3.5 DYSON Equation
The DYSON equation can be achieved by classifying the various contributions in
arbitrary FEYNMAN diagrams. DYSON's equation summarizes the
FEYNMAN-DYSON perturbation theory in a particularly compact form.
The exact GREEN's function can be written as the non-interacting GREEN's function
plus all connected terms with a non-interacting GREEN's function at each end, see (3.40).
This structure is shown in Fig. 3.5,
where the double line denotes
and the single line
.
Figure 3.5:
The GREEN's function expanded in terms of
connected diagrams.
|
By introducing the concept of self-energy
the structure in Fig. 3.5
takes the form shown Fig. 3.6.
Figure 3.6:
FEYNMAN diagrams showing the general structure of
.
|
The corresponding analytic expression is given by
![\begin{displaymath}\begin{array}{l}\displaystyle
\displaystyle G({\bf {r}},t;{\...
...r}},t;1) \ \Sigma(12) \ G_{0}(2;{\bf {r'}},t') \ ,
\end{array}\end{displaymath}](img577.png) |
(3.41) |
where the abbreviation
and
is used. The self-energy
describes the renormalization
of single-particle states due to the interaction with the surrounding
many-particle system and the DYSON equation determines the
renormalized GREEN's function.
Another important concept is the proper self-energy insertion which is a
self-energy insertion that can not be separated into two pieces by cutting a
single-particle line. By definition, the proper self-energy is the sum of all
proper self-energy insertions, and will be denoted by
. Using the
perturbation expansion, one can define the proper self-energy
as an
irreducible part of the GREEN's function. Based on this definition
first-order proper self-energies, which are resulted from the first-order
expansion of the GREEN's function (see Section 3.4.2), are shown
in Fig. 3.7. These diagrams are irreducible parts
of Fig. 3.4-b and Fig. 3.4-c and are referred to as
the HARTREE (
) and the FOCK (
)
self-energies.
Figure 3.7:
FEYNMAN diagrams of the
first-order proper self-energies.
|
The self-energy can also in principle be introduced
variationally [203]. A variational derivation of the
self-energies for the electron-electron and electron-phonon interactions is
presented in Appendices F.1 and F.2,
respectively. It follows from these definitions that the self-energy consists
of a sum of all possible repetitions of the proper self-energy
![\begin{displaymath}\begin{array}{l}\displaystyle
\displaystyle \Sigma({\bf {r}}...
...2) \
\Sigma^{*}(2;{\bf {r'}},t') \ + \ \ldots \ .
\end{array}\end{displaymath}](img585.png) |
(3.42) |
Correspondingly, the GREEN's function in (3.41) can be rewritten as
![\begin{displaymath}\begin{array}{l} \displaystyle
\displaystyle G({\bf {r}},t;{...
...{*}(12) \
G_{0}(2;{\bf {r'}},t') \ + \ \ldots \ ,
\end{array}\end{displaymath}](img586.png) |
(3.43) |
which can be summed formally to yield an integral equation
(DYSON equation) for the exact GREEN's function which is shown in Fig. 3.8.
Figure 3.8:
FEYNMAN diagrams representing DYSON's equation.
|
The corresponding analytic expression is given by
![\begin{displaymath}\begin{array}{l} \displaystyle
\displaystyle G({\bf {r}},t;{...
...{r}},t;1) \ \Sigma^{*}(12) \ G(2;{\bf {r'}},t')\ .
\end{array}\end{displaymath}](img588.png) |
(3.44) |
The validity of (3.44) can be verified by iterating the right
hand-side, which reproduces (3.43) term by term. In a similar
manner, one can show that the DYSON equation can be also written as
![\begin{displaymath}\begin{array}{l}
\displaystyle G({\bf {r}},t;{\bf {r'}},t') ...
...}},t;1) \ \Sigma^{*}(12) \ G_0(2;{\bf {r'}},t')\ .
\end{array}\end{displaymath}](img589.png) |
(3.45) |
M. Pourfath: Numerical Study of Quantum Transport in Carbon Nanotube-Based Transistors