TOMBO Ver.2 Manual

2.2 The Green function 14
2.2 The Green function
The propagation of one particle through the system is described by the one particle
Green function. The one particle Green function is defined as:
∣ [
]∣ ⟩
∣∣T
G(x 1 ,t 1 ,x 2 ,t 2 ) = i
⟨Ψ N ψ(x 1 ,t 1 )ψ † ∣∣ΨN
(x 2 ,t 2 )
(2.4)
It contains the information on energy and lifetimes of the quasiparticle, and also on
the ground state energy of the system and the momentum distribution. Here, x = (x,t) =
(r,σ,t). Ψ N is the Heisenberg ground state vector of the interacting N-electron system satisfying
the engenvalue equation H|Ψ N ⟩ = E|Ψ N ⟩,ψ H and ψ † H
are respectively the annihilation
and creation field operator and T is the Wick time ordering operator.
ψ(x,t) = e iHt ψ(x)e −iHt (2.5)
and the field operator satisfy the anti-commutation relation:
ψ † (x,t) = e −iHt ψ † (x)e iHt (2.6)
{
}
ψ(x),ψ † (x ′ ) = δ(x − x ′ ) (2.7)
and:
{
ψ(x),ψ(x ′ ) } {
}
= ψ † (x),ψ † (x ′ ) = 0 (2.8)
T [ψ(x 1 ,t 1 )ψ † (x 2 ,t 2 )] =
{
ψ(x 1 ,t 1 )ψ † (x 2 ,t 2 ) if t 1 > t 2
ψ(x 2 ,t 2 )ψ † (x 1 ,t 1 ) if t 1 < t 2
(2.9)
Therefore, the Green function describes the probability amplitude for the propagation
of an electron (hole) from position r 2 at t 2 to r 1 at time t 1 for t 1 > t 2 (t 1 < t 2 ) (See Fig.2.1).
Insert a complete set of N+1 and N-1 particle states, ∑ j |ψ N±1 | = 1, we perform a
Fourier transform in energy space we obtain:
j
⟩⟨ψ N±1
j
G(x 1 ,x 2 ;ω) = ∑
s
f s (x 1 fs ∗ (x 2 )
ω − ε s + iηsgn(ε s − ε F )
(2.10)