Recall that the self-energyΣ
is defined as a sum of Feynman diagrams,
which each have an order n equal to the number of interaction lines.
We consider the self-energy in the context of jellium,
so the interaction lines W represent Coulomb repulsion,
and we use imaginary time.
Let us non-dimensionalize the Feynman diagrams in the self-energy,
by measuring momenta in units of ℏkF,
and energies in ϵF=ℏ2kF2/(2m).
Each internal variable then gives a factor kF5,
where kF3 comes from the 3D momentum integral,
and kF2 from the energy 1/β:
An nth-order diagram in Σ contains n interaction lines,
2n−1 fermion lines, and n integrals,
so in total it evolves as 1/kFn−2.
In jellium, we know that the electron density is proportional to kF3,
so for high densities we can rest assured that higher-order terms in Σ
converge to zero faster than lower-order terms.
However, at a given order n, not all diagrams are equally important.
In a given diagram, due to momentum conservation,
some interaction lines carry the same momentum variable.
small k make a large contribution,
and the more interaction lines depend on the same k,
the larger the contribution becomes.
In other words, each diagram is dominated by contributions
from the momentum carried by the largest number of interactions.
At order n, there is one diagram
where all n interactions carry the same momentum,
and this one dominates all others at this order.
The random phase approximation consists of removing most diagrams
from the defintion of the full self-energy Σ,
leaving only the single most divergent one at each order n,
i.e. the ones where all n interaction lines
carry the same momentum and energy:
Where we have defined the screened interactionWRPA,
denoted by a double wavy line:
Rearranging the above sequence of diagrams quickly leads to the following
In Fourier space, this equation’s linear shape
means it is algebraic, so we can write it out:
Where we have defined the pair-bubbleΠ0 as follows,
with an internal wavevector q, fermionic frequency iωmF, and spin s.
We isolate the Dyson equation for WRPA,
which reveals its physical interpretation as a screened interaction:
the “raw” interaction W=e2/(ε0∣k∣2)
is weakened by a term containing Π0:
Where we have used that nF(ε+iℏωnB)=nF(ε).
Analogously to extracting the retarded Green’s function GR(ω)
from the Matsubara Green’s function G0(iωnF),
we replace iωnF→ω+iη,
where η→0+ is a positive infinitesimal,
yielding the retarded pair-bubble Π0R: