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