMyrta GrueningLudger WirtzOliva Pulci

Prof. Angel RubioProf. Rodolfo Del Sole

“ a shiny pot of fun and happiness”

[C. Hogan]

http://people.roma2.infn.it/~marini