By construction “[•, G] is the generator with respect to the ¬eld, •, of two-particle

irreducible vertex functions. For example, δ 2 “[•, G]/δ•1 δ•2 evaluated at vanishing

¬eld, • = 0, is the diagrammatic expansion for the inverse two-state Green™s function

with two-particle reducible diagrams absent and lines representing the full Green™s

function, i.e.

δ 2 “[•, G]

= G’1 = (G(0) ’ Σ[G])’1 . (10.136)

12 12

δ•1 δ•2

•=0

10.5.1 The 2PI loop expansion of the e¬ective action

In this section we shall take the discussion of Section 10.4 to the next level, the two-

particle irreducible (2PI) level and following Cornwall, Jackiw and Tomboulis obtain

the expression for the e¬ective action in terms of two-particle irreducible vacuum

diagrams [53]. We shall use the path integral representations of the generators to

¬rst get a useful path integral expression for the two-particle irreducible e¬ective

action which has an explicit diagrammatic expansion. In the two-particle irreducible

description of the previous section, physical quantities are expressed in terms of the

average ¬eld and the full Green™s functions. The generating functional with one-

and two-particle sources, f and K, corresponding to the diagrammatic expansion in

Figure 10.4 is

i

Dφ exp iS[φ] + iφf + φKφ = eiW [f,K] .

Z[f, K] = (10.137)

2

The normalization constant is chosen so that Z[f = 0, K = 0] = 1.

The derivatives of the generating functional generate the average ¬eld

δW ¯

= φ1 (10.138)

δf1

and the 2-state Green™s function according to

δW 1 ¯

= φ φ2 + iG12 , (10.139)

21

δK12

where

iG12 = φ1 φ2 ’ φ1 φ2 (10.140)

10.5. Two-particle irreducible e¬ective action 347

and we use for short

i

Dφ φ1 φ2 exp iS[φ] + iφf + φKφ

φ1 φ2 = (10.141)

2

for the amplitude A12 .

The two-particle irreducible e¬ective action, the double Legendre transform of

the generating functional of connected Green™s functions, Eq. (10.132)

1 i

“[φ, G] = W [f, K] ’ f φ ’ φKφ ’ GK (10.142)

2 2

ful¬lls

δ“

= ’f ’ Kφ (10.143)

δφ

and

δ“ i

= ’ K. (10.144)

δG 2

The double Legendre transformation can be performed sequentially, i.e. we ¬rst

de¬ne for ¬xed K

“K [φ] = (W [f, K] ’ φf ) δW [f,K]/δf =φ (10.145)

and then de¬ne G according to

δ“K [φ] 1

= (φ φ + iG) (10.146)

δK 2

and the e¬ective action according to

1 i

“[φ, G] = “K [φ] ’ φKφ ’ GK . (10.147)

2 2

That the two de¬nitions of the Green™s function and the e¬ective action are identical

follows from the identity

δ“K [φ] δW [f, K] δf δW [f, K] δf

’φ

= +

δK δf δK δK δK δW [f,K]/δf =φ

δW [f, K]

= . (10.148)

δK δW [f,K]/δf =φ

Considering K as ¬xed, “K [φ] is the e¬ective action for the theory governed by

the action

1

S K [φ] = S[φ] + φKφ. (10.149)

2

We therefore consider the generating functional

K

Dφ eiS

Z K [f ] = [φ]+iφf

(10.150)

348 10. E¬ective action

and observe

Z K [f ] = Z[f, K] . (10.151)

The generating functional of connected Green™s functions, for ¬xed K, is

W K [f ] = ’i ln Z K [f ] (10.152)

with the corresponding e¬ective action

“K [φ] = W K [f ] ’ φf . (10.153)

We can now use the method of functional evaluation of the e¬ective action of Section

10.4 and obtain

“K [φ] = S K [φ] + W1 [φ] ,

K

(10.154)

where

δS K [φ] K

δW1 [φ]

= ’i ln Dφ exp i S [φ + φ] ’ S [φ] ’ φ ’φ

K K K

W1 .

δφ δφ

(10.155)

Introducing the functional “2 according to the equation

i i i

“[φ, G] = S[φ] + Tr ln G’1 + TrD’1 [φ]G + “2 [φ, G] ’ Tr1 (10.156)

2 2 2

with the inverse of the propagator D de¬ned as

δ 2 S[φ]

’1

D [φ] ≡ (10.157)

δφ δφ

and using Eqs. (10.147) and (10.154) we have

1 i i

“2 [φ, G] = ’ Tr iD’1 [φ] + K G ’ Tr ln G’1 + W1 [φ] + Tr1 .

K

(10.158)

2 2 2

Lastly, we want to show that “2 is the sum of all the two-particle irreducible

vacuum graphs in a theory with vertices determined by the action

∞

δ N S[φ]

1

φ1 · · · φN ,

Sint [φ; φ] = (10.159)

N ! δφ1 · · · δφN

N =3

and propagator lines by the full Green™s function G. In order to do so we ¬rst

eliminate the two-particle source K

δ“[φ, G] δ“2 [φ, G]

= G’1 ’ D’1 [φ] + 2i

K = 2i . (10.160)

δG δG

Using Eqs. (10.147), (10.154) and (10.155), the e¬ective action, “[φ, G], can be

rewritten as a functional integral

δS K [φ] K

δW1 [φ]

Dφ exp i S [φ + φ] ’ S [φ] ’ φ ’φ

i“[φ,G] K K

e =

δφ δφ

K K

[φ]’ 1 φKφ’ 2 GK) [φ]’ 1 φKφ’ 2 GK)

i i

— ≡ ei(S

ei(S K

Z1 [φ] . (10.161)

2 2

10.5. Two-particle irreducible e¬ective action 349

Introducing the generator

δS K [φ]

˜ Dφ exp i (S K [φ + φ] ’ S K [φ] ’ φ

Z K [φ, J] = + φJ (10.162)

δφ

a calculation similar to the one of Section 10.4 gives

˜

δ Z K [φ, J]

= 0. (10.163)

δJ K

J=’δW1 /δφ

The average value of φ has thus been shown to vanish in the theory governed by the

action

δS K

S [φ, φ] = S [φ + φ] ’ S [φ] ’ φ

K K K

(10.164)

δφ

˜

when the source takes the value J = ’δW1 /δφ. If the generating functional Z K [φ, J]

K

is multiplied by a factor depending on G and φ the average value of φ is still zero.

Using Eqs. (10.143), (10.147) and (10.154) we therefore have

δS K [φ] δW1 [φ]

K

f+ + =0 (10.165)

δφ δφ

and obtain the following functional integral expression for the two-particle irreducible

e¬ective action

ei“[φ,G] = e’ 2 φKφ+ 2 GK Dφ ei(S

K

1

i

[φ+φ]+f φ)

. (10.166)

Using Eqs. (10.143) and (10.144) to eliminate the source f

δ“ δ“

f=’ ’ 2i φ (10.167)

δG

δφ

we obtain

δ“[φ, G]

“[φ, G] ’ G = ’i ln Dφ eiS[φ,G;φ] (10.168)

δG

where

δ“[φ, G] δ“[φ, G]

S[φ, G; φ] = S[φ + φ] ’ φ + iφ φ. (10.169)

δG

δφ

Di¬erentiating Eq. (10.168) with respect to G we obtain

δ 2 “[φ, G] δ 2 “[φ, G] δ 2 “[φ, G]

G’ φ’

0= i φφ , (10.170)

δG δG δG δG

δG δφ

where the angle brackets denote the average with respect to the action S[φ, G; φ].

The action in Eq. (10.164) with the source term ’δW1 /δφ added and the action

K

appearing in Eq. (10.169) di¬er only by an irrelevant constant, ’S[φ], and we can

350 10. E¬ective action

conclude that the average value of the ¬eld is zero for the action S[φ, G; φ], i.e.

φ = 0, and we obtain that

G = ’i φφ (10.171)

i.e. G is the full Green™s function for the theory governed by the action S[φ, G; φ].

Finally we rewrite Eq. (10.160)

G’1 = D’1 [φ] + K ’ Σ[φ, G] , (10.172)

where

δ“2 [φ, G]

Σ[φ, G] = 2i . (10.173)

δG

Since D’1 [φ] + K is the free inverse Green™s function and G’1 is the inverse full

Green™s function for the theory governed by the action in Eq. (10.169), we conclude

that Σ is the self-energy, and Eq. (10.172) thereby the Dyson equation. Since the

self-energy, Σ, is the sum of one-particle irreducible connected vacuum diagrams,

we therefore ¬nally conclude that “2 is given by the sum of two-particle irreducible

connected vacuum diagrams.

We have thus shown that the e¬ective action can be written in the form

i i i

“[φ, G] = S[φ] + Tr ln G’1 + TrD’1 [φ] G + “2 [φ, G] ’ Tr1, (10.174)

2 2 2

where “2 [φ, G] is the sum of all two-particle irreducible connected vacuum diagrams

in the theory with action φG’1 φ/2 + Sint [φ : φ], i.e.

“2 [φ, G] = ’i ln eiSint [φ;φ] 2PI

G, (10.175)

where the superscript and subscript on the angle bracket indicate that the func-

tional integral is restricted to the two-particle irreducible vacuum diagrams and the

propagator lines are the full Green™s function.

In general amplitudes or physical quantities can not be calculated exactly, and

an approximation scheme must be invoked. If no small dimensionless expansion

parameter is available we are at a loss. Furthermore, if non-perturbative e¬ects

are prevalent we are left without a general tool to obtain information. To cope

with such situations, approximate self-consistent or mean ¬eld theories have been

useful, although they are uncontrollable as not easily analytically characterized by a

small parameter. The e¬ective action approach can be used to systematically study

correlations order by order in the loop expansion. It is thus the general starting

point for constructing self-consistent approximations. An important feature of the

loop expansion is that it is capable of capturing important nonlinearities of a theory.

In practice one must at a certain order break the chain of correlations described by the

e¬ective action by brute force, a felony we are quite used to in kinetic theory. The

rationale behind this scheme working quite well for calculating average properties

such as densities and currents is that higher-order correlations average out when

interest is in such low-correlation probes. We shall use the e¬ective action approach

to study classical statistical dynamics in Chapter 12, but ¬rst we apply it in the

quantum context, viz. for the study of Bose gases.

10.6. E¬ective action approach to Bose gases 351

10.6 E¬ective action approach to Bose gases

In this section, the e¬ective action formalism is applied to a gas of bosons.17 The

equations describing the condensate and the excitations are obtained by using the

loop expansion for the e¬ective action. For a homogeneous gas, the expansion in

terms of the diluteness parameter is identi¬ed in terms of the loop expansion. The

loop expansion and the limits of validity of the well-known Bogoliubov and Popov

equations are examined analytically for a homogeneous dilute Bose gas and numeri-

cally for a gas trapped in a harmonic-oscillator potential. The expansion to one-loop

order, and hence the Bogoliubov equation, we shall show to be valid for the zero-

temperature trapped gas as long as the characteristic length of the trapping potential

exceeds the s-wave scattering length.

10.6.1 Dilute Bose gases

The dilute Bose gas has been subject to extensive study for more than half a century,

originally in an attempt to understand liquid Helium II, but also as an interesting

many-body system in its own right. In 1947, Bogoliubov showed how to describe

Bose“Einstein condensation as a state of broken symmetry, in which the expecta-

tion values of the ¬eld operators are non-vanishing due to the single-particle state

of lowest energy being macroscopically occupied, i.e. the annihilation and creation

operators for the lowest-energy mode can be treated as c-numbers [55]. In modern

terminology, the expectation value of the ¬eld operator is the order parameter and

describes the density of the condensed bosons. In Bogoliubov™s treatment, the phys-

√

ical quantities were expanded in the diluteness parameter n0 a3 , where n0 denotes

the density of bosons occupying the lowest single-particle energy state, and a is the

s-wave scattering length, and Bogoliubov™s theory is therefore valid only for homo-

geneous dilute Bose gases. The inhomogeneous Bose gas was studied by Gross and

Pitaevskii, who independently derived a nonlinear equation determining the conden-

sate density [56] [57]. A ¬eld-theoretic diagrammatic treatment was applied by Beli-

aev to the zero-temperature homogeneous dilute Bose gas, showing how to go beyond

Bogoliubov™s approximation in a systematic expansion in the diluteness parameter

√

n0 a3 ; and also showing how repeated scattering leads to a renormalization of the

interaction between the bosons [58, 59]. This renormalization in Beliaev™s treatment

was a cumbersome issue, where diagrams expressed in terms of the propagator for

the non-interacting particles are intermixed with diagrams where the propagator con-

tains the interaction potential. Beliaev™s diagrammatic scheme was extended to ¬nite

temperatures by Popov and Faddeev [60], and was subsequently employed to extend

the Bogoliubov theory to ¬nite temperatures by incorporating terms containing the

excited-state operators to lowest order in the interaction potential [61, 62].

A surge of interest in the dilute Bose gas due to the experimental creation of

gaseous Bose“Einstein condensates occurred in the mid-1990s [63]. The atomic con-

densates in the experiments are con¬ned in external potentials, which poses new

theoretical challenges; especially, the Beliaev expansion in the diluteness parameter

√

n0 a3 is questionable when the density is inhomogeneous. Experiments on trapped

17 In this section we essentially follow reference [54].

352 10. E¬ective action

Bose gases employ Feshbach resonances to probe the regime of large scattering length,

and hence large values of the diluteness parameter. It is therefore of importance to

understand the low-density approximations to the exact equations of motion and the

corrections thereto. In the following, we shall employ the two-particle irreducible ef-

fective action approach, and show that it provides an e¬cient systematic scheme for

dealing with both homogeneous Bose gases and trapped Bose gases. We demonstrate

how the e¬ective action formalism can be used to derive the equations of motion for

the dilute Bose gas, and more importantly, that the loop expansion can be used to

determine the limits of validity of approximations to the exact equations of motion

in the trapped case.

10.6.2 E¬ective action formalism for bosons

A system of spinless non-relativistic bosons is according to Eq. (3.68) and Eq. (10.37)

described by the action

S[ψ, ψ † ] drdt ψ † (r, t) [i‚t ’ h(r) + μ] ψ(r, t)

=

1

drdr dt ψ † (r, t)ψ † (r , t) U (r ’ r ) ψ(r , t)ψ(r, t) , (10.176)

’

2

where ψ is the scalar ¬eld describing the bosons, and μ the chemical potential. The

one-particle Hamiltonian, h = p2 /2m + V (r), consists of the kinetic term and an

external potential, and U (r) is the potential describing the interaction between the

bosons. As usual we introduce a matrix notation whereby the ¬eld and its complex

conjugate are combined into a two-component ¬eld φ = (ψ, ψ † ) = (φ1 , φ2 ).

The correlation functions of the bose ¬eld are obtained from the generating func-

tional

i

Z[·, K] = Dφ exp iS[φ] + i· † φ + φ† Kφ (10.177)

2

by di¬erentiating with respect to the source · † = (·, · — ) = (·1 , ·2 ). Here ·(r, t)

denotes a complex scalar ¬eld, not a Grassmannn variable, as we are considering

bosons. A two-particle source term, K, has been added to the action in the generating

functional in order to obtain equations involving the two-point Green™s function in a

two-particle irreducible fashion as discussed in Section 10.5.1.

The generator of the connected Green™s functions is

W [·, K] = ’i ln Z[·, K], (10.178)

and the derivative

δW ¯

= φi (r, t) (10.179)

δ·i (r, t)

gives the average ¬eld, φ, with respect to the action S[φ] + · † φ + φ† Kφ/2,

¯

i

¦(r, t)

Dφ φ(r, t) exp iS[φ] + i· † φ + φ† Kφ

¯

φ(r, t) = = = φ(r, t) .

¦— (r, t) 2

(10.180)

10.6. E¬ective action approach to Bose gases 353

The average ¬eld ¦ is seen to specify the condensate density and is referred to as the

condensate wave function.18

The derivative of W with respect to the two-particle source is (recall Figure 10.7)

δW 1¯ i

¯

= φi (r, t) φj (r , t ) + Gij (r, t, r , t ) , (10.181)

δKij (r, t; r , t ) 2 2

where G is the full connected two-point matrix Green™s function describing the bosons