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15 Random-Phase Approximation (RPA)
15.1 Derivation and Properties of the RPA Equations
The random-phase approximation (RPA) for the polarization propagator is established by a specific subset of diagrams shown in Fig. 15.1, referred to as RPA diagrams. The RPA diagrams are complete through first order, whereas in second and
higher order they represent only a small fraction of the full set of diagrams. As
seen in Fig. 13.3, the RPA diagram is one of altogether five second-order diagrams.
The RPA diagrams can be summed completely through infinite order owing to their
simple recursive construction principle, which is depicted in Fig. 15.2. In energy
representation, this recursion relation takes on the simple algebraic form (see Exercise 15.1)
R P A
(ω) =
(0)
(ω) +
(0)
(ω)R
R P A
(ω)
(15.1)
Here,
(0)
(ω) is the free (or zeroth-order) polarization propagator (13.19), reading
in energy representation
(0)
rs,r s (ω) = δ rr δ ss
¯
n r n s
ω− r + s +iη
−
n r ¯
n s
ω− r + s −iη
(15.2)
Fig. 15.1 Series of RPA diagrams through third order
Fig. 15.2 Recursive construction of the RPA diagram series