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

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

(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)