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not in the condensate
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

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