X X 2e2 y y

H¼ E k nk þ c c 0 c 0c :

q2 kÀq k þq k k

0

k;k ;q

k

Now

X X

cy cy 0 þq ck 0 ck cy cy 0 þq ck ck 0

¼À

kÀq k kÀq k

k;k 0 ;q k;k 0 ;q

X

cy °k;k 0 þq À ck cy 0 þq Þck 0

¼À kÀq k

k;k 0 ;q

XX X

X

cy 0 ck 0 þ cy ck cy 0 þq ck 0

¼À k kÀq k

k 0 ;q k0

q k

XX X

¼À nk 0 þ 2

q Àq :

0

q q

k

60 Second quantization and the electron gas

P

nk 0 ¼ N, the total number of particles, we have

But since k0

X X 2e2 2 y

H¼ E k nk þ ° q q À NÞ:

q2

q

k

Note that the summation over q cannot be performed for each component of

the interaction separately, as these would not converge.

2.7 The random phase approximation and screening

Let us suppose that there is an operator By that creates an excitation of a

many-body system. If the ground-state wavefunction is jÉi then we should

have

HjÉi ¼ E 0 jÉi

and

HBy jÉi ¼ °E 0 þ E b ÞBy jÉi;

since the excited state of the system, By jÉi, is also an eigenfunction of H,

having an energy that is greater than the ground state by an amount E b , the

excitation energy. Thus

HBy jÉi À By HjÉi ¼ E b By jÉi:

If in particular

HBy À By H ¼ E b By

then we can say that By creates an excitation of energy E b irrespective of which

eigenstate of H it operates on (provided By jÉi ¼ 0). We abbreviate this con-

6

dition by

½H; By ¼ E b By ;

where ½H; By is known as the commutator of H and By , in distinction to the

anticommutator, in which the minus sign is replaced by a plus.

Let us now examine the case where

By ¼ cy cp ;

pþq

61

2.7 The random phase approximation and screening

and see under what conditions this would create an excitation in the electron

gas. In this case

X X

E k cy ck Vq 0 cy 0 cy 0 þq 0 ck 0 ck ;

H¼ þ 1

k kÀq k

2

k;k 0 ;q 0

k

and it takes but a tedious half-hour to show that

½H; cy cp ¼ °E pþq À E p Þcy cp

pþq pþq

X Vq 0 y

½°c pþqÀq 0 cp À cy cpþq 0 Þy 0

þ pþq q

2

q0

þ q 0 °cy 0 þq cp À cy cpÀq 0 Þ: °2:7:1Þ

pþq pþq

We recall that for the noninteracting system the operator cy cp creates a

pþq

particle“hole pair when it operates on the ground state of a system in which

the pth state is occupied and the °p þ qÞth state is empty. We do not know

what the ground state of the interacting system is, but we can see under what

conditions the operator cy cp will create an excitation in it. Let us consider

pþq

this operator when jqj is much greater than the Fermi radius, kF (Fig. 2.7.1).

Then for any occupied p we can choose q such that E pþq À E p is as large as we

like. Then we can ignore the second term in (2.7.1), and to a good approx-

imation the commutator of H and cy cp is a number times this operator

pþq

itself. This means that where large momentum transfers between particles are

concerned, we are justi¬ed in considering quasiparticle excitations of the

system. But now let us look at the case where q is small. Then the argument

that the ¬rst term will dominate is no longer valid, indicating that the quasi-

particle picture may not be appropriate where small momentum transfers are

Figure 2.7.1. For large q the change in kinetic energy on scattering from p to p þ q is

generally large compared with the matrix element of the potential for this process.

62 Second quantization and the electron gas

concerned. Accordingly we go on to ask whether y , the creation operator for

q

a density ¬‚uctuation, might not be the one we are after.

We can form the commutator of y with the Hamiltonian simply by sum-

q

ming (2.7.1) over all p. The interaction terms cancel exactly, since

X X

cy cy cpþq 0 ¼ y 0 ;

¼

pþqÀq 0 cp pþq qÀq

p p

which leaves us with

X

½H; y À1

°E pþq À E p Þcy cp :

¼

q pþq

p

Although this does not appear to be proportional to y , we cannot be sure

q

that it will not have a similar e¬ect when acting on the wavefunction of the

interacting system. Accordingly we optimistically persevere, and once again

take the commutator with the Hamiltonian H. If ½H; y is e¬ectively equiva-

q

y

lent to 0!q then we should have

½H; ½H; y ¼ °0!Þ2 y :

q q

In fact, our tenacity will be rewarded. We ¬nd

X

°E pþq À E p Þcy cp

½H; ½H; y ¼ À1 H;

q pþq

p

X

°E pþq À E p Þ½H; cy cp

À1

¼ pþq

p

X X Vq 0

E p Þ2 cy cp

À1

¼ °E pþq À þ °E pþq À E p Þ

pþq

2

p;q 0

p

‚ ½°cy y y y y

pþqÀq 0 cp À cpþq cpþq 0 Þq 0 þ q 0 °cpþq 0 þq cp À cpþq cpÀq 0 Þ:

This simpli¬es when we remember that

02 02 p2 02

E pþq ¼ °p þ qÞ ; E p ¼

2

; E pþq À E p ¼ °2p Á q þ q2 Þ:

2m 2m 2m

63

2.7 The random phase approximation and screening

This means that

X

°E pþq À E p Þ°cy y

pþqÀq 0 cp À cpþq cpþq 0 Þ

p

02 X

°2p Á q þ q2 Þcy

¼ pþqÀq 0 cp

2m p

02 X

½2°p 0 À q 0 Þ Á q þ q2 cy 0 þqÀq 0 cp 0 ;

À p

2m p 0

where we have written p 0 for p þ q 0 . This reduces to

02 X 0 02 q 0 Á q y

y

2q Á qcp 0 þqÀq 0 cp 0 ¼ qÀq 0 ;

2m p 0 m

so that

X 02 2

°2p Á q þ q2 Þ cy cp =

°0!Þ2 y ¼

q pþq

2m

p

X Vq 0 0 2 q Á q 0

°q 0 Àq y 0 À y 0 ÀqÀq 0 Þ:

þ °2:7:2Þ

Àq

q

2 m

q0

Now the zero Fourier component of the density, 0 , plays a very di¬erent

role from all the other components. It is just the average density of particles

in the system. Consider, for instance, a box of electrons of average density 0 ,

the box being of length L (Fig. 2.7.2). The ¬rst nonzero Fourier component

Figure 2.7.2. When q ¼ 2=L the operator q measures a quantity approximately

equal to the di¬erence in the number of particles in the two halves of the container.

64 Second quantization and the electron gas

of will be approximately equal to the di¬erence between the number of

particles in the left- and right-hand sides of the box, since it is just

°

2ix

À1 °rÞ exp dr:

L

When the number of particles present is large the di¬erence between numbers

in the two halves will be very small compared with the total number, and so

0 will be the most important term in the summation over q 0 in (2.7.2).

Because the term with q 0 ¼ 0 is omitted, it is only when q 0 ¼ Æq that such

a term will appear. The neglect of all terms for which q 0 6¼ Æq is known as the

random phase approximation, or RPA. This, combined with neglect of the

summation over p, which is small when q is small, leaves us with

4e2 02 q2

°0!Þ2 y °0 y þ y 0 Þ;

¼

q q q

2

2q m

and since all the q commute (Problem 2.7) we have

4e2 0

!¼2

;

m

which gives just the classical plasma frequency, !p .

We thus see that the relevant excitations of low wavenumber are not par-

ticle“hole pairs, but collective motions of the electron gas. Bohm and Pines

argue that we should consider the electrons as interacting through a matrix

element Vq that is equal to 4e2 =q2 only when q is greater than some

characteristic value qc . Below qc the interactions contribute only to the

plasma oscillations and can be left out of the particle interaction terms in

the Hamiltonian. We expect qÀ1 to be of the order of the average interparticle

c

distance, since plasma waves can only exist when their wavelength is greater

than this value. Thus we put Vq ¼ 0 for q < qc . This means that the inter-

action potential in con¬guration space, which is the Fourier transform of Vq ,

will be

X 4e2

iq Á r

V°rÞ ¼ :

e

q2

q>qc

This gives a function rather like the Yukawa potential,

°e2 =rÞ exp °Àqc rÞ;

65

2.7 The random phase approximation and screening

which is an example of a screened Coulomb potential. An electron at a point

tends to repel all the others from its vicinity, which e¬ectively gives a region

of net positive charge surrounding each electron. This partially cancels (or

screens) the mutual repulsion of the electrons at large distances.

We can understand the concept of screening within the framework of

perturbation theory by considering the e¬ect of a weak sinusoidal potential

applied to the electron gas. The total Hamiltonian would then be

H ¼ H0 þ V þ U;

with V the Coulomb interaction of the electrons, and U the externally

applied potential, being given by

U ¼ 2Uq cos q Á r;

which in the notation of second quantization is

X

°cy cp þ cy cp Þ:

U ¼ Uq °2:7:3Þ

pþq pÀq

p

Perturbation theory can then be used to express the wavefunction and energy

as a power series in °U þ VÞ, which we can then rearrange in the form of a

power series in U. In the Rayleigh“Schrodinger expansion for the wavefunc-

¨

tion, for example, we have

jÉi ¼ jÈi þ °E 0 À H0 ÞÀ1 °1 À jÈihÈjÞ°U þ VÞjÈi þ Á Á Á

¼ ½jÈi þ °E 0 À H0 ÞÀ1 °1 À jÈihÈjÞVjÈi þ Á Á Á

þ °E 0 À H0 ÞÀ1 °1 À jÈihÈjÞUjÈi þ °E 0 À H0 ÞÀ1

‚ °1 À jÈihÈjÞV°E 0 À H0 ÞÀ1 °1 À jÈihÈjÞUjÈi þ Á Á Á

þ ½terms of order U 2 þ Á Á Á :

We investigate the response of the system to weak applied ¬elds by examining

those terms that are linear in U. We notice that we could write the sum of

these contributions in the form

°E 0 À H0 ÞÀ1 °1 À jÈihÈjÞUeff jÈi

66 Second quantization and the electron gas

if we were to de¬ne an e¬ective potential Ueff by the equation

Ueff ¼ U þ V°E 0 À H0 ÞÀ1 °1 À jÈihÈjÞU

þ U°E 0 À H0 ÞÀ1 °1 À jÈihÈjÞV þ Á Á Á : °2:7:4Þ

Let us now substitute for V, the Coulomb interaction, and for U, and sim-

plify U by considering only the ¬rst part of the summand in expression

(2.7.3). Then the second term on the right-hand side of Eq. (2.7.4), for

example, becomes

X X

Vq 0 cy 0 cy 0 þq 0 ck 0 ck °E 0 À H0 ÞÀ1 °1 À jÈihÈjÞ Uq cy cp : °2:7:5Þ

1

kÀq k pþq

2

k;k 0 ;q 0 p

This component of Ueff is thus a sum of terms that annihilate the electrons in

states p; k, and k 0 , and create them again in states p þ q; k 0 þ q 0 , and k À q 0 .

Such complicated processes could be represented by diagrams like Fig. 2.7.3,

and are not easily interpreted in physical terms.

There are, however, some terms from this sum that contribute in a special

way to Ueff , and whose e¬ect has a simple interpretation. Let us look, for

example, at the term in which q ¼ q 0 and p þ q ¼ k. Then we can join

together the two parts of Fig. 2.7.3 and represent the scattering in the

form shown in Fig. 2.7.4. We note the interesting fact that the net result of

Figure 2.7.3. In this diagram an electron is scattered by the externally applied poten-

tial U, and then two other electrons interact through their Coulomb repulsion V.

67

2.7 The random phase approximation and screening

Figure 2.7.4. In this special case of the preceding diagram the same electron parti-

cipates in both scattering processes.

these interactions is that an electron is scattered from k 0 to k 0 þ q. The

physical interpretation of this is that the externally applied potential U causes

a density ¬‚uctuation in the electron gas, and it is this density ¬‚uctuation that

scatters the electron originally in the state k 0 . The contribution to expression

(2.7.5) from these processes is

X X

À1

Uq cy 0 þq ck 0 ;

Vq np °1 À npþq Þ°E p À E pþq Þ

1

k

2

k0

p

the energy denominator °E p À E pþq ÞÀ1 coming from the e¬ect of U on the

state È. There is also a set of terms for which q ¼ Àq 0 and p þ q ¼ k 0 , which

contribute an equal amount again. To these must then be added a set of

terms from the third component of the right-hand side of expression (2.7.4) in

which V acts ¬rst, followed by U. From these we select the terms shown in

Fig. 2.7.5, which contribute an amount

X X

Uq cy ck :

À1

Vq npþq °1 À np Þ°E pþq À E p Þ kþq

p k

We identify the sums over k and k 0 as just being equal to U itself, and so our

approximation for Eq. (2.7.4) becomes

X

npþq À np

Ueff ™ U þ Vq U þ ÁÁÁ:

E pþq À E p

p

If we make similar approximations for the terms of higher order in this series

we shall have contributions of the form shown in Fig. 2.7.6, which can be

68 Second quantization and the electron gas

Figure 2.7.5. Here an electron is ¬rst scattered by another electron and then by the

applied potential U.

Figure 2.7.6. In this diagram an electron scattered by the externally applied potential

passes its extra momentum to another electron through a chain of Coulomb inter-

actions.

given a simpler aspect if we think of the scattering of the electron from p þ q

to p as the creation of a particle“hole pair, and represent the hole of wave-

number p þ q by an arrow pointing backwards. Figure 2.7.6 can then be

redrawn as in Fig. 2.7.7. All these complicated diagrams will have the net

e¬ect of scattering just one electron and increasing its wavenumber by an

69

2.7 The random phase approximation and screening

Figure 2.7.7. This redrawing of Fig. 2.7.6. depicts the absence of an electron in a

given k-state as a line pointing backwards.

amount q. They in fact form a geometric series, which allows us to write

X X

npþq À np npþq À np 2

™ 1 þ Vq þ Vq

2

þ ÁÁÁ U

Ueff

E pþq À E p E pþq À E p

p p

U

¼ ;

°qÞ

where

X npþq À np

°qÞ ¼ 1 À Vq : °2:7:6Þ

E pþq À E p

p

De¬ned in this way, °qÞ plays the role of a dielectric constant in that it is the

factor by which the applied ¬eld, which may be likened to the electric dis-

placement D, exceeds the actual ¬eld E within the electron gas. Because a

conductor like the electron gas cannot support a steady uniform electric ¬eld

70 Second quantization and the electron gas

we expect °qÞ to become in¬nite as q ! 0. This does indeed occur, since Vq

varies as qÀ2 while the summation over p remains ¬nite.

Because any potential can be analyzed into its Fourier components, this

theory gives us an approximate result for the modi¬cation by the electron gas

of a potential of any shape. If, for example, we put a charged impurity into

the electron gas the potential U would be ÀZe2 =r. This is the sum of Fourier

components À4Ze2 =q2 , each of which would be screened in our linear

approximation by the dielectric constant °qÞ. The result would be a screened

potential of Fourier transform

À4Ze2 =q2

X

Ueff °qÞ ™ :

1 À °4e2 =q2 Þ ½°npþq À np Þ=°E pþq À E p Þ

p

This expression remains ¬nite as q ! 0, and thus represents a potential that

again has some similarity to the Yukawa potential.

Improvements on this theory are fairly arduous, even in the linear approx-

imation. The most obvious correction would be to include exchange scatter-

ing in our analysis by considering processes of the type shown in Fig. 2.7.8 in

addition to those of Fig. 2.7.4. In higher orders, however, these processes do

not reduce to simple products that can be summed as geometric series, and

their investigation lies beyond the scope of this book.

Figure 2.7.8. Exchange processes such as this one, in which the ¬nal k-state of one

electron is identical to the initial state of another electron, are neglected in deriving

Eq. (2.7.6).

71

2.8 Spin waves in the electron gas

2.8 Spin waves in the electron gas

An interesting application of the random phase approximation occurs in the

theory of metallic ferromagnets. We saw in Section 2.4 that in the Hartree“

Fock approximation the exchange energy is negative. It is illustrated in

Problem 2.3 that at low electron densities this exchange energy becomes

large enough in comparison to the kinetic energy that a magnetized phase,

in which all the electron spins are pointing in the same direction, appears the

most stable. While we are aware of the failings of the Hartree“Fock approx-

imation and should not accept its predictions unquestioningly, we are led to

the conclusion that in a metal such as nickel it is the presence of some e¬ective

electron interaction that gives rise to ferromagnetism. We cannot accept

an alternative model of the type we assumed in Section 1.4, in which each

spin is localized at a lattice site, because measurements show there to be a

nonintegral number of spins per atom in this metal. We thus assume a

Hamiltonian of the form

X

E k cy ck;s

H¼ k;s

k;s

X

1 y y

þ Vq 0 c kÀq 0 ;s ck 0 þq 0 ;s 0 ck 0 ;s 0 ck;s ;

2 k;k 0 ;q 0 ;s;s 0

which is identical to our previous form for the Hamiltonian of the electron

gas except that we shall take Vq 0 to be an e¬ective interaction, and not

necessarily the pure Coulomb interaction. We make the assumption that

the ground state of this system is magnetized, so that N# , the total number

of electrons with spin down, is greater than N" .

We now look for collective excitations of this system that can be inter-

preted as spin waves. We ¬rst consider the commutator of the Hamiltonian

with the operator

By ¼ cy cp# : °2:8:1Þ

p pþq"

This di¬ers from the operator considered in the previous section in that it

reverses the spin of the electron on which it acts, and can thus change the

magnetization of the electron gas. We ¬nd

X

½H; By ¼ °E pþq À E p ÞBy þ Vq 0 °cy y

pþqÀq 0 " c kþq 0 ;s ck;s cp#

p p

q 0 ;k;s

þ cy cy 0 ;s cpÀq 0 # ck;s Þ: °2:8:2Þ

pþq" kÀq

72 Second quantization and the electron gas

Figure 2.8.1. Of these scattering processes we retain only those in which one of the

¬nal electron states is the same as one of the initial ones.

The summation is thus over interactions of the form shown in Fig. 2.8.1.

We now make the random phase approximation by retaining only those

processes in which one electron leaves in a state identical to one of the

original states. We thus select from Fig. 2.8.1(a) only those processes for

which k þ q 0 ; s ¼ p# or for which k; s ¼ p þ q À q 0 ". With a similar selection

from the processes of Fig. 2.8.1(b) we ¬nd that Eq. (2.8.2) becomes

X

½H; By ™ °E pþq À E p ÞBy þ Vq 0 ½°npÀq 0 # À npÀq 0 þq" ÞBy

p p p

q0

þ °npþq" À np# ÞBy 0 ; °2:8:3Þ

pÀq

where

By 0 ¼ cy 0 þq" cpÀq 0 # :

pÀq pÀq

This can be characterized as a random phase approximation because it

retains only those terms involving the number operators, and it is the sum

73

2.8 Spin waves in the electron gas

of the number operators that gives the zeroth Fourier component of the

density.

The fact that the right-hand side of Eq. (2.8.3) involves terms in By 0pÀq

y

shows that Bp does not create eigenstates of H when acting on the ground

state. It does, however, suggest that we once again form a linear combination

of these operators by writing

X

By ¼ p By ;

p

p

where the p are constants. If this operator does indeed create spin waves of

energy 0!q we shall ¬nd

X X

p ½H; By ¼ 0!q p By :

p p

p p

We substitute in this relation from Eq. (2.8.3) and equate the coe¬cients of

By to ¬nd

p

X

°0!q À E pþq þ E p Þp ¼ Vq 0 ½°npÀq 0 # À npÀq 0 þq" Þp

q0

þ °npþqþq 0 " À npþq 0 # Þpþq 0 :

At this point we simplify the problem by assuming that Vq 0 can be taken as a

positive constant V. We can then write

X

°0!q À E pþq þ E p À VN# þ VN" Þp ¼ V °np 0 þq" À np 0 # Þp 0 ;

p0

where we have written p 0 for p þ q 0 . This can be solved by noting that the

right-hand side is independent of p. We can thus multiply both sides of this

equation by a factor

npþq" À np#

;

0!q À E pþq þ E p À VN# þ VN"

sum over p, and ¬nd

X npþq" À np#

¼ 1:

V

0!q À E pþq þ E p þ V°N" À N# Þ

p

74 Second quantization and the electron gas

This equation determines !q . For small q, which is the regime in which the

random phase approximation is best justi¬ed, we can expand the left-hand

side binomially to ¬nd

X 0!q À E pþq þ E p

1

°npþq" À np# Þ 1 À

N" À N# p V°N" À N# Þ

2

0!q À E pþq þ E p

þ À Á Á Á ¼ 1:

V°N" À N# Þ

If E p is just the free-electron energy, 02 p2 =2m, and we retain only terms of

order q2 or greater, then this reduces to

X npþq" À np# 02

04 °p Á qÞ2

0!q ¼ °2p Á q þ q2 Þ þ 2

N" À N# m V°N" À N# Þ

2m

p

02 q2

¼ ° þ Þ; °2:8:4Þ

2m

where is a constant of order unity and independent of V, while is

inversely proportional to V. These constants are most simply evaluated by

considering the ground state of the system to consist of two ¬lled Fermi

spheres in momentum space “ a large one for the down-spin electrons and

a small one for the up-spin electrons. The form of the magnetic excitation

spectrum is then as shown in Fig. 2.8.2, and consists of two branches. The

spin waves have an energy 0!q that increases as q2 for small q, and they

represent the collective motion of the system. There are, however, also the

quasiparticle excitations of energy around V°N# À N" Þ that are represented

by the diagonal terms in Eq. (2.8.3).

This calculation presents a very much oversimpli¬ed picture of magnons in

a metal, and should not be taken too seriously. It has the disadvantage, for

instance, that states of the system in which spin waves are excited are eigen-

states of S" , the total spin in the up direction, but not of S 2 , the square of the

total spin angular momentum. The model su¬ces to show, however, the

possibility of the existence of a type of magnon quite dissimilar to that

introduced in the localized model of Section 1.4. It is also interesting to

note that there are some materials, such as palladium, in which the inter-

actions are not quite strong enough to lead to ferromagnetism, but are

strong enough to allow spin ¬‚uctuations to be transmitted an appreciable

distance before decaying. Such critically damped spin waves are known as

paramagnons.

75

Problems

Figure 2.8.2. The spectrum of elementary excitations of the ferromagnetic electron

gas. The lower branch shows the magnons while the upper band represents quasi-

particle excitations.

Problems

2.1 Using the de¬nitions of cp and cy given, verify that

p

fcp ; cy 0 g ¼ pp 0 ; fcp ; cp 0 g ¼ fcy ; cy 0 g ¼ 0:

p pp

2.2 Verify the statement that dW=dp as de¬ned by Eq. (2.4.5) becomes

in¬nite as p ! pF , the radius of the Fermi surface.

2.3 In the Hartree“Fock approximation the energy of the electron gas is

composed of kinetic and exchange energies. In a certain set of units the

kinetic energy per electron is 2.21 rydbergs and the exchange energy

À0:916 rydbergs when the gas is at unit density and zero temperature,

and the up- and down-spin levels are equally populated. Estimate the

density at which a magnetic phase, in which all spins are pointing up,

becomes the more stable one.

2.4 The operators

k0 and

k1 are de¬ned in terms of electron annihilation

and creation operators by the relations

k0 ¼ uk ck" À vk cy ;

k1 ¼ vk cy þ uk cÀk# ;

Àk# k"

where cy , for instance, creates an electron of wavenumber k with spin

k"

76 Second quantization and the electron gas

up, and uk and vk are real constants such that u2 þ v2 ¼ 1. What are the

k k

y

various anticommutation relations of the

and

?

2.5 Verify that

X

nk ; H ¼ 0

k

for the electron gas.

2.6 Verify Eq. (2.7.1).

2.7 Verify that ½k ; k 0 ¼ 0.

2.8 Calculate the contribution to Ueff as de¬ned in Eq. (2.7.4) of the

exchange scattering processes shown in Fig. 2.7.8.

2.9 The theory of the dielectric constant of the electron gas can be general-

ized to include the responses to applied ¬elds that vary with time. If a

potential U°rÞeÀi!t is applied then scattering of an electron occurs by

absorption of a photon of energy 0!, and the energy denominator of

Eq. (2.7.6) is modi¬ed to give

X npþq À np

°q; !Þ ¼ 1 À Vq :

E pþq À E p þ 0!

p

Show that for vanishingly small q the dielectric constant itself vanishes

when ! is the plasma frequency !p .

2.10 Evaluate the constants and of Eq. (2.8.4).

2.11 If the sum of coe¬cients þ in Eq. (2.8.4) becomes negative, then the

magnetic system will be unstable. Use your answer to Problem 2.10 to

¬nd the minimum value that V°N# À N" Þ=E F must have to ensure that

the magnet is stable. (Here E F is the Fermi energy of the unmagnetized

system.) Does your result agree qualitatively with the semiclassical

argument that N# À N" should be equal to the di¬erence between the

integrated densities of states N °E F# Þ À N °E F" Þ?

2.12 Sketch a contour map of °q; !Þ as determined from the expression

given in Problem 2.9. That is, estimate the sign and magnitude of

°q; !Þ for various q and !, and plot lines of constant in the q“! plane.

77

Problems

2.13 Consider a system consisting of a large number N of spinless interacting

fermions in a large one-dimensional box of length L. There are periodic

boundary conditions. The particles interact via a delta-function poten-

tial, and so the Hamiltonian is

X X

y y y

H¼ Ak2 ck ck þ °V=2LÞ ckÀq ck 0 þq ck 0 ck

k;k 0 ;q

k

with A and V constants. The sums proceed over all permitted values of

k, k 0 , and q. That is, the terms with q ¼ 0 are not excluded from the

sum.

(a) Calculate the energy of the ground state of the noninteracting

system.

(b) Calculate the energy of the ground state of the interacting system in

the Hartree“Fock approximation.

(c) State in physical terms why the answer you obtained to part (b)

must be an exact solution of the problem.

Chapter 3

Boson systems

3.1 Second quantization for bosons

In the formalism that we developed for dealing with fermions the number

operator, np , played an important role, as we found that the Hamiltonian for

the noninteracting system could be expressed in terms of it to give

X

H0 ¼ E k nk :

k

Now we turn to the consideration of systems in which we can allow more

than one particle to occupy the same state. This time we shall need to de¬ne a

number operator that has not only the eigenvalues 0 and 1, but all the

nonnegative integers. The wavefunctions È that describe the noninteracting

system will no longer be determinants of one-particle states, but will be

symmetrized products of them, such that È remains unaltered by the inter-

change of any two particles.

In analogy with the fermion case we de¬ne annihilation and creation

operators for boson systems

X p¬¬¬¬¬

ap ¼ np jn1 ; . . . °np À 1Þ; . . .ihn1 ; . . . np ; . . . j °3:1:1Þ

fni g

and

X p¬¬¬¬¬

ay ¼ np jn1 ; . . . np ; . . .ihn1 ; . . . °np À 1Þ; . . . j: °3:1:2Þ

p

fni g

The summation is understood to be over all possible sets of numbers ni ,

including np , with the sole condition that np > 0. These operators reduce or

p¬¬¬¬¬

increase by one the number of particles in the pth state. The factor of np is

78

79

3.1 Second quantization for bosons

included so that the combination ay ap will correspond to the number opera-

p

tor and have eigenvalues np . We do, in fact, ¬nd that with these de¬nitions

X

ay ap ¼ np j . . . np ; . . .ih. . . np ; . . . j;

p

fni g

so that

ay ap j . . . np . . .i ¼ np j . . . np . . .i:

p

On the other hand,

ap ay j . . . np . . .i ¼ °np þ 1Þj . . . np . . .i

p

and so

ap ay À ay ap ¼ 1:

p p

We write this as

½ap ; ay ¼ 1;

p

which says that the commutator of ap and ay is equal to unity. We can further

p

show that

½ay ; ay 0 ¼ ½ap ; ap 0 ¼ 0

p p

½ap ; ay 0 ¼ pp 0 :

p

These results are in terms of commutators rather than anticommutators

because of the fact that we form the same wavefunction irrespective of the

order in which we create the particles.

We can then write the Hamiltonian for a noninteracting system of bosons

in the form

X X

y

H0 ¼ E k a k ak ¼ E k nk :

k k

An important di¬erence between the fermion and boson systems that we

consider is that while for the Fermi systems the total number of particles,

N, is constant in time, this is not generally so for true Bose systems, where N

may be determined by thermodynamic considerations; for example, the total

80 Boson systems

number of phonons present in a solid may be increased by raising the tem-

perature of the system. On the other hand, there are also systems such as

atoms of 4 He for which N is conserved but which behave as pseudobosons in

that their behavior is approximately described by boson commutation rela-

tions. This distinction will be made clearer in Section 3.3.

3.2 The harmonic oscillator

The simplest example of a system of noninteracting bosons is provided by the

case of the three-dimensional harmonic oscillator, where a particle of mass m

is imagined to be in a potential 1 m!2 r2 . The Hamiltonian is

2

X1

3

H¼ ½ p2 þ °m!xi Þ2 ;

2m i

i¼1

where the three components of momentum and position obey the commuta-

tion relations

½xi ; pj ¼ i0 ij :

We then de¬ne

r¬¬¬¬¬¬¬¬¬¬¬¬ r¬¬¬¬¬¬¬¬¬¬¬¬

1 1

ay ¼

ai ¼ °m!xi þ ipi Þ; °m!xi ¼ ipi Þ; °3:2:1Þ

i

2m!0 2m!0

and are not in the least surprised to ¬nd that

y

½ai ; a j ¼ ij

and that

X y

þ ai ay Þ

H¼ 2 0!°a i ai

1

i

i

X

¼ °ni þ 1Þ0!;

2

i

y

where ni ¼ a i ai .

In Section 3.1 we started our discussion of boson systems with the assump-

tion that there was a number operator whose eigenvalues were the positive

integers and zero, and deduced the commutation relations for the a and ay .

What we could have done in this section is to proceed in the reverse direction,

81

3.2 The harmonic oscillator

starting with the commutation relations and hence deducing that the eigen-

values of ni are the natural numbers. Our solution to the harmonic oscillator

problem is then complete. The eigenfunctions are constructed from the

y

ground state by creating excitations with the a i ,

Y

3

y

jÈi ¼ A °ai Þni j0i;

i¼1

P

the energy eigenvalues of these states being simply i °ni þ 1Þ0!. (Here A is

2

some normalizing constant.)

As an exercise in using boson annihilation and creation operators we shall

now consider a simple example “ the anharmonic oscillator in one dimension.

To the oscillator Hamiltonian,

1

H0 ¼ ½ p2 þ °m!xÞ2 ;

2m

we add a perturbation

V ¼ x3 :

Since

r¬¬¬¬¬¬¬¬¬¬

0

°ay þ aÞ;

x¼ °3:2:2Þ

2m!

then

3=2

0

°ay þ aÞ3

x ¼

3

2m!

and

3=2

0

1

H ¼ 0! ay a þ °ay þ aÞ3 :

þ

2 2m!

We try to ¬nd the energy levels of the anharmonic oscillator by using per-

turbation theory. For the unperturbed state jÈi, which we can write as jni

(since it is characterized solely by its energy (n þ 1Þ0!Þ, the perturbed energy

2

to second order in V will be

1 1

E ¼ nþ 0! þ hnjVjni þ hnjV °3:2:3Þ

Vjni:

E n À H0

2

82 Boson systems

In this particular case the ¬rst-order energy change, hnjVjni, will be zero,

since V is a product of an odd number of annihilation or creation operators,

and cannot recreate the same state when it operates upon jni. In second

order, however, we shall ¬nd terms like

3

0 1

hnjay aa ay aay jni;

2 °3:2:4Þ

E n À H0

2m!

which will give a contribution. From the de¬nitions (3.1.1) and (3.1.2) we

have

ay aay jni ¼ °n þ 1Þ3=2 jn þ 1i;

so that

°E n À H0 ÞÀ1 ay aay jni ¼ ½n0! À °n þ 1Þ0!À1 °n þ 1Þ3=2 jn þ 1i:

Thus the expression (3.2.4) is equal to

3

0 n°n þ 1Þ2 2 0

22

À 2

¼ Àn°n þ 1Þ :

0! 8m3 !4

2m!

The energy to second order will be the sum of a handful of terms similar to

this, and is easily enough evaluated. Note that had we been using wavefunc-

tions Èn °xÞ instead of the occupation-number representation jni we would

have had to calculate the energy shift by forming integrals of the form

°

È* °xÞx3 Èn 0 °xÞ dx;

n

which would have required knowledge of the properties of integrals of

Hermite polynomials. Note also that in this particular case the perturbation

series must eventually diverge, because the potential x3 becomes inde¬nitely

large and negative for large negative x. This does not detract from the useful-

ness of the theory for small x.

3.3 Quantum statistics at ¬nite temperatures

In the last section we saw that the excited energy levels of a harmonic oscil-

lator could be regarded as an assembly of noninteracting bosons. It is clear

that for such systems the total number of bosons present is not constant,

83

3.3 Quantum statistics at finite temperatures

since exciting the oscillator to a higher level is equivalent to increasing the

number of bosons present. While we were considering the electron gas we

always had a Hamiltonian that conserved the total number of particles, so

that we were then able to write the equation

!

X

H; nk ¼ 0:

k

This was because any term in H that contained annihilation operators always

contained an equal number of creation operators. In the boson case this was

not so, as it is easily veri¬ed that

½°ay þ aÞ3 ; ay a 6¼ 0;

and so the interactions in the anharmonic oscillator change the total number

of particles. This leads us to the consideration of systems at temperatures

di¬erent from zero, for if the system of noninteracting bosons is at zero

temperature, then there are no bosons present, and we have nothing left to

study. At a ¬nite temperature the system will not be in its ground state, but

will have a wavefunction in which the various k-states are occupied according

to the rules of statistical mechanics. This contrasts with the system of non-

interacting fermions, where at zero temperature

jÈi ¼ j1; 1; . . . 1; 0; 0; . . .i:

If the density of fermions is reasonably large, as in the case of electrons in a

metal, the average energy per particle is large compared with thermal ener-

gies. Thermal excitation is then only of secondary importance in determining

the total energy of the system. In the case of bosons, however, the thermal

energy is of primary interest, and so we now turn brie¬‚y to a consideration of

the form we expect jÈi to take at a ¬nite temperature T.

A fundamental result of statistical mechanics is that the probability of

a system being in a state jii of energy E i is proportional to eÀE i where

¼ 1=kT and k is Boltzmann™s constant. Thus the average value of a

quantity A that has values Ai in the states jii is given by

X

Ai eÀE i

"

A ¼ X ÀE :

i

ei

i

84 Boson systems

This can be expressed as

ÀH

Þ

" Tr°Ae

A¼ °3:3:1Þ

Tr°eÀH Þ

where the operation of taking the trace is de¬ned by

X

ÀH

h jjAeÀH j j i:

Þ¼

Tr°Ae

j

We show this by using the identity operator (2.1.6) to write

X

ÀH

h jjAjiihijeÀH j j i

Þ¼

Tr°Ae

i; j

X

h jjAjiieÀE j ij

¼

i; j

X

Ai eÀE i :

¼

i

Fortunately, a trace is always independent of the choice of basis functions,

and so here we have chosen the most convenient set, j j i, the eigenfunctions

of the Hamiltonian.

When the Hamiltonian refers to a system of interacting bosons whose total

number N is not conserved, the operation of taking the trace must include

summing over all possible values of N, as these are all valid states of the

system. But while it is the case that for true bosons N may not be constant,

there are some systems in which the total number of particles is conserved,

and whose commutation relations are very similar to those for bosons. We

shall see in the theory of superconductivity that an assembly of bound pairs

of electrons has some similarity to a Bose gas. If we de¬ne the operator that

creates an electron in the state k" and one in the state Àk# by

y y y

b k ¼ ck" cÀk#

then we can show that

y

½bk ; b k 0 ¼ kk 0 °1 À nk" À nÀk# Þ:

Note that it is the commutator, and not the anticommutator, that vanishes

when k 6¼ k 0 . When k ¼ k 0 , the commutator is not the same as when the b

are boson operators, and so in the case of superconductivity it is necessary to

85

3.3 Quantum statistics at finite temperatures

use these special commutation relations for electron pairs. An even better

example in which it is suitable to approximate the operators for composite

particles by boson operators is the case of liquid 4 He. The isotope of helium

of atomic mass 4 is composed of an even number of fermions, has no net spin

or magnetic moment, and “ what is most important “ is very tightly bound,

so that the wavefunctions are well localized. This means that operators for

atoms at di¬erent locations will commute. If it is valid to treat helium atoms

as noninteracting bosons, then we should expect that at zero temperature all

the atoms are in the state k ¼ 0, and we should have

jÈi ¼ jN; 0; 0; . . .i:

At ¬nite temperatures we should expect to use Eq. (3.3.1) to predict the

various properties of the system. There is, however, a di¬culty involved in

this in that we must choose a zero of energy for the single-particle states. That

is, the Hamiltonian,

X

H¼ E k nk þ Hinteractions ;

k

could equally well be written as

X

^

H¼ °E k À Þnk þ Hinteractions ¼ H À N;

k

as we have no obvious way of deciding the absolute energy of a single-particle

state. This was not a problem when N was not conserved, for then we knew

exactly the energy E k required to create a phonon or a magnon, and we could

take as being zero. The approach we take, which corresponds to the con-

cept of the grand canonical ensemble in statistical mechanics, is to choose in

such a way that Eq. (3.3.1) predicts the correct result, N1 , for the average

value of the operator N when the trace includes a summation over all

possible N. That is, we choose such that

Tr NeÀ°HÀNÞ

N1 ¼

Tr eÀ°HÀNÞ

1@

ln Tr eÀ°HÀNÞ :

¼

@

The energy is known as the chemical potential.

86 Boson systems

We are now in a position to calculate explicitly various temperature-

dependent properties of a system of independent bosons or fermions. For

"

np , the average number of bosons in the pth single-particle state, for example,

we ¬nd

Tr ay ap eÀ°HÀNÞ

p

"

np ¼

Tr eÀ°HÀNÞ

Tr ap eÀ°HÀNÞ ay

p

¼ :

Tr eÀ°HÀNÞ

We are allowed to permute cyclically the product of which we are taking the

trace because the exponential makes the sum converge. Now

eÀ°HÀNÞ ay ¼ ay eÀ°HÀNÞ eÀ°E p ÀÞ ;

p p

since ay increases N by one, and alters the eigenvalues of H by an amount E p .

p

Thus

Tr ap ay eÀ°HÀNÞ eÀ°E p ÀÞ

p

"

np ¼

Tr eÀ°HÀNÞ

Tr °1 þ ay ap ÞeÀ°HÀNÞ eÀ°E p ÀÞ

p

¼

Tr eÀ°HÀNÞ

¼ eÀ°E p ÀÞ °1 þ np Þ;

"

from which

1

"

np ¼ : °3:3:2Þ

e°E p ÀÞ À 1

For fermions the anticommutator leads to a positive sign, giving

1

"

np°fermionsÞ ¼ : °3:3:3Þ

e°E p þÞ þ 1

"

These functions np give the average value taken by the operator np .

The form of the boson distribution function, (3.3.2), has an interesting

consequence for a system of independent particles in which the total number

N is conserved, as in the case of 4 He. We have

X X 1

"

N¼ np ¼ :

e°E p ÀÞ À 1

p p

87

3.3 Quantum statistics at finite temperatures

"

We know that 0, because if it were not then some np would be negative,

which would be nonsense. Thus

X 1

: °3:3:4Þ

N

eE p À 1

p

Because we know (from Section 2.1) that the density of states in wave-

number space is =83 we can change the sum to an integral and write

°1

4k2 dk

"

n0 þ 3 :

N

exp °02 k2 =2mÞ À 1

8 0þ

"

We consider nk for k ¼ 0 separately, since this term is not de¬ned in (3.3.4).

The integral is well behaved, and gives a number which we shall call N0 °TÞ.

As T tends to zero, N0 °TÞ, which represents an upper bound to the number

of particles in excited states, becomes inde¬nitely small, as illustrated in

Fig. 3.3.1. When N0 °TÞ < N it follows from the inequality that all the rest

of the particles must be in the state for which k ¼ 0. Thus there is a tem-

perature Tc , de¬ned by N0 °Tc Þ ¼ N, below which the zero-energy state is

occupied by a macroscopic number of particles. This phenomenon is

known as the Bose“Einstein condensation, and is remarkable in being a

phase transition that occurs in the absence of interparticle forces.

We might expect the introduction of forces between particles to destroy

the transition to a condensed phase, but this is not the case. Bose“Einstein

Figure 3.3.1. This curve represents the greatest possible number of particles that can

be in excited states. When it falls below N, the actual number of particles present, we

know that a macroscopic number of particles, n0 °TÞ, must be in the state for which

k ¼ 0.

88 Boson systems

condensation is observed in a wide variety of systems, including not only 4 He

but also spin-polarized atomic hydrogen and gases of alkali atoms like 23 Na,

which consist of an even number of fermions. We must now develop the

formalism with which to attack this problem.

3.4 Bogoliubov™s theory of helium

As early as 1946 Bogoliubov developed a theory of a system of interacting

bosons of the number-conserving kind by making use of the fact that n0 may

be very large. He was able in this way to provide an insight into how a weak

interaction may totally change the nature of the excitation spectrum of a

system and also increased our understanding of the phenomenon of super-

¬‚uidity.

We treat liquid 4 He as a system of interacting bosons. The Hamiltonian

will look just like that for the spinless electron gas, except that we shall have

to replace every c and cy by an a or an ay , and, of course, the form of the

interaction will be di¬erent. We have

X 1X

E k ay ak þ Vq ay ay 0 þq ak 0 ak :

H¼ k kÀq k

2 k;k 0 ;q

k

The single-particle energies E k will be just

02 k2

Ek ¼ ;

2M

with M being the mass of the helium atom, and Vq the Fourier transform of a

short-range potential.

We can immediately arrive at an expression for the energy of this system at

zero temperature by employing the same procedure that we used in deriving

the Hartree“Fock approximation for the electron gas. That is, we write

E H ¼ hÈjHjÈi;

where È is the wavefunction in the absence of interactions. Because all N

particles are in the state having k ¼ 0 we simply ¬nd

yy

E H ¼ NE 0 þ 1 V0 hÈja 0 a0 a0 a0 jÈi

2

¼ NE 0 þ 1 N°N À 1ÞV0 :

2

89

3.4 Bogoliubov™s theory of helium

We recall from the de¬nition (2.3.14) that V0 is inversely proportional to the

volume . If we restore this factor by writing V0 ¼ V00 = and approximate

N À 1 by N we ¬nd

N 2 V00

E H ™ NE 0 þ :

2

Note that there are no exchange terms present in this approximation, as only

one state is occupied.

We can now predict from the dependence of the energy on the volume that

this system will support longitudinal sound waves of small wavenumber. For

a classical ¬‚uid the velocity of sound is given by

s¬¬¬¬

R

v¼ ;

where R is the bulk modulus, À°@P=@ÞN , and is the mass density. If we

interpret the pressure P as À°@E H =@ÞN we ¬nd that

s¬¬¬¬¬¬¬¬¬¬¬¬

@2 E H r¬¬¬¬¬¬¬¬¬¬¬¬

0

@2 ¼ NV0 :

v¼

M

Thus for a system of unit volume we expect there to be boson excitations for

small k having energies 0!k such that

NV0 k2

!2 ¼ :

k

M

Bogoliubov™s method shows how these excitations arise as a modi¬cation of

the single-particle excitation spectrum.

In looking for the ground-state solution of this problem we invoke the fact

that in the noninteracting system all the particles are in the state for which

k ¼ 0. We make the assumption that even in the interacting system there is

still a macroscopic number of particles in the zero-momentum state. The num-

y

ber still in the zero-momentum state is the expectation value of a 0 a0 , which

we write as N0 . Because we expect N0 to be large we treat it as a number

yy

rather than an operator, and similarly take a 0 a0 to be equal to N0 . In fact, the

yy

operator a0 a0 operating on a wavefunction with N0 particles in the k ¼ 0

p¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬

state would give °N0 þ 1Þ°N0 þ 2Þ times the state with N0 þ 2 particles, but

90 Boson systems

since 2 ( N0 , we ignore this di¬erence. We next rewrite the Hamiltonian

dropping all terms that are of order less than N0 . Provided Vq is equal to

VÀq this leaves us

X X X y

E k ay ak ay ak

H™ þ 12

þ N0 V0 þ N0

2 N0 V0 Vk 0 a k 0 ak 0

k k

k0

k k

X

Vq °aq aÀq þ ay ay Þ;

þ 1

2 N0 Àq q

q

where the sums exclude the zero term. We then put

X

ay ak ¼ N;

N0 þ N0 Vk ¼ k ; E k þ k ¼ 0k ;

k

k

and make the assumption that N À N0 ( N0 (an assumption that is of