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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
Np = ω + µp ’ μ + Σ11 (’p) . (10.212)
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

|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


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

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
(15N0 )1/5
nnc (0) a

= (10.237)
n0 (0) aosc
3π 2 2
as ¬rst obtained in reference [68]. The LDA is a valid approximation when the gas
locally resembles that of a homogeneous system, i.e. when the condensate wave
function changes little on the scale of the coherence length ξ, which according to the
Gross“Pitaevskii equation is ξ = (8πn0 (0)a)’1/2 . For a trapped cloud of bosons in the
ground state, its radius R determines the rate of change of the density pro¬le. Since
R is a factor g 2/5 larger than ξ [67], we expect the agreement between the LDA and
˜
the exact results to be best in the strong-coupling regime. The fractional depletion of
the condensate at the trap center as a function of the dimensionless coupling strength
g = 4πN0 a/aosc is shown in Figure 10.19. In Figure 10.19 are displayed both the
˜
local-density result Eq. (10.235) with the numerically computed condensate density
inserted, and the Thomas“Fermi approximation Eq. (10.237), showing that the LDA
indeed is valid when the coupling is strong. Furthermore, inspection of Eq. (10.237)
reveals that the LDA coe¬cient and exponent agree with the numerically found
result of Eq. (10.234), which is valid for strong coupling. However, when g ˜ 10,
the LDA prediction for the depletion deviates signi¬cantly from the numerically
computed depletion. Inserting the numerically obtained condensate density into the
LDA instead of the Thomas“Fermi approximation is seen not to substantially improve
the result, as seen in Figure 10.19.
The relation for the fractional depletion, Eq. (10.234), is in agreement with the
results of reference [69], where the leading-order corrections to the Gross“Pitaevskii
equation were considered in the one-particle irreducible e¬ective action formalism,
employing physical assumptions about the relevant length scales in the problem.
These leading-order corrections were found to have the same power-law dependence
on N0 and a/aosc . A direct comparison of the prefactors cannot be made, because
the objective of reference [69] was to estimate the higher-loop correction terms to the
Gross“Pitaevskii equation and not to the self-energy.
(2a)
The two-loop term Σ11 can, at zero temperature, according to Eq. (10.232) be
ignored as long as nnc
˜ n0 , which is true in a wide, experimentally relevant param-
˜
eter regime. The one-loop result for the fractional depletion Eq. (10.234) depends
10.6. E¬ective action approach to Bose gases 369


very weakly on N0 , so as long as N0 does not exceed 109 , which is usually ful¬lled
in experiments, we can restate the criterion for the validity of Eq. (10.234) into the
condition a aosc . In experiments on atomic rubidium and sodium condensates,
this condition is ful¬lled, except in the instances where Feshbach resonances are used
to enhance the scattering length [70].
In Section 10.6.3 we showed that for a homogeneous gas all two-loop diagrams are
equally important in the sense that they are all of the same order in the diluteness

parameter n0 a3 . The situation in a trapped system is not so clear, since the den-
sity is not constant. We shall therefore compare the ¬ve normal self-energy diagrams
(2a’e)
Σ11 in Figure 10.18, to see whether they display the same parameter dependence
and whether any of the terms can be neglected. In particular, the Popov approx-
(2a)
imation corresponds to keeping the diagram Σ11 but neglects all other two-loop
diagrams, and we will now determine its limits of validity at zero temperature. Since
diagram (2a) contains a delta function, we shall integrate over one of the spatial
arguments of the self-energy terms and keep the other one ¬xed at the origin, r = 0.
We denote by R(j) the ratio between the integrated self-energy terms (j) and (2a),
(j)
dr Σ11 (0, r, ω = 0)
(j)
R = . (10.238)
(2a)
dr Σ11 (0, r, ω = 0)

In Figure 10.20, we display the ratios R(j) for the di¬erent integrated self-energy
contributions corresponding to the diagrams where (j) represents (2b) and (2c).




Figure 10.20 Ratio between di¬erent two-loop self-energy terms as functions of the
dimensionless coupling strength g = 4πN0 a/aosc . Asterisks denote the ratio R(2b) as
˜
de¬ned in Eq. (10.238) and circles denote the ratio R(2c) . The terms R(2d) and R(2e)
are equal and turn out to be equal in magnitude to R(2b) , and are not displayed.

The contributions from the diagrams (2d) and (2e) are equal and within our
numerical precision turn out to be equal to the contribution from diagram (2c).
Furthermore, inspection of the diagrams in Figure 10.18 reveals that when the con-
(2a) (2d)
densate wave function is real, the anomalous contribution Σ12 is equal to Σ11 , the
370 10. E¬ective action


(2b) (2c) (2c) (2d) (2b)
diagrams Σ12 and Σ12 are equal to Σ11 , and Σ12 is equal to Σ11 .
In the parameter regime displayed in Figure 10.20, the contribution from diagram
(2a) is larger than the others by approximately a factor of ten, and displays only a
weak dependence on the coupling strength. In the weak-coupling limit, g 1, it is
˜
seen that the terms corresponding to diagrams (2b)“(2e) can be neglected as in the
Popov approximation, with an error in the self-energy of a few per cent. When the
coupling gets stronger, this correction becomes more important. A power-law ¬t to
the ratio R(2c) in the regime where the log“log curve is straight yields the dependence

R(2c) ≈ 0.065 g 0.14 ,
˜ (10.239)

which is equal to 0.5 when g ≈ 106 ; for g greater than this value, the Popov ap-
˜ ˜
proximation is seen not to be valid. If the ratio between the oscillator length and
the scattering length is equal to one hundred, aosc = 100a, the Popov approximation
deviates markedly from the two-loop result when N0 exceeds 107 , which is often the
case experimentally.
In order to investigate the importance of higher-order terms in the loop expansion,
we proceed to study the three-loop self-energy diagrams. We have found the number
of summations over Bogoliubov levels to be prohibitively large for most three-loop
(3a) (3a)
terms; however, we have been able to compute the two diagrams Σ11 and Σ12 ,
displayed in Figure 10.21, for the case where one of the spatial arguments is placed
at the origin thereby avoiding a summation over l = 0 components.




(3a) (3a)
Σ Σ
= =
11 12




Figure 10.21 Self-energy diagrams to three-loop order which are evaluated numer-
ically.

(3a) (3a)
We compare the diagrams Σ11 and Σ12 to the two-loop diagrams. As we have
(2b) (2c) (2d)
seen, diagrams Σ11 , Σ11 , and Σ11 in Figure 10.18 are similar in magnitude and
dependence on g , as are of the same order of magnitude and have similar depen-
˜
(2a’2d)
dence on g, and equivalently for the anomalous two-loop diagrams Σ12
˜ ; we have
(2b) (2a)
therefore chosen to evaluate only diagrams Σ11 and Σ12 . The results for the ra-
10.6. E¬ective action approach to Bose gases 371


˜ (3a) ˜ (2b) ˜ (3a) ˜ (2a)
tios Σ11 (0, r, ω = 0)/Σ11 (0, r, ω = 0) and Σ12 (0, r, ω = 0)/Σ12 (0, r, ω = 0),
evaluated for di¬erent choices of r, are shown in Figure 10.22.




Figure 10.22 Ratio of three-loop to two-loop self-energy diagrams as a function of
the dimensionless coupling strength g = 4πN0 a/aosc . Asterisks denote the ratio of
˜
(3a) (2b)
the normal self-energy terms N0 Σ11 /Σ11 evaluated at the point (0, aosc , ω = 0),
open circles denote the same ratio evaluated at (0, 0.5aosc, ω = 0), and diamonds
denote the same ratio evaluated at (0, 1.5aosc , ω = 0). Crosses denote the ratio of
(3a) (2a)
anomalous self-energy terms N0 Σ12 /Σ12 at (0, aosc , ω = 0).

A linear ¬t to the log“log plot gives, for the normal terms, the coe¬cient 0.016 and
the exponent 0.76 when r = 0.5aosc and the coe¬cient 0.0029 and the exponent 0.78
when r = aosc , and for the anomalous terms with the choice r = aosc the coe¬cient
is 0.0015 and the exponent 0.82. Restoring dimensions according to Eq. (10.233) we
obtain
0.8
(3a)
Σ11 (0, aosc , ω = 0) a
’0.2
≈ 0.15N0 . (10.240)
(2b) aosc
Σ (0, aosc , ω = 0)
11
The ratio between three- and two-loop self-energy terms in the homogeneous case was

in Section 10.6.3 found to be proportional to n0 a3 . A straightforward application
of the LDA, substituting the central density n0 (0) for n0 , yields the dependence
(3a) (2b)
Σ11 /Σ11 ∝ N0 (a/aosc )1.2 . This is not in accordance with the numerical result
0.2

Eq. (10.240) although the self-energies were evaluated at spatial points close to the
trap center. The discrepancy between the LDA and the numerical three-loop result
is attributed to the fact that we ¬xed the spatial points in units of aosc while varying
the coupling g , although the physical situation at the point r = aosc (and r = 1 aosc
˜ 2
3
and r = 2 aosc respectively) varies when g is varied. It is possible that the agreement
˜
with the LDA had been better if the length scales had been ¬xed in units of the
actual cloud radius (as given by the Thomas“Fermi approximation) rather than the
oscillator length. However, the present calculation agrees fairly well with the LDA as
long as the number of atoms in the condensate lies within reasonable bounds. Since
(3a) (2b)
N0 > 1 in the condensed state, Eq. (10.240) yields that Σ11 Σ11 whenever
372 10. E¬ective action


the s-wave scattering length is much smaller than the trap length. We conclude that
only when this condition is not ful¬lled is it necessary to study diagrams of three-loop
order and beyond.
We have shown that by employing the two-particle irreducible e¬ective action
approach to a condensed Bose gas, Beliaev™s diagrammatic expansion in the dilute-
ness parameter and the t-matrix equations are expediently arrived at with the aid of
the e¬ective action formalism. The parameter characterizing the loop expansion for
a homogeneous Bose gas turned out to equal the diluteness parameter, the ratio of
the s-wave scattering length and the inter-particle spacing. For a Bose gas contained
in an isotropic, three-dimensional harmonic-oscillator trap at zero temperature, the
small parameter governing the loop expansion was found to be almost proportional
to the ratio between the s-wave scattering length and the oscillator length of the
trapping potential, and to have a weak dependence on the number of particles in the
condensate. The expansion to one-loop order, and hence the Bogoliubov equation,
is found to provide a valid description for the trapped gas when the oscillator length
exceeds the s-wave scattering length. We compared the numerical results with the
local-density approximation, which was found to be valid when the number of par-
ticles in the condensate is large compared to the ratio between the oscillator length
and the s-wave scattering length. The physical consequences of the self-energy cor-
rections considered are indeed possible to study experimentally by using Feshbach
resonances to vary the scattering length. Furthermore, we found that all the self-
energy terms of two-loop order are not equally large for the case of a trapped system:
in the limit when the number of particles in the condensate is not large compared
with the ratio between the oscillator length and the s-wave scattering length, the
Popov approximation was shown to be a valid approximation.


10.7 Summary
In this chapter we have considered the e¬ective action. To study its properties and di-
agrammatic expansions, we introduced the functional integral representations of the
generators. We showed how to express the e¬ective action in terms of one-particle
and two-particle irreducible loop vacuum diagram expansions. As an application, we
applied the two-particle irreducible e¬ective action approach to a condensed Bose
gas, and showed that it allows for a convenient and systematic derivation of the
equations of motion both in the homogeneous and trapped case. We chose in ex-
plicit calculations to apply the formalism to the situation where the temperature was
zero, but the formalism is with equal ease capable of dealing with systems at ¬nite
temperatures and general non-equilibrium states.
11

Disordered conductors

Quantum corrections to the classical Boltzmann results for transport coe¬cients
in disordered conductors can be systematically studied in the expansion parameter
/pF l, the ratio of the Fermi wavelength and the impurity mean free path, which
typically is small in metals and semiconductors. The quantum corrections due to
disorder are of two kinds, one being the change in interactions e¬ects due to disorder,
and the other having its origin in the tendency to localization. When it comes to
an indiscriminate probing of a system, such as the temperature dependence of its
resistivity, both mechanisms are e¬ective, whereas when it comes to the low-¬eld
magneto-resistance only the weak localization e¬ect is operative, and it has therefore
become an important diagnostic tool in material science. We start by discussing the
phenomena of localization and (especially weak localization) before turning to study
the in¬‚uence of disorder on interaction e¬ects.


11.1 Localization
In this section the quantum mechanical motion of a particle at zero temperature in a
random potential is addressed. In a seminal paper of 1958, P. W. Anderson showed
that a particle™s motion in a su¬ciently disordered three-dimensional system behaves
quite di¬erently from that predicted by classical physics according to the Boltzmann
theory [71]. In fact, at zero temperature di¬usion will be absent, as particle states are
localized in space because of the random potential. A su¬ciently disordered system
therefore behaves as an insulator and not as a conductor. By changing the impurity
concentration, a transition from metallic to insulating behavior occurs, the Anderson
metal“insulator transition.
In a pure metal, the Bloch or plane wave eigenstates of the Hamiltonian are
extended states and current carrying

ˆ dx p| ˆ
= j(x)|p = e vp . (11.1)
j ext


In a su¬ciently disordered system, a typical energy eigenstate has a ¬nite extension,

373
374 11. Disordered conductors


and does not carry any average current

ˆ =0. (11.2)
j loc


The last statement is not easily made rigorous, and the phenomenon of localization
is quite subtle, a quantum phase transition at zero temperature in a non-equilibrium
state.1
Astonishing progress in the understanding of transport in disordered systems
has taken place since the introduction of the scaling theory of localization [72]. A
key ingredient in the subsequent development of the understanding of the transport
properties of disordered systems was the intuition provided by diagrammatic pertur-
bation theory. We shall bene¬t from the physical intuition provided by the developed
real-time diagrammatic technique in the present chapter, where it will provide the
physical interpretation of the weak localization e¬ect and the di¬usion enhancement
of interactions. We start by considering the scaling theory of localization.2

11.1.1 Scaling theory of localization
We shall consider a macroscopically homogeneous conductor, i.e. one with a spatially
uniform impurity concentration, at zero temperature. By macroscopically homoge-
neous we mean that the impurity concentration on the macroscopic scale, i.e. much
larger than the mean free path, is homogeneous. The conductance of a d-dimensional
hypercube of linear dimension L is, according to Eq. (6.57), proportional to the con-
ductivity

G(L) = Ld’2 σ(L) . (11.3)

The central idea of the scaling theory of localization is that the conductance rather
than the conductivity is the quantity of importance for determining the transport
properties of a macroscopic sample. The conductance has dimension of e2 / , inde-
pendent of the spatial dimension of the sample, and we introduce the dimensionless
conductance of a hypercube

G(L)
g(L) ≡ . (11.4)
e2


The one-parameter scaling theory of localization is based on the assumption that
the dimensionless conductance solely determines the conductivity behavior of a dis-
ordered system. Consider ¬tting nd identical blocks of length L, i.e. having the same
impurity concentration and mean free path (assumed smaller than the size of the
system, l < L) into a hypercube of linear dimension nL. The d.c. conductance of
the hypercube g(nL) is then related to the conductance of each block, g(L), by
1 For
a discussion of wave function localization we refer the reader to chapter 9 of reference [1].
2 Thescaling theory of localization has its inspiration in the original work of Wegner [73] and
Thouless [74].
11.1. Localization 375




g(nL) = f (n, g(L)) . (11.5)

This is the one-parameter scaling assumption, the conductance of each block solely
determines the conductance of the larger block; there is no extra dependence on L
or microscopic parameters such as l or »F .
For a continuous variation of the linear dimension of a system, the one-parameter
scaling assumption results in the logarithmic derivative being solely a function of the
dimensionless conductance
d ln g
= β(g) . (11.6)
d ln L
This can be seen by di¬erentiating Eq. (11.5) to get

d ln g(L) L dg(L) L dg(nL) 1 dg(nL) 1 df (n, g)
≡ β(g(L)) .
= = = =
d ln L g dL g dL g dn g dn
n=1 n=1 n=1
(11.7)
The physical signi¬cance of the scaling function, β, is as follows. If we start out with
a block of size L, with a value of the conductance g(L) for which β(g) is positive, then
the conductance according to Eq. (11.6) will increase upon enlarging the system, and
vice versa for β(g) negative. The β-function thus speci¬es the transport properties
at that degree of disorder for a system in the in¬nite volume limit.
In the limit of weak disorder, large conductance g 1, we expect metallic con-
duction to prevail. The conductance is thus described by classical transport theory,
i.e. Ohm™s law prevails G(L) = Ld’2 σ0 , and the conductivity is independent of the
linear size of the system, and we obtain according to Eq. (11.6) the limiting behavior
for the scaling function
β(g) = d ’ 2 , g 1, (11.8)
the scaling function having an asymptotic limit depending only on the dimensionality
of the system.
In the limit of strong disorder, small conductance g 1, we expect with Anderson
[71] that localization prevails, so that the conductance assumes the form g(L) ∝
e’L/ξ , where ξ is called the localization length, the length scale beyond which the
resistance grows exponentially with length.3 In the low-conductance, so-called strong
localization, regime we thus obtain for the scaling function, c being a constant,

β(g) = ln g + c , g 1, (11.9)

a logarithmic dependence in any dimension.
Since there is no intrinsic length scale to tell us otherwise, it is physically reason-
able in this consideration to draw the scaling function as a monotonic non-singular
function connecting the two asymptotes. We therefore obtain the behavior of the
scaling function depicted in Figure 11.1.
3 At this point we just argue that if the envelope function for a typical electronic wave function
is exponentially localized, the conductance will have the stated length dependence, where ξ is the
localization length of a typical wave function in the random potential, as it is proportional to the
probability for the electron to be at the edge of the sample. For a justi¬cation of these statements
within the self-consistent theory of localization we refer the reader to chapter 9 of reference [1].
376 11. Disordered conductors




Figure 11.1 The scaling function as function of ln g. Reprinted with permission
from E. Abrahams, P. W. Anderson, D. C. Licciardello, and T. V. Ramakrishnan,
Phys. Rev. Lett., 42, 673 (1979). Copyright 1979 by the American Physical Society.



This is precisely the picture expected in three and one dimensions. In three
dimensions the unstable ¬x-point signals the metal“insulator transition predicted by
Anderson. The transition occurs at a critical value of the disorder where the scaling
function vanishes, β(gc ) = 0. If we start with a sample with conductance larger
than the critical value, g > gc , then upon increasing the size of the sample the
conductance increases since the scaling function is positive. In the thermodynamic
limit, the system becomes a metal with conductivity σ0 . Conversely, starting with a
more disordered sample with conductance less than the critical value, g < gc , upon
increasing the size of the system, the conductance will ¬‚ow to the insulating regime,
since the scaling function is negative. In the thermodynamic limit the system will be
an insulator with zero conductance. This is the localized state. In one dimension it
can be shown exactly, that all states are exponentially localized for arbitrarily small
amount of disorder [75, 76, 77, 78], and the metallic state is absent, in accordance
with the scaling function being negative. An astonishing prediction follows from the
scaling theory in the two-dimensional case where the one-parameter scaling function
is also negative. There is no true metallic state in two dimensions.4
The prediction of the scaling theory of the absence of a true metallic state in
4 In this day and age, low-dimensional electron systems are routinely manufactured. For example,
a two-dimensional electron gas can be created in the inversion layer of an MBE grown GaAs“AlGaAs
heterostructure. Two-dimensional localization e¬ects provide a useful tool for probing material
characteristics, as we discuss in Section 11.2.
11.1. Localization 377


two dimensions was at variance with the previously conjectured theory of minimal
metallic conductivity. The classical conductivity obtained from the Boltzmann theory
has the form, in two and three dimensions (d = 2, 3),5

e2 kF l d’2
σ0 = k . (11.10)
dπ d’1 F
According to Mott [79], the conductivity in three (and two) spatial dimensions should
decrease as the disorder increases, until the mean free path becomes of the order of the
Fermi wavelength of the electron, l ∼ »F . The minimum metallic conductivity should
thus occur for the amount of disorder for which kF l ∼ 2π, and in two dimensions
should have the universal value e2 / . Upon further increasing the disorder, the
conductivity should discontinuously drop to zero.6 This is in contrast to the scaling
theory, which predicts the conductivity to be a continuous function of disorder. The
metal“insulator transition thus resembles a second-order phase transition, a quantum
phase transition at zero temperature, in contrast to Mott™s ¬rst-order conjecture
(corresponding to a scaling function represented by the dashed line in Figure 11.1).7
The phenomenological scaling theory o¬ers a comprehensive picture of the con-
ductance of disordered systems, and predicts that all states in two dimensions are
localized irrespective of the amount of disorder. To gain con¬dence in this surprising
result, one should check the ¬rst correction to the metallic limit. We therefore calcu-
late the ¬rst quantum correction to the scaling function and verify that it is indeed
negative.

11.1.2 Coherent backscattering
In this section we apply the standard diagrammatic impurity Green™s function tech-
nique to calculate the in¬‚uence of quenched disorder on the conductivity.8 In dia-
grammatic terms, the quantum corrections to the classical conductivity are described
by conductivity diagrams, as discussed in Section 6.1.3, where impurity lines connect-
ing the retarded and advanced propagator lines cross. Such diagrams are nominally
smaller, determined by the quantum parameter /pF l, than the classical contribu-
tion. The subclass of diagrams, where the impurity lines cross a maximal number
of times, is of special importance since their sum exhibits singular behavior. Such a
type of diagram is illustrated in Eq. (11.11).
5 In one dimension, the Boltzmann conductivity is σ0 = 2e2 l/π . However, the conclusion to
be drawn from the scaling theory is that even the slightest amount of disorder invalidates the
Boltzmann theory in one and two dimensions.
6 In three dimensions in the in¬nite volume limit, the conductance drops to zero at the critical

value according to the scaling theory.
7 The impressive experimental support for the existence of a minimal metallic conductivity in

two dimensions is now believed either to re¬‚ect the cautiousness one must exercise when attempting
to extrapolate measurements at ¬nite temperature to zero temperature, or to invoke a crucial
importance of electron“electron interaction in dirty metals even at very low temperatures.
8 For a detailed description of the standard impurity average Green™s function technique we refer

the reader to reference [1].
378 11. Disordered conductors


R


(11.11)

A




The maximally crossed diagrams describe the ¬rst quantum correction to the classical
conductivity, the weak-localization or coherent backscattering e¬ect, a subject we
discuss in detail in Section 11.2.
In the frequency and wave vector region of interest, each insertion in a maximally
crossed diagram is of order one.9 Diagrams with maximally crossing impurity lines
are therefore all of the same order of magnitude and must accordingly all be summed
( Q ≡ p + p ):


p+ p+ p+
p+ p+ p+ p+

+ + ... .
qω qω qω qω
p’ p’
p’
p’
Q’p+ Q’p+ Q’p+ (11.12)



From the maximally crossed diagrams, we obtain analytically, by applying the Feyn-
man rules for conductivity diagrams, the correction to the conductivity of a degen-
10
erate Fermi gas, ω, kT F,




e dp dp
2
˜
p± pβ Cp,p ( F , q, ω) GR (p+ ,
δσ±,β (q, ω) = + ω)
F
(2π )d (2π )d
m π

— GR (p + , + ω)GA (p ’ , A
F )G (p’ , F ) . (11.13)
F



To describe the sum of the maximally crossed diagrams, we have introduced the
9 This is quite analogous to the case of the ladder diagrams important for the classical conduc-
tivity, recall Exercise 6.1 on page 163, and for details see chapter 8 of reference [1].
10 In fact we shall in this section assume zero temperature as we shall neglect any in¬‚uence on the

maximally crossed diagrams from inelastic scattering. Interaction e¬ects will be the main topic of
Section 11.3.
11.1. Localization 379


˜ ≡
+
so-called Cooperon C,11 corresponding to the diagrams ( + ω):
F
F




p+ p +
˜
C
˜
Cp,p ( F , q, ω) ≡

p’ p ’




R R R
p+ p+ p+ p+
+ + +
F p+ F p+ F p+


≡ + + ...
A A A
p’ p’ p’ p’
Q’p+ Q’p+ Q’p+
F F F




R
p+ p+
+
F p+


=
A
p’ p’
Q’p+
F




R R
p+ p+
+ +
F p+ F p+


+ + ... . (11.14)
A A
p’ p’
Q’p+ Q’p+
F F

In the last equality we have twisted the A-line around in each of the diagrams, and
by doing so, we of course do not change the numbers being multiplied together.
Let us consider the case where the random potential is delta correlated12
= u2 δ(x ’ x ) .
V (x)V (x ) (11.15)
11 The nickname refers to the singularity in its momentum dependence being for zero total momen-
tum, as is the case for the Cooper pairing correlations resulting in the superconductivity instability
as discussed in Chapter 8.
12 For the case of a short-range potential, the only change being the appearance of the transport

time instead of the momentum relaxation time. For details we refer the reader to reference [1].
380 11. Disordered conductors


Since the impurity correlator in the momentum representation then is a constant,
u2 , all internal momentum integrations become independent. As a consequence, the
dependence of the Cooperon on the external momenta will only be in the combination
˜
p+p , for which we have introduced the notation Q ≡ p+p , as well as Cω (p+p ) ≡
˜ ˜
Cp,p ( F , 0, ω) ≡ Cω (Q), and we have


R R R
p+ p+ p+ p+
+ + +
F p+ F p+ F p+


˜
Cω (Q) = + + ...
A A A
p’ p’ p’ p’
Q’p+ Q’p+ Q’p+
F F F




R R


+ +
F p+ F p+



= 1 +



A A

Q’p+ Q’p+
F F





R R


+ +
F p+ F p+


...⎟
+ +



A A

Q’p+ Q’p+
F F




R
p+ p+
+
F p+


C
≡ . (11.16)
A
p’ p’
Q’p+
F




For convenience we have extracted a factor from the maximally crossed diagrams
which we shortly demonstrate, Eq. (11.24), is simply the constant u2 in the relevant
11.1. Localization 381


parameter regime. We shall therefore also refer to the quantity C as the Cooperon.
Diagrammatically we obtain according to Eq. (11.16)


R

p+
C = 1 + C . (11.17)
A

Q’p+



Analytically the Cooperon satis¬es the equation
dp
’ Q,
Cω (Q) = 1 + u2 GR (p + ω)GA (p
+, F F) Cω (Q) . (11.18)
+
d
(2π )
It is obvious that a change in the wave vector of the external ¬eld can be compensated
by a shift in the momentum integration variable, leaving the Cooperon independent
of any spatial inhomogeneity in the electric ¬eld, which is smooth on the atomic
scale.
The Cooperon equation is a simple geometric series that we can immediately sum
(1 + ζ(Q, ω) + ζ 2 (Q, ω) + ζ 3 (Q, ω) + ... )
Cω (Q) =

= 1 + ζ(Q, ω) Cω (Q)

1
= , (11.19)
1 ’ ζ(Q, ω)
where we have for the insertion
dp
+ ω)GA (p ’ Q,
ζ(Q, ω) = u2 GR (p , F) . (11.20)
F
d
(2π )
Diagrammatically we can express the result

1
Cω (Q) = . (11.21)
R
+
F p+


1’
A

Q’p+
F




The insertion ζ(Q, ω), Eq. (11.20), is immediately calculated for the region of
1, and we have13
interest, ω„, Ql
ζ(Q, ω) = 1 + iω„ ’ D0 „ Q2 (11.22)
13 Fordetails we refer the reader to [1], where the relation between the Di¬uson and its twisted
diagrams, the Cooperon, in the case of time-reversal invariance, is also established.
382 11. Disordered conductors


and for the Cooperon
1

Cω (Q) = . (11.23)

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