We then have

Dp = Np N’p ’ Σ12 (p)Σ21 (p) = D’p (10.213)

and the contribution from diagram (b) can be rewritten in the form

Σ12 (p)

d4 p Up G12 (p) d4 p Up

=

Np N’p ’ Σ12 (p)Σ21 (p)

10.6. E¬ective action approach to Bose gases 361

Σ12 (p)Np N’p Σ12 (p)Σ21 (p)Σ12 (p)

’

d4 p Up

= 2 2

Dp Dp

d4 p Up Σ12 (p)G11 (p)G11 (’p) ’ Σ21 (p)G12 (p)2

=

d4 p Up (n0 Up G11 (p)G11 (’p) + [Σ12 (p) ’ n0 Up ]

=

— G11 (p)G11 (’p) ’ Σ21 (p)G12 (p)2 . (10.214)

In Figure 10.16 the last two rewritings are depicted diagrammatically.

Figure 10.16 Diagrammatic representation of the last two rewritings in Eq. (10.214)

which lead to the conclusion that the diagram (b) of Figure 10.15 implicitly contains

the ladder contribution to diagram (a). The anomalous self-energy Σ12 is represented

by an oval with two in-going lines, Σ21 is represented by an oval with two outgoing

lines, and the sum of the second- and higher-order contributions to Σ12 is represented

by an oval with the label “2.”

We see immediately that the ¬rst term on the right-hand side corresponds to the

¬rst ladder contribution to diagram (a), and since to one-loop order, Σ12 (p) = n0 Up ,

the other terms in Eq. (10.214) are of two- and higher-loop order. The self-energy in

the second term on the right-hand side can be expanded to second loop order, and

by iteration this yields all the ladder terms, and the remainder can be kept track

of analogously to the way in which it is done in Eq. (10.214). The resulting ladder

resummed diagrammatic expression for the chemical potential is displayed in Figure

10.17.

362 10. E¬ective action

Figure 10.17 The chemical potential to one-loop order after the ladder summation

has been performed and the resulting t-matrix been replaced by its expression in the

dilute limit, the constant g.

In the same manner, the self-energies are resummed. For Σ11 , a straightforward

ladder resummation of all terms is possible, while for Σ12 , the same procedure as the

one used for diagrams (a) and (b) in Figure 10.15 for the chemical potential has to be

performed. In Figure 10.18, we show the resulting ladder resummed diagrams for the

self-energies Σ11 and Σ12 to two-loop order in the dilute limit where T (p, p , q) ≈ g.

Figure 10.18 Normal Σ11 and anomalous Σ12 self-energies to two-loop order after

the ladder summation has been performed and the resulting t-matrix been replaced

by its expression in the dilute limit, the constant g.

In reference [61] a diagrammatic expansion in the potential was performed, which

(2a)

yields to ¬rst order the diagram Σ11 in Figure 10.18, but not the other two-loop

diagrams. This approximation, where the normal self-energy is taken to be Σ11 =

(1a) (2a) (1a)

Σ11 + Σ11 , the anomalous self-energy to Σ12 = Σ12 , and the diagrams displayed

in Figure 10.17 are kept in the expansion of the chemical potential, is referred to

as the Popov approximation. Although we showed at the end of Section 10.6.3 that

all the two-loop diagrams of Figure 10.18 are of the same order of magnitude in the

√

diluteness parameter n0 a3 at zero temperature, the Popov approximation applied

at ¬nite temperatures is justi¬ed, when the temperature is high enough, kT gn0 .

10.6. E¬ective action approach to Bose gases 363

Below, we shall investigate the limits of validity at zero temperature of the Popov

approximation in the trapped case.

In this and the preceding section we have shown how the expressions for the self-

energies and chemical potential for a homogeneous dilute Bose gas are conveniently

obtained by using the e¬ective action formalism, where they simply correspond to

working to a particular order in the loop expansion of the e¬ective action. We

have established that an expansion in the diluteness parameter is equivalent to an

expansion of the e¬ective action in the number of loops. Furthermore, the method

provided a way of performing a systematic expansion, and the results are easily

generalized to ¬nite temperatures. We now turn to show that the e¬ective action

approach provides a way of performing a systematic expansion even in the case of an

inhomogeneous Bose gas.

10.6.5 Inhomogeneous Bose gas

We now consider the experimentally relevant case of a Bose gas trapped in an exter-

nal static potential, thereby setting the stage for the numerical calculations in the

next section. In this case, the Bose gas will be spatially inhomogeneous. The e¬ective

action formalism is equally capable of dealing with the inhomogeneous gas, in which

case all quantities are conveniently expressed in con¬guration space, as presented

in Section 10.6.2. We show that the Bogoliubov and Gross“Pitaevskii theory corre-

sponds to the one-loop approximation to the e¬ective action. The one-loop equations

will be exploited further in the next section.

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

with respect to ¦— (r, t), we obtain the equation of motion for the condensate wave

function

¯

δ “2

2

(i ‚t ’ h + μ)¦(r, t) = g |¦(r, t)| ¦(r, t) + 2igG11 (r, t, r, t)¦(r, t) ’ .

δ¦— (r, t)

(10.215)

To zero-loop order, where only the ¬rst term on the right-hand side appears, the equa-

tion is the time-dependent Gross“Pitaevskii equation. We have already, as elaborated

in the previous section, performed the ladder summation by which the potential is

renormalized and the t-matrix appears and substituted its lowest-order approxima-

tion in the diluteness parameter, the constant g. Since the t-matrix in the momentum

variables is a constant in the dilute limit, it becomes in con¬guration space a product

of three delta functions,

T (r1 , r2 , r3 , r4 ) = g δ(r1 ’ r4 ) δ(r2 ’ r4 ) δ(r3 ’ r4 ) . (10.216)

¯

The quantity “2 is de¬ned as the e¬ective action obtained from “2 by summing the

ladder terms whereby U is replaced by the t-matrix, and its diagrammatic expansion

is topologically of two-loop and higher order.

The Dyson“Beliaev equation, Eq. (10.204), and the equation determining the

condensate wave function, Eq. (10.215), form a set of coupled integro-di¬erential self-

consistency equations for the condensate wave function and the Green™s function, with

364 10. E¬ective action

the self-energy speci¬ed in terms of the Green™s function through the e¬ective action

according to Eq. (10.202). The Green™s function can be conveniently expanded in

the amplitudes of the elementary excitations. We write the Dyson“Beliaev equation,

Eq. (10.204), in the form

dr dt [i σ3 ‚t δ(r ’ r )δ(t ’ t ) + σ3 L(r, t, r , t )] G(r , t , r , t )

1δ(r ’ r )δ(t ’ t ) ,

= (10.217)

where we have introduced the matrix operator

L(r, t, r , t ) = σ3 h δ(r ’ r )δ(t ’ t ) + σ3 Σ(r, t, r , t ) (10.218)

and σ3 is the third Pauli matrix. Up to one-loop order, the matrix Σ is diagonal in

the time and space coordinates and we can factor out the delta functions and write

L(r, t, r , t ) = δ(t ’ t ) δ(r ’ r ) L(r), where

h ’ μ + 2g|¦(r)|2 g¦(r)2

L(r) = . (10.219)

’g¦— (r)2 ’h + μ ’ 2g|¦(r)|2

The eigenvalue equation for L are the Bogoliubov equations. The Bogoliubov op-

erator L is not hermitian, but the operator σ3 L is, which renders the eigenvectors

of L the following properties. For each eigenvector •j (r) = (uj (r), vj (r)) of L with

eigenvalue Ej , there exists an eigenvector •j (r) = (vj (r), u— (r)) with eigenvalue

—

˜ j

’Ej . Assuming the Bose gas is in its ground state, the normalization of the positive-

eigenvalue eigenvectors can be chosen to be •j , •k = δjk , where we have introduced

the inner product

dr •† (r)σ3 •k (r) = dr (u— (r)uk (r) ’ vj (r)vk (r)).

—

•j , •k = (10.220)

j

j

It follows that the inner product of the negative-eigenvalue eigenvectors • are

˜

dr •† (r)σ3 •k (r) = dr (vj (r)vk (r) ’ uj (r)u— (r)) = ’δjk

—

•j , •k

˜˜ = ˜j ˜ (10.221)

k

and the eigenvectors • and • are mutually orthogonal, •j , •k = 0. By virtue of

˜ ˜

—

the Gross“Pitaevskii equation, the vector •0 (r) = (¦(r), ’¦ (r)) is an eigenvector

of the Bogoliubov operator L with zero eigenvalue and zero norm. In order to obtain

a completeness relation, we must also introduce the vector •a (r) = (¦a (r), ’¦— (r))

a

satisfying the relation L•a = ±•0 , where ± is a constant determined by normaliza-

tion, •0 , •a = 1. The resolution of the identity then becomes

•j (r)•† (r ) ’ •j (r)•† (r ) σ3 + •a (r)•† (r ) + •0 (r)•† (r ) σ3 = 1δ(r ’ r )

˜ ˜j a

0

j

j

(10.222)

10.6. E¬ective action approach to Bose gases 365

where the prime on the summation sign indicates that the zero-eigenvalue mode •0

is excluded from the sum. Using the resolution of the identity, Eq. (10.222), allows us

to invert Eq. (10.217) to obtain the Bogoliubov spectral representation of the Green™s

function

1 1

•j (r)•† (r ) ’ •j (r)•† (r ) .

G(r, r , ω) = ˜ ˜j (10.223)

’ ω + Ej ’ ω ’ Ej

j

j

It follows from the spectral representation of the Green™s function that the eigenvalues

Ej are the elementary excitation energies of the condensed gas (here constructed

explicitly to one-loop order). Using Eq. (10.223), we can at zero temperature express

the non-condensate density or the depletion of the condensate, nnc = n’ n0 , in terms

of the Bogoliubov amplitudes

dω

|vj (r)|2 .

nnc (r) = i G11 (r, r, ω) = (10.224)

2π j

The results obtained in this section form the basis for the numerical calculations

presented in the next section.

10.6.6 Loop expansion for a trapped Bose gas

We now turn to determine the validity criteria for the equations obtained to various

orders in the loop expansion for the ground state of a Bose gas trapped in an isotropic

harmonic potential V (r) = 1 mωt r2 . To this end, we shall numerically compute the

2

2

self-energy diagrams to di¬erent orders in the loop expansion.

Working consistently to one-loop order, we need only employ Eq. (10.215) to

zero-loop order, providing the condensate wave function, which upon insertion into

Eq. (10.219) yields the Bogoliubov operator L to one-loop order, from which the

Green™s function to one-loop order is obtained from Eq. (10.223). The resulting

Green™s function is then used to calculate the various self-energy terms numerically.

In order to do so, we make the equations dimensionless with the transformations

’3/2

˜ ˜

r = aosc r , ¦ = N0 /a3 ¦, uj = aosc uj , Ej = ωt Ej , and g = ( ωt a3 /N0 )˜,

˜ ˜ g

osc osc

where aosc = /mωt is the characteristic oscillator length of the harmonic trap,

and N0 is the number of bosons in the condensate.

To zero-loop order, the time-independent Gross“Pitaevskii equation on dimen-

sionless form reads

1 ˜ 1˜ ˜ ˜ ˜ ˜ ˜˜

’ ∇2 ¦ + r2 ¦ + g |¦|2 ¦ = μ¦. (10.225)

r

˜

2 2

We solve Eq. (10.225) numerically with the steepest-descent method, which has

proven to be su¬cient for solving the present equation [65]. The result thus ob-

˜

tained for ¦ is inserted into the one-loop expression for the Bogoliubov operator

L, Eq. (10.219), in order to calculate the Bogoliubov amplitudes uj and vj and

˜ ˜

˜ ˜

the eigenenergies Ej . Since the condensate wave function for the ground state, ¦,

is real and rotationally symmetric, the amplitudes uj , vj in the Bogoliubov equa-

˜˜

tions can be labeled by the two angular momentum quantum numbers l and m,

and a radial quantum number n, and we write unlm (˜, θ, φ) = unl (˜)Ylm (θ, φ),

˜ r ˜r

366 10. E¬ective action

vnlm (˜, θ, φ) = vnl (˜)Ylm (θ, φ). The resulting Bogoliubov equations are linear and

˜ r ˜r

one-dimensional

˜˜ r ˜˜ r v r ˜˜ r

L unl (˜) + g ¦2 (˜)˜nl (˜) = Enl unl (˜) (10.226)

and

˜˜ r ˜˜ r u r ˜˜ r

L vnl (˜) + g ¦2 (˜)˜nl (˜) = ’ Enl vnl (˜) (10.227)

where

1 1 ‚2 1 l(l + 1) 1

˜ g˜ r

’ + r2 ’ μ + 2˜¦2 (˜)

L= r+

˜ ˜ ˜ . (10.228)

2 2

2 r ‚˜

˜r 2r ˜ 2

We note that the only parameter in the problem is the dimensionless coupling pa-

rameter g = 4πN0 a/aosc . Solving the Bogoliubov equations reduces to diagonalizing

˜

the band diagonal 2M — 2M matrix L, where M is the size of the numerical grid.

The value of M in the computations was varied between 180 and 240, higher values

for stronger coupling, and the grid constant has been chosen to 0.05 aosc giving a

maximum system size of 18 aosc .

In the following, we shall estimate the orders of magnitude and the parameter

dependence of the di¬erent two- and three-loop self-energy diagrams, and to this

end we shall use the one-loop results for the amplitudes u, v and the eigenenergies

˜˜

˜ obtained numerically. When working to two- and three-loop order, one must

E

also consider the corresponding corrections to the approximate t-matrix g. These

contributions have been studied in reference [66], and their inclusion will not lead to

any qualitative changes of the results.

Let us ¬rst compare the one-loop and two-loop contributions to the normal self-

energy. The only one-loop term is

(1a)

Σ11 (r, r , ω) = 2g|¦(r)|2 δ(r ’ r ) = 2gn0 (r) δ(r ’ r ). (10.229)

(1a)

We ¬rst compare Σ11 with the two-loop term which is proportional to a delta

function, i.e. the diagram (2a) in Figure 10.18. We shall shortly compare this

diagram to the other two-loop diagrams. For diagram (2a) we have

dω

(2a)

Σ11 (r, r , ω) = 2igδ(r ’ r ) G(r, r, ω ) = 2gnnc(r) δ(r ’ r ). (10.230)

2π

The ratio of the two-loop to one-loop self-energy contributions at the point r is thus

equal to the fractional depletion of the condensate at that point. In Figure 10.19

we show the numerically computed dimensionless fractional depletion at the origin,

nnc (0)/˜ 0 (0), where we have introduced the dimensionless notation

˜ n

˜r

n0 (˜) = |¦(˜)|2 |˜j (˜)|2 .

˜r , nnc (˜) =

˜r vr (10.231)

j

We have chosen to evaluate the densities at the origin, r = 0, in order to avoid a

prohibitively large summation over the l = 0 eigenvectors.

10.6. E¬ective action approach to Bose gases 367

Figure 10.19 Fractional depletion of the condensate N0 nnc /n0 at the trap center

as a function of the dimensionless coupling strength g = 4πN0 a/aosc . Asterisks

˜

represent the numerical results, circles represent the local-density approximation with

the numerically computed condensate density inserted, and the line is the local-

density approximation using the Thomas“Fermi approximation for the condensate

density.

As apparent from Figure 10.19, the log“log curve has a slight bend initially, but

becomes almost straight for coupling strengths g ˜ 100. A logarithmic ¬t to the

straight portion of the curve gives the relation

nnc (0)

˜

0.0019 g 1.2.

˜ (10.232)

n0 (0)

˜

When we reintroduce dimensions, the power-law relationship Eq. (10.232) is multi-

’1

plied by the reciprocal of the number of bosons in the condensate N0 because the

actual and dimensionless self-energies are related according to

ωt a3 ˜ (s)

osc

Σ(s) = s’1 Σ , (10.233)

N0

where s denotes the loop order in question. The ratio between di¬erent loop orders of

the self-energy is thus not determined solely by the dimensionless coupling parameter

g = 4πN0 a/aosc , but by N0 and a/aosc separately. We thus obtain for the fractional

˜

depletion in the strong-coupling limit, g 100,

˜

1.2

nnc (0) 1 nnc (0)

˜ a

≈ 0.041N0

0.2

= . (10.234)

n0 (0) N0 n0 (0)

˜ aosc

It is of interest to compare our numerical results with approximate analytical

results such as those obtained by using the local density approximation (LDA). The

368 10. E¬ective action

LDA amounts to substituting a coordinate-dependent condensate density in the ex-

pressions valid for the homogeneous gas. The homogeneous-gas result for the frac-

tional depletion is [55]

nnc 8

=√ n 0 a3 . (10.235)

n0 3π

In the strong-coupling limit we can use the Thomas“Fermi approximation for the

condensate density

2/5 2/5

r2

1 15N0 a aosc

1’

n0 (r) = , (10.236)

8πa2 a a2

aosc 15N0 a

osc osc

which is obtained by neglecting the kinetic term in the Gross“Pitaevskii equation

[67]. For the fractional depletion at the origin there results in the local density

approximation

6/5