contributes in the sum of vacuum diagrams to W [J = 0], is absent in the series

expansion of the e¬ective action.

Figure 10.3 One-particle reducible diagram contributing to W [J = 0].

The above functional evaluation of the e¬ective action generates the one-particle

irreducible loop expansion in terms of skeleton diagrams, and in¬nite partial sum-

mation of naive perturbation theory diagrams is thus already done. A virtue of the

above expansion is that at each loop level for the e¬ective action it contains far fewer

diagrams than the naive perturbation expansion.

In Section 10.6, where the e¬ective action approach is applied to a Bose gas,

and in Chapter 12, where the theory of classical statistical dynamics is applied to

vortex dynamics in a superconductor, we shall need to take the loop expansion to

the next level where only two-particle irreducible vacuum diagrams will appear. We

therefore ¬rst go back to the generating functional technique, but now we will include

a two-particle source.

10.5. Two-particle irreducible e¬ective action 339

10.5 Two-particle irreducible e¬ective action

The e¬ective action can be taken to the next level in irreducibility in which only two-

particle irreducible vertices appear. To construct such a description, we introduce

a two-particle source K12 in addition to the one-particle source J1 , and a generator

of Green™s functions where the connected Green™s functions of the theory now are

contracted on both types of sources, i.e. de¬ned according to the diagrammatic

expansion in terms of the two sources as depicted in Figure 10.4.

+ 1

, + ···

=

2

+ +

1 1

2 2

+ 1

+ 1

+ ···

2 2

+ 1

+ ···

2

Figure 10.4 Diagrammatic expansion of the generator, W [J, K], in the presence of

one- and two-particle sources.

The diagrammatic notation for the two-particle source is thus as displayed in

Figure 10.5.

K12 = 2

1

Figure 10.5 Diagrammatic notation for the two-particle source.

340 10. E¬ective action

The diagrammatic notation for the generator makes explicit the feature that it

depends on both a one- and a two-particle source as stipulated in Figure 10.6.

,

W [J, K] =

Figure 10.6 Diagrammatic notation for the generator in the presence of one- and

two-particle sources.

The generator consists of the same one-particle source terms as the previous

generator of Section 9.5, and therefore generates the connected amplitudes or Green™s

functions of the theory according to

δ N W [J, K]

G12...N = . (10.120)

δJ1 δJ2 · · · δJN

J=0,K=0

In addition the generator contains two-particle source terms and mixed terms.

We notice the new feature of the presence of the two-particle source, that dif-

ferentiating with respect to the two-particle source can lead to the appearance of

disconnected diagrams; for example, see Figure 10.7.

1

1

+

δW [J,K]

2 =

δK12 J=K=0

2

2

Figure 10.7 Removing a two-particle source can create disconnected diagrams.

Taking the derivative with respect to the one-particle source

δW [J, K]

•1 = (10.121)

δJ1

10.5. Two-particle irreducible e¬ective action 341

we can, analogous to the procedure of Section 9.6, exploit the topological features

of diagrams to construct the diagrammatic expansion in terms of two-particle irre-

ducible vertices, as depicted in Figure 10.8. In the following we leave out in the

diagrammatic notation the implicit source dependences of quantities.

= + +

+ +

+ +

1 1

2 3!

+ ···

+ +

Figure 10.8 Two-particle irreducible expansion of the 1-state Green™s function.

The topological arguments for the diagrammatic equation displayed in Figure 10.8

is: the particle state exposed can propagate directly to either a one-particle source

or a two-particle source. In the latter case its other state can end up in anything

342 10. E¬ective action

connected, and these two classes of diagrams are depicted as the two ¬rst diagrams

on the right in Figure 10.8. Or the exposed state can enter into a two-particle

irreducible diagram, giving the class of diagrams represented by the third diagram

on the right, or into a two-particle reducible diagram. A two-particle irreducible

vertex is by de¬nition a vertex diagram which can not be cut in two by cutting only

two lines, otherwise it is two-particle reducible. In the case of entering into a two-

particle reducible diagram, the exposed state can enter into a two-particle irreducible

vertex which emerges by one line into anything connected, accounting for the fourth

diagram on the right containing the self-energy in the skeleton representation where

it is two-particle irreducible (recall the topological discussion of diagrams in Section

4.5.2). Or it can enter into a two-particle irreducible vertex which emerges by two

lines into anything connected, which can be done in the two ways as depicted in the

¬fth and sixth displayed diagrams on the right, or three or four, etc., lines as depicted

in Figure 10.8. We note that, from a two-particle irreducible vertex, three lines can

not emerge into a connected 3-state diagram since such a part is already included in

the vertex owing to its two-particle irreducibility.

Analytically we have, according to the diagrammatic equation depicted in Figure

10.8, the equation

1

(0)

•1 = G12 J2 + K23 •3 + “2 + Σ23 •3 + “2(34) G34 + “234 •3 •4

2

1 (0)

+ “2345 •3 •4 •5 + “23(45) •3 G45 + “2(34)(56) G34 G56 + ... .

3!

(10.122)

Operating on both sides of Eq. (9.79) with the inverse free propagator gives

1

J1 + K12 •2 + “1 + (’G’1 + Σ)12 •2 + “1(23) G23 +

0= “123 •2 •3

0

2

1

“1234 •2 •3 •4 + “12(34) •2 G34 + “1(23)(45) G23 G45 + · · ·

+ ,

3!

(10.123)

which corresponds to the diagrammatic equation depicted in Figure 10.9.

10.5. Two-particle irreducible e¬ective action 343

0 = + +

+ +

+1 +

2

+ ···

+

1 1 δ“[φ,G]

≡ +

+ δφ1

Figure 10.9 Two-particle irreducible vertices and source relation.

The last equality de¬nes the ¬eld-derivative of the two-particle irreducible e¬ective

action, i.e, just as the diagrams in Figure 9.48 lead to the introduction of the one-

particle irreducible e¬ective action, we collect the two-particle irreducible vertex

344 10. E¬ective action

functions into the two-particle irreducible e¬ective action

∞

1

“[•, G] ≡ “12...N •1 •2 · · · •N

N!

N

“1(23) •1 G23 + “1(23)(45) •1 G23 G45 + · · · ,

+ (10.124)

which in addition to the ¬eld is a functional of the full propagators.

In the two-particle irreducible action, we encounter two di¬erent types of vertices,

viz. only ¬eld attachment vertices

δ N “[•, G]

“12...N = (10.125)

δ•1 · · · δ•N

φ=0,G=0

which are two-particle irreducible and for which we introduce the diagrammatic no-

tation depicted in Figure 10.10.

2

1

“12...N =

N

Figure 10.10 The 2PI vertex with only ¬eld attachments.

In addition we encounter vertices with also propagator attachments, for example

δ δ δ δ

“1(23)4(56) = “[•, G] (10.126)

δ•1 δG23 δ•4 δG56

φ=0,G=0

for which we introduce the diagrammatic notation depicted in Figure 10.11.

4

5

3

“1(23)4(56) =

2 6

1

Figure 10.11 Vertex with both ¬eld and propagator attachments.

In terms of the two-particle irreducible e¬ective action, we can diagrammatically

represent the equation depicted in Figure 10.9 as depicted in Figure 10.12 (rede¬ning

“12 ≡ (’G’1 + Σ)12 ).

0

10.5. Two-particle irreducible e¬ective action 345

0 = + +

Figure 10.12 Sources and 2PI e¬ective action relation.

Analytically, the diagrammatic relationship depicted in Figure 10.12 is

δ“[•, G]

= ’J1 ’ K12 •2 . (10.127)

δ•1

By diagrammatic construction we have analogously to the one-particle irreducible

case, G1 ≡ •1 ,

δW [J, K]

G1 = (10.128)

δJ1

but now in addition

δW [J, K] 1

= (G12 + •1 •2 ) (10.129)

K12 2

and these two relationships give implicitly the sources as functions of the ¬eld and

the full Green™s function

J = J[•, G] (10.130)

and

K = K[•, G] . (10.131)

Since the sources are independent, so are • and G. We then have the two generators

being related by the double Legendre transformation, i.e. with respect to two sources,

1 1

W [J, K] ’ • J ’ •K • ’ GK

“[•, G] = (10.132)

2 2

J=J[•,G],K=K[•,G]

and we obtain the second relation for the two-particle irreducible e¬ective action and

the sources

δ“[•, G] 1

= ’ K12 . (10.133)

δG12 2

For K = 0 we encounter the usual Legendre transformation and e¬ective action,

i.e. “[•] = “[•, G(0) ] for the value of the Green™s function for which

δ“[•, G(0) ]

= 0. (10.134)

(0)

δG12

By construction “[•, G] is the generator, in the ¬eld variable •, of the two-

particle irreducible vertices with lines representing the full Green™s function, G, and

346 10. E¬ective action

“[• = 0, G] is thus the sum of all two-particle irreducible connected vacuum diagrams.

Using Eq. (10.132) and Eq. (10.133) we have

δ“[0, G] δ“[0, G]

’ i ln Dφ φ exp i S[φ] + φJ (0) ’ φ

“[0, G] = Tr G φ

δG δG

i ln Dφ φ exp {iS0 [φ]} ,

+ (10.135)

where J (0) is the value of the source for which δW [J, K]/δJ vanishes, i.e. tadpoles