PHYS07200604007 Manas Kumar Dala - Homi Bhabha National ...

Experimental Techniques 28
A(k, E) = π −1 Im{G(k, E)} (2.12)
Here A(k, E) describes the probability of removing (below the E F ) or adding
(above the E F ) an electron with the energy E and the wave vector k from (to) the
N-electron system (Interacting).
For a non interacting system with the one electron energy Ek,
0
G 0 (K, E) =
where ǫ is very small
and
1
(2.13)
E − Ek 0 − iǫ,
A 0 (k, E) = δ(E − E 0 k) (2.14)
which means the spectral function is a δ-function at E = Ek 0 . This result is
identical to that of equation (2.6) as in a noninteracting system the binding energy
is by definition the Koopman’s binding energy.
In an interacting electron system the electron energy gets renormalized by the
so-called self energy Σ(k, E) = Re{Σ(k, E)} + iIm{Σ(k, E)}, yielding
and
G(K, E) =
1
E − E 0 k − Σ(k, E) (2.15)
A(k, E) = 1 π
Im{Σ(k, E)}
[E − E 0 k − Re{Σ(k, E)}]2 + [Im{Σ(k, E)}] 2 (2.16)
If we assume that the spectrum of the interacting system is not too different
from that of the noninteracting system (Im{Σ(k, E)}), then the poles in the Green’s
function of the interacting one occur at
with the condition
E 1 k = Re{E1 k } + iIm{E1 k } (2.17)
E 1 k − E0 k − Σ(k, E1 k ) = 0 (2.18)
This allows to separate the Greens function into two parts, a pole part and a
non-singular term G inc , i.e,