δ2W

’

Gij (r, t, r , t ) =

δ·i (r, t) δ·j (r , t )

δψ(r, t)δψ † (r , t ) δψ(r, t)δψ(r , t )

’i

= , (10.182)

δψ † (r, t)δψ † (r , t ) δψ † (r, t)δψ(r , t )

where δψ(r, t) is the deviation of the ¬eld from its mean value, δψ = ψ ’ ¦. Likewise,

¯

we shall write φ = φ + δφ for the two-component ¬eld. We recall that in the path

integral representation, averages over ¬elds, such as in Eq. (10.182), are automatically

time ordered.

We then introduce the e¬ective action for the bosons, “, the generator of the two-

particle irreducible vertex functions, through the Legendre transform of the generator

of connected Green™s functions, W ,

¯ 1¯ ¯ i

“[φ, G] = W [·, K] ’ · † φ ’ φ† K φ ’ TrGK.

¯ (10.183)

2 2

The e¬ective action satis¬es according to section 10.5.1 the equations

δ“ δ“ i

¯

¯ = ’· ’ K φ , = ’ K. (10.184)

δG 2

δφ

In a physical state where the external sources vanish, · = 0 = K, the variations of

¯

the e¬ective action with respect to the ¬eld averages φ and G vanish, yielding the

equations of motion

δ“

¯=0 (10.185)

δφ

and

δ“

= 0. (10.186)

δG

18 Indeed, as pointed out by Penrose and Onsager, Bose“Einstein condensation is associated

with o¬-diagonal long-range order in the two-point correlation function limr’∞ ψ† (r) ψ(0) =

ψ† (r) ψ(0) = 0 [64]. For a conventional description of bosons in terms of ¬eld operators we refer

to reference [15]. We note that, in the presented e¬ective-action approach, the inherent additional

necessary considerations associated with the macroscopic occupation of the ground state in the

conventional description is conveniently absent.

354 10. E¬ective action

According to Section 10.5.1, the e¬ective action can be written in the form

i i i

“[φ, G] = S[φ] + Tr ln G0 G’1 + Tr(G’1 ’ Σ(1) )G ’ Tr1 + “2 [φ, G] , (10.187)

¯ ¯ ¯

0

2 2 2

where G0 is the non-interacting matrix Green™s function,

i‚t ’ h + μ 0

G’1 (r, t, r , t ) = ’ δ(r ’ r )δ(t ’ t ) (10.188)

’i‚t ’ h + μ

0 0

and the matrix

δ2S

+ G’1 (r, t, r , t )

(r, t, r , t ) = ’ †

(1)

Σ (10.189)

0

δφ (r, t)δφ(r , t ) ¯

φ=φ

will turn out to be the self-energy to one-loop order (see Eq. (10.204)). Using the

action describing the bosons, Eq. (10.176), we obtain for the components

(1) (1)

Σij (r, t, r , t ) = δ(t ’ t ) Σij (r, r ) (10.190)

where

δ(r ’ r) dr U (r ’ r )|¦(r , t)|2 + U (r ’ r )¦— (r , t)¦(r, t)

(1)

Σ11 (r, r ) =

(10.191)

and

(1)

Σ12 (r, r ) = U (r ’ r )¦(r, t)¦(r , t) (10.192)

and

Σ21 (r, r ) = U (r ’ r )¦— (r, t)¦— (r , t)

(1)

(10.193)

and

δ(r ’ r) dr U (r ’ r )|¦(r , t)|2 + U (r ’ r )¦— (r, t)¦(r , t) .

(1)

Σ22 (r, r ) =

(10.194)

The delta function in the time coordinates re¬‚ects the fact that the interaction is

instantaneous. Finally, the quantity “2 in Eq. (10.187) is

¯

“2 = ’i ln eiSint [φ,δφ] 2PI

, (10.195)

G

¯ ¯

where Sint [φ, δφ] denotes the part of the action S[φ + δφ] which is higher than second

order in δφ in an expansion around the average ¬eld. The quantity “2 is conveniently

¯

described in terms of the diagrams generated by the action Sint [φ, δφ], and consists

10.6. E¬ective action approach to Bose gases 355

of all the two-particle irreducible vacuum diagrams as indicated by the superscript

“2PI”, and the diagrams will therefore contain two or more loops. The subscript

indicates that propagator lines represent the full Green™s function G, i.e. the brackets

with subscript G denote the average

†

G’1 δφ iSint [φ,δφ]

= (det iG)’1/2 D(δφ) e 2 δφ

¯ ¯

i

eiSint [φ,δφ] e . (10.196)

G

The diagrammatic expansion of “2 corresponding to the action for the bosons,

Eq. (10.176), is illustrated in Figure 10.13 where the two- and three-loop vacuum

diagrams are shown.

“2 = + +

+ + +...

Figure 10.13 Two-loop (upper row) and three-loop vacuum diagrams (lower row)

contributing to the e¬ective action.

Since matrix indices are suppressed, the diagrams in Figure 10.13 are to be un-

derstood as follows. Full lines represent full boson Green™s functions and in the cases

where we display the di¬erent components explicitly, G11 will carry one arrow (ac-

cording to Eq. (10.182) G22 can be expressed in terms of G11 and thus needs no

special symbol), G12 has two arrows pointing inward and G21 carries two arrows

pointing outward. Dashed lines represent the condensate wave function and can also

be decorated with arrows, directed out from the vertex to represent ¦, or directed

towards the vertex representing ¦— . The dots where four lines meet are interaction

vertices, i.e. they represent the interaction potential U (which in other contexts will

be represented by a wiggly line). When all possibilities for the indices are exhausted,

subject to the condition that each vertex has two in-going and two out-going par-

ticle lines, we have represented all the terms of “2 to a given loop order. Finally,

the expression corresponding to each vacuum diagram should be multiplied by the

356 10. E¬ective action

factor is’2 , where s is the number of loops the diagram contains. In the e¬ective

action approach, the appearance of the condensate wave function in the diagrams

is automatic, and as noted generally in Section 9.6.1, the approach is well suited to

describe broken-symmetry states.

The expansion of the e¬ective action in loop orders was shown in Section 10.3

to be an expansion in Planck™s constant. The ¬rst term on the right-hand side of

¯

Eq. (10.187), S[φ], the zero-loop term, is proportional to 0 , and the terms where

the trace is written explicitly, the one-loop terms, are proportional to 1 . We stress

that the e¬ective action approach presented in this chapter is capable of describ-

ing arbitrary states, including non-equilibrium situations where the external poten-

tial depends on time. Although we in the following in explicit calculations shall

limit ourselves to study a Bose gas at zero temperature the theory is straightfor-

wardly generalized to ¬nite temperatures. The equations of motion, Eq. (10.185)

and Eq. (10.186), together with the expression for the e¬ective action, Eq. (10.187),

form the basis for the subsequent calculations.

10.6.3 Homogeneous Bose gas

In this section we consider the case of a homogeneous Bose gas in equilibrium. The

equilibrium theory of a dilute Bose gas is of course well known, but the e¬ective action

formalism will prove to be a simple and e¬cient tool which permits one to derive

the equations of motion with particular ease, and to establish the limits of validity

for the approximate descriptions often used. For the case of a homogeneous Bose

gas in equilibrium, the general theory presented in the previous section simpli¬es

considerably. The single-particle Hamiltonian, h, is then simply equal to the kinetic

term, h(p) = p2 /2m ≡ µp , and the condensate wave function ¦(r, t) is a time- and

√

coordinate-independent constant whose value is denoted by n0 , so that n0 denotes

the condensate density. The ¬rst term in the e¬ective action, Eq. (10.187), is then

1

S[¦] = (μn0 ’ U0 n2 ) drdt 1 (10.197)

0

2

where

U0 = dr U (r) (10.198)

is the zero-momentum component of the interaction potential. For a constant value

√

of the condensate wave function, ¦(r, t) = n0 , Eq. (10.194) yields

n0 (U0 + Up ) n0 Up

Σ(1) (p) = . (10.199)

n0 U p n0 (U0 + Up )

Varying, in accordance with Eq. (10.185), the e¬ective action, Eq. (10.187), with

respect to n0 yields the equation for the chemical potential

d4 p

i

μ= n0 U 0 + [(U0 + Up )(G11 (p) + G22 (p)) + Up (G12 (p) + G21 (p))]

(2π)4

2

δ“2

’ (10.200)

δn0

10.6. E¬ective action approach to Bose gases 357

where the notation for the four-momentum, p = (p, ω), has been introduced. The

¬rst term on the right-hand side is the zero-loop result, which depends only on the

condensate fraction of the bosons. The second term on the right-hand side is the one-

loop term which takes the noncondensate fraction of the bosons into account. The

term involving the anomalous Green™s functions G12 and G21 will shortly, in Section

10.6.4, be absorbed by the renormalization of the interaction potential. From the

last term originate the higher-loop terms, which will be dealt with at the end of this

section.

The equation determining the Green™s function is obtained by varying the e¬ec-

tive action with respect to the matrix Green™s function G(p), in accordance with

Eq. (10.186), yielding

δ“ i

= ’ ’G’1 + G’1 + Σ(1) + Σ

0= , (10.201)

0

δG 2

where

δ“2

Σij = 2i . (10.202)

δGji

Introducing the notation for the matrix self-energy

Σ = Σ(1) + Σ (10.203)

Eq. (10.201) is seen to be the Dyson equation

G’1 = G’1 ’ Σ . (10.204)

0

In the context of the dilute Bose gas, this equation is referred to as the Dyson“Beliaev

equation.

The Green™s function in momentum space is obtained by simply inverting the 2—2

matrix G’1 (p) ’ Σ(p) resulting in the following components

0

ω + µp ’ μ + Σ22 (p) ’Σ12 (p)

G11 (p) = , G12 (p) = (10.205)

Dp Dp

and

’Σ21 (p) ’ω + µp ’ μ + Σ11 (p)

G21 (p) = , G22 (p) = (10.206)

Dp Dp

all having the common denominator

Dp = (ω + µp ’ μ + Σ22 (p))(ω ’ µp + μ ’ Σ11 (p)) + Σ12 (p)Σ21 (p). (10.207)

From the expression for the matrix Green™s function, Eq. (10.182), it follows that in

the homogeneous case its components obey the relationships

G22 (p) = G11 (’p) , G12 (’p) = G12 (p) = G21 (p) . (10.208)

The corresponding relations hold for the self-energy components. We note that the

results found for μ and G to zero- and one-loop order coincide with those found

358 10. E¬ective action

√

in reference [58] to zeroth and ¬rst order in the diluteness parameter n0 a3 . For

example, according to Eq. (10.199) we obtain for the components of the matrix

Green™s function to one loop-order

’n0 Up

ω + µ p + n0 U p

(1) (1)

G11 (p) = , G12 (p) = , (10.209)

ω 2 ’ µ2 ’ 2n0 Up µp ω 2 ’ µ2 ’ 2n0 Up µp

p p

which are the same expressions as the ones in reference [58]. As we shortly demon-

strate, the loop expansion for the case of a homogeneous Bose gas is in fact equivalent

to an expansion in the diluteness parameter. From Eq. (10.209) we obtain for the

single-particle excitation energies to one-loop order

µ2 + 2n0 Up µp

Ep = (10.210)

p

which are the well-known Bogoliubov energies [55].

Di¬erentiating with respect to n0 the terms in “2 corresponding to the two-

loop vacuum diagrams gives the two-loop contribution to the chemical potential.

Functionally di¬erentiating the same terms with respect to Gji gives the two-loop

contributions to the self-energies Σij . The diagrams we thus obtain for the chemical

potential μ and the self-energy Σ are topologically identical to those found originally

by Beliaev [59]; however, the interpretation di¬ers in that the propagator in the

vacuum diagrams of Figure 10.13 is the exact propagator, whereas in reference [59]

the propagator to one-loop order appears.

In order to establish that the loop expansion for a homogeneous Bose gas is an

√

expansion in the diluteness parameter n0 a3 , we examine the general structure of the

vacuum diagrams comprised by “2 . Any diagram of a given loop order di¬ers from

any diagram in the preceding loop order by an extra four-momentum integration,

the condensate density n0 to some power k, the interaction potential U to the power

k + 1, and k + 2 additional Green™s functions in the integrand. We can estimate the

contribution from these terms as follows. The Green™s functions are approximated

by the one-loop result Eq. (10.209). The additional frequency integration over a

product of k + 2 Green™s functions yields k + 2 factors of n0 U (where U denotes the

typical magnitude of the Fourier transform of the interaction potential), divided by

2k + 3 factors of the Bogoliubov energy E. The range of the momentum integration

provided by the Green™s functions is (mn0 U )1/2 . The remaining three-momentum

’k+1/2 3/2 ’k+1/2

integration therefore gives a factor of order n0 mU , and provided the

Green™s functions make the integral converge, the contribution from an additional

loop is of the order (n0 m3 U 3 )1/2 . This is the case except for the ladder diagrams,

in which case the convergence need to be provided by the momentum dependence of

the potential. The ladder diagrams will be dealt with separately in the next section

where we show that they, through a renormalization of the interaction potential,

lead to the appearance of the t-matrix which in the dilute limit is proportional to

the s-wave scattering length a and inversely proportional to the boson mass. The

renormalization of the interaction potential will therefore not change the estimates

performed above, but change only the expansion parameter. Anticipating this change

we conclude that the expansion parameter governing the loop expansion is for a √

homogeneous Bose gas indeed identical to Bogoliubov™s diluteness parameter n0 a3 .

10.6. E¬ective action approach to Bose gases 359

10.6.4 Renormalization of the interaction

Instead of having the interaction potential appear explicitly in diagrams, one should

work in the skeleton diagram representation where diagrams are partially summed

so that the four-point vertex appears instead of the interaction potential, thus ac-

counting for the repeated scattering of the bosons. In the dilute limit, where the

inter-particle distance is large compared to the s-wave scattering length, the ladder

diagrams give as usual the largest contribution to the four-point vertex function. The

ladder diagrams are depicted in Figure 10.14.

Figure 10.14 Summing all diagrams of the ladder type results in the t-matrix, which

to lowest order in the diluteness parameter is a momentum-independent constant g,

diagrammatically represented by a circle.

On calculating the corresponding integrals, it is found that an extra rung in

√

a ladder contributes with a factor proportional not to n0 m3 U 3 , as was the case

for the type of extra loops considered in the previous section, but to k0 mU , where

k0 is the upper momentum cut-o¬ (or inverse spatial range) of the potential, as

¬rst noted by Beliaev [59]. The quantity k0 mU is not necessarily small for the

atomic gases under consideration here. Hence, all vacuum diagrams which di¬er

only in the number of ladder rungs that they contain are of the same order in the

diluteness parameter, and we have to perform a summation over this in¬nite class

of diagrams. The ladder resummation results in an e¬ective potential T (p, p , q),

referred to as the t-matrix and is a function of the two ingoing momenta and the

four-momentum transfer. Owing to the instantaneous nature of the interactions, the

t-matrix does not depend on the frequency components of the in-going four-momenta,

but for notational convenience we display the dependence as T (p, p , q). To lowest

order in the diluteness parameter, the t-matrix is independent of four-momenta and

proportional to the constant scattering amplitude, T (0, 0, 0) = 4π 2 a/m = g. This

is illustrated in Figure 10.14, where we have chosen an open circle to represent g.

Iterating the equation for the ladder diagrams we obtain the t-matrix equation

d4 q Uq G11 (p + q ) G11 (p ’ q ) T (p + q , p ’ q , q ’ q ). (10.211)

T (p, p , q) = Uq + i

At ¬nite temperatures, the t-matrix takes into account the e¬ects of thermal popu-

lation of the excited states.

360 10. E¬ective action

We shall now show how the ladder resummation alters the diagrammatic repre-

sentation of the chemical potential and the self-energy. In Figure 10.15 are displayed

some of the terms up to two-loop order contributing to the chemical potential μ.

Figure 10.15 Diagrams up to two-loop order contributing to the chemical potential.

Only the two-loop diagrams relevant to the resummation of the ladder diagrams are

displayed. The two-loop diagrams not displayed are topologically identical to those

shown, but di¬er in the direction of arrows or the presence of anomalous instead of

normal propagators.

The ¬rst two terms in Eq. (10.200) is represented by diagrams (a)“(d), and the

two-loop diagrams (e)“(f) originate from “2 . The diagrams labeled (e) and (f) are

formally one loop order higher than (c) and (d), but they di¬er only by containing

one additional ladder rung. Hence, the diagrams (c), (d), (e), and (f), and all the

diagrams that can be constructed from these by adding ladder rungs, are of the same

√

order in the diluteness parameter n0 a3 as just shown above. They are therefore

resummed, and as discussed this leads to the replacement of the interaction potential

U by the t-matrix.

We note that no ladder counterparts to the diagrams (a) and (b) in Figure 10.15

appear explicitly in the expansion of the chemical potential, since such diagrams are

two-particle reducible and are by construction excluded from the e¬ective action “2 .

However, diagram (b) contains implicitly the ladder contribution to diagram (a). In

order to establish this we ¬rst simplify the notation by denoting by Np the numerator

of the exact normal Green™s function G11 (p), which according to Eq. (10.205) is