electrons with two ¬‚ux quanta attached to each electron. According to the

statistics that we worked out earlier for the two-quasiparticle wavefunctions,

each ¬‚ux quantum that we attach to each electron will add a phase of to the

exchange phase of the electron. So if we add two ¬‚ux quanta to each electron,

the composite particles consisting of an electron plus two ¬‚ux quanta must be

fermions. These are the composite fermions. For a general fractional quan-

tum Hall state ¼ n=°2np Æ 1Þ, we form the appropriate composite fermions

by attaching 2p ¬‚ux quanta to each electron.

We can now write down a general recipe for constructing composite

fermion states. Start with a fractional quantum Hall state at some ¬lling

factor ¼ n=°2np Æ 1Þ. Then form composite fermions by letting each

electron in this state gobble up 2p ¬‚ux quanta of the total magnetic ¬‚ux

penetrating the system. The e¬ective magnetic ¬eld B* experienced by the

composite fermions is then the ¬eld that is left over, and is given by

B* ¼ B À 2pÈ0 ;

where is the density of the electrons in the fractional quantum Hall state.

Note that B* can actually be negative, in which case the e¬ective ¬eld acting

on the composite fermions would point in the direction opposite to that

acting on the electrons. The original ¬lling factor of the electrons was

È0

¼ ;

B

and the ¬lling factor * of the composite fermions is similarly given by

È0

* ¼ :

jB*j

Combining these two equations we can relate and *:

*

¼ ;

2p* Æ 1

where we choose the minus sign in the denominator if B* is antiparallel to B.

In this way, we can quickly construct the equivalent composite fermion

states of all fractional quantum Hall states. For example, as we already

mentioned, ¼ 1=3 maps to * ¼ 1 when p ¼ 1. The next fraction in this

sequence, ¼ 2=5, maps to * ¼ 2. The particle“hole conjugate ¼ 2=3 of

¼ 1=3 maps to * ¼ À1. Similarly, the Laughlin state at ¼ 1=5 maps to

* ¼ 1 by attaching four ¬‚ux quanta to each electron so that p ¼ 2, and so on.

380 The quantum Hall effect

Approximate wavefunctions for fractional ¬lling factor ground states and

elementary excited states can then be generated by inverting this mapping.

One then starts with a composite fermion ground state or elementary excited

state at an integer ¬lling factor * and maps this onto a fractional ¬lling

Q

factor by applying factors of i<j °zi À zj Þp and by ¬nally projecting the

resulting wavefunctions onto the lowest Landau level.

As we have already stated, it turns out that the wavefunctions constructed

in this way are excellent approximations to exact wavefunctions, provided we

are careful and project them onto the lowest Landau level. Some of the

reasons for this are that the composite fermion wavefunctions are all

uniquely determined, by which we mean that there is no variational freedom

left to tinker with the composite fermion wavefunction once the mapping has

been done. This can then also be extended to composite fermion excited

states, which can be mapped to excitations in fractional quantum Hall states.

So long as the composite fermion mapping provides a unique one-to-one

correspondence between the states, the composite fermion prescription is

remarkably accurate.

Another reason for its success lies in how the motions of the electrons are

correlated in fractional quantum Hall states. According to Laughlin™s wave-

function, the electrons like to attach extra powers in the relative coordinates

of electrons so that the wavefunction vanishes as quickly as possible when

two particles are brought together. This corresponds to attaching vortices, or

¬‚ux quanta, to the electrons. It is a good approximation to assume that these

¬‚ux quanta are rigidly bound to the electrons and that the electrons only

experience a residual average ¬eld. The e¬ective interactions between com-

posite fermions are then given by the di¬erence between the average residual

¬eld and the actual ¬eld experienced by the composite fermions. The actual

¬eld consists of the real applied ¬eld minus the ¬eld of the ¬‚ux quanta at the

locations of the composite fermions. As one can easily imagine, this actual

¬eld is rather complicated and has all kinds of unpleasant singularities stem-

ming from the singular nature of the ¬‚ux quanta attached to the electrons.

Nature is kind to us in that these interactions are rather weak.

Problems

10.1 Consider a square sample with edges at x ¼ Æa; y ¼ Æa. Electrodes are

attached to the left and right edges to make the electrostatic potential

¼ ÀV at x ¼ Àa and ¼ V at x ¼ a. Assume that xx ¼ yy is very

small, but nonzero, and that xy ¼ Àyx is constant. A steady-state

current I ¬‚ows through the sample.

381

Problems

(a) By using the continuity equation and Maxwell™s equation show that

@Ex =@x þ @Ey =@y ¼ 0.

(b) Show that the current ¬‚ows in through one corner and out of the

diagonally opposite one.

(c) Show that the Hall voltage is equal to the source-to-drain voltage.

(This problem is due to Rendell and Girvin.)

10.2 A complete solution to Problem 10.1 in the limit where xx and yy are

vanishingly small is found by solving Laplace™s equation, r2 ¼ 0. The

electrostatic potential is found to be of the form

X

1

¼ bn ½cos°kn xÞ sinh°kn yÞ þ cos°kn yÞ sinh°kn xÞ;

n¼0

where kn ¼ °2n þ 1Þ=2a. Solve for bn and hence ¬nd Ey °x ¼ 0; y ¼ 0Þ

and Ey °x ¼ 0; y ¼ aÞ.

10.3 Consider a noninteracting two-dimensional electron gas in a strong

magnetic ¬eld B ¼ B^ and in the presence of a uniform electric ¬eld

z

^

E ¼ E x. Far from the edges of the system, there is a short-ranged elastic

scatterer. Impose periodic boundary conditions along the y-direction.

Show that far away from the scatterer, the only e¬ect it has on extended

states is to give rise to a phase shift n °kÞ, with the states enumerated

according to their Landau level index and asymptotic wavenumber.

Compare the allowed values of k with and without a scatterer using

the phase shift n °kÞ. What has happened to the number of extended

states if the phase shift decreases by 2M, where M is an integer, as we

go through all the allowed states k for a given n? The total number of

states must be the same with and without the scatterer, so what has

happened to the rest of the states? Show that the phase shift is related to

the transit time for a scattering state to traverse the system from one end

to the other. What happens to this transit time for states that experience

a nonzero phase shift?

10.4 Consider the two lowest Landau levels n ¼ 0 and n ¼ 1 in a non-

^

interacting system with an applied electric ¬eld E ¼ E x. Ignore edge

e¬ects. Now suppose that the electrons can interact with acoustic pho-

nons having a dispersion relation 0!°qÞ ¼ 0vs jqj, with q the wavevector

of a phonon and vs the speed of sound. What conditions must the

electric ¬eld satisfy in order to make phonon emission by the electrons

382 The quantum Hall effect

energetically permissible? The transition rate for phonon emission can

be approximated using Fermi™s Golden Rule. Assume that the electron-

p¬¬¬

phonon interaction potential has the form V°qÞ / q and write down

an expression for the transition rate. Estimate how large the electric

¬eld would have to be for phonon emission to be of consequence.

10.5 Skyrmion solutions were ¬rst discovered as solutions to models of clas-

sical magnetic systems, so-called nonlinear sigma models. In a dimen-

sionless form, the magnetic ¬eld corresponding to these skyrmions can

be written

°4x; Æ4y; r2 À 42 Þ

B°rÞ ¼ :

r2 þ 42

Here r2 ¼ x2 þ y2 and is a parameter having the dimension of a length

that sets the scale of the size of the skyrmion. Consider the exchange

Hamiltonian H0 ¼ ÀB°rÞ Á s°rÞ. Substitute for sx and sy using spin low-

ering and raising operators, and use the complex coordinate z ¼ x þ iy

to rewrite the exchange Hamiltonian in terms of spin operators. Show

that this Hamiltonian must have eigenstates of the form of the skyrmion

states (10.7.1) and (10.7.2). [Hint: Use the fact that z and z* act as

angular momentum raising and lowering operators.]

Chapter 11

The Kondo e¬ect and heavy fermions

11.1 Metals and magnetic impurities

We argued in Section 8.4 that, at temperatures much lower than the Debye

temperature, the resistivity of a metal should be given by an expression of the

form

°TÞ ¼ 0 þ AT 5 :

Here the T 5 behavior comes from electron“phonon scattering, while the

impurities in the metal give rise to the constant term 0 . This assumed that

the scattering caused by the impurities was elastic. If the scattering is inelas-

tic, then a variety of interesting phenomena may occur. In this chapter, we

discuss some of the processes that can occur when a metal is doped with

magnetic impurities, whose spin states introduce extra degrees of freedom

into the scattering problem. Interaction between the spins of the conduction

electrons and those of the impurities then provides a mechanism for inelastic

scattering of the conduction electrons.

The magnetic impurities that one can ¬nd in a metal fall into three classes.

There are the transition metals, such as manganese and iron, the rare earths,

such as cerium, and the actinides, of which the most important is uranium.

The magnetic character of these classes of elements originates in the fact they

have partially ¬lled inner shells. For the transition metals it is the 3d shell, for

the rare earths the 4f shell, and for the actinides the 5f shell that remains only

partially ¬lled, even though the outer valence states (4s, 5s, and 6s, respec-

tively) also contain electrons. This circumstance arises because the centrifugal

force experienced by an electron in a state of higher orbital angular momen-

tum, like a 3d state in a transition metal, for example, causes its wavefunction

to vanish at the nucleus, and so it resides in a region where the Coulomb

potential of the nucleus is partially screened. Although the wavefunction of

383

384 The Kondo effect and heavy fermions

the 4s state extends much further from the nucleus than that of the more

localized 3d state, it su¬ers no centrifugal force. Its wavefunction has a

nonvanishing amplitude at the nucleus, where the potential is unscreened,

and its energy is correspondingly lowered below that of the 3d state. In

addition, the antisymmetry of the several-electron wavefunction lowers the

mutual Coulomb interaction of electrons with parallel spin, resulting in one

of Hund™s rules, which stipulates that, other things being equal, electron spins

in atoms prefer parallel alignment.

In Section 4.4 we studied band structure by using the method of tight

binding. There we saw that when we assembled a periodic array of atoms

to form a crystal, the width of the resulting energy bands was proportional to

an integral involving the overlap of atomic wavefunctions on adjacent sites.

The electrons in the partially ¬lled d shells of the transition metals have

strongly localized orbitals, and this results in rather ¬‚at energy bands. The

4s conduction band, on the other hand, arises from the considerable overlap

of wavefunctions on neighboring atoms, and consequently spans an energy

range that includes the energies of 3d states. In the electronic density of states

one then sees a tall narrow d band superimposed on the low ¬‚at s band, as in

Fig. 11.1.1. The actual electron Bloch states are mixtures of s- and d-like

components, particularly in the region of overlap.

In this chapter, we begin by exploring the properties of a metal containing

a dilute concentration of magnetic impurities, and then look at the conse-

quences of increasing the impurity concentration. In addition to the inelastic

scattering that the impurity sites provide for the conduction electrons, there

are two di¬erent and competing phenomena that appear as the impurity

concentration is increased. One is the conduction-electron-mediated inter-

action between di¬erent magnetic impurities. This is the so-called RKKY (or

Ruderman“Kittel“Kasuya“Yosida) interaction, and occurs because a mag-

netic moment on one impurity site polarizes the conduction electrons, which

propagate the polarization to another impurity site, in a manner somewhat

Figure 11.1.1. In the density of states of a transition metal the narrow peak com-

posed of localized d states overlaps the much broader peak of the s states.

385

11.2 The resistance minimum and the Kondo effect

analogous to the phonon-mediated electron“electron interaction responsible

for BCS superconductivity. The important facts for us are that the interaction

strength falls o¬ as the cube of the separation distance between impurities,

and that the interaction tends to order systems magnetically. One complica-

tion (which fortunately is not important to us here) is that the sign of the

interaction, which determines whether it tends to order moments ferro-

magnetically or antiferromagnetically, oscillates with distance. The second

relevant phenomenon is a result of the increasing overlap of the wavefunc-

tions of electrons bound to adjacent impurity sites as the concentration is

increased. This overlap leads to the formation of something akin to Bloch

states, which then occupy impurity bands.

The low-concentration limit of a metal in which the conduction electrons

interact with impurity atoms, each of which has two localized electron spin

states, up and down, is known as the Kondo limit. Here, the impurities are

far enough apart that the RKKY interaction can be neglected, as can the

overlap between di¬erent impurity states. The impurities can then be treated

as independent, and we can study these systems by considering a single

magnetic impurity in a nonmagnetic metal. In the opposite limit of high

concentration, we really have a crystal consisting of both species of atoms

(host plus impurity) with conduction bands formed from electrons of both

species, and strong RKKY interactions. As we may expect, such systems can

display very complicated magnetic and transport behavior, and are described

in terms of the theory of heavy fermions. It turns out that some of the

important ingredients in the physics of heavy fermions stem from the prop-

erties of a single impurity in a sea of conduction electrons, and so we start our

examination in the dilute, or Kondo, limit.

11.2 The resistance minimum and the Kondo effect

While the resistivity of many pure metals does appear to vary as T 5 at low

temperatures, adding a small amount of magnetic impurities can yield a very

di¬erent behavior. When small amounts of iron, chromium, manganese,

molybdenum, rhenium, or osmium are added to copper, silver, gold, magne-

sium, or zinc, for example, the resistivity generally exhibits a minimum. The

temperature Tmin at which this occurs is usually quite low, and does not

appear to be related to the Fermi temperature TF ¼ E F =k or to the Debye

temperature ‚, and seems to vary with impurity concentration c roughly as

c1=5 , while the depth of the minimum, °0Þ À °Tmin Þ, is proportional to c.

Since °0Þ itself is proportional to c, the relative depth of the minimum is

roughly independent of c, and usually of the order of one tenth of °0Þ.

386 The Kondo effect and heavy fermions

The explanation for this e¬ect was provided by Kondo in 1964. He realized

that when magnetic impurities are present the conduction electrons may

su¬er a change of spin as they scatter, and that higher orders of perturbation

theory than the ¬rst have to be treated very carefully, since the commutation

relations of not only the annihilation and creation operators of the conduc-

tion electrons but also of the spin raising and lowering operators have to be

taken into account. In other words, the Exclusion Principle must be used in

calculating any scattering process that passes through an intermediate state

when the internal degrees of freedom of the scatterer are involved. To show

this, we follow Kondo™s calculation, which starts with the part of the per-

turbing potential containing the magnetic interaction, s Á S, where s is the spin

of the conduction electron, and S the spin of the localized electron in the d or

f shell of the impurity. While the localized spin can have a total spin di¬erent

from 1=2, we here for simplicity consider localized spin-1/2 states. We impli-

citly use a high-temperature theory in that we assume that the spins of the

localized and conduction electrons are uncorrelated and independently have

equal probability of initially being either up or down. While this will directly

reveal the onset of the resistance minimum as the temperature is lowered, it

will fail as the temperature is decreased further, since at the lowest tempera-

tures conduction and localized spins form bound singlet pairs.

Following the notation of Section 3.10 we de¬ne spin raising and lowering

operators sþ , sÀ , Sþ , and SÀ for the Bloch and localized electrons, respec-

tively, and ¬nd

À Á

s Á S ¼ 1 sþ S À þ sÀ S þ þ sz Sz :

2

In the notation of second quantization for the Bloch states we can then write

the perturbation as

X y

°Vkk 0 =02 Þ ck 0 0 ck h 0 j ½1 °sþ SÀ þ sÀ S þ Þ þ sz Sz ji:

H1 ¼ 2

k;k 0 ; ; 0

In this expression and 0 refer to the spin states of the Bloch electrons and

Vkk 0 is the matrix element between Bloch conduction-band states of the

spatial part V°rÞ of the perturbing spin-dependent interaction due to an

impurity at the origin,

°

Vkk 0 ¼ *°rÞV°rÞ dr:

k0

k

387

11.2 The resistance minimum and the Kondo effect

Because V°rÞ extends over only about one unit cell, while the wavefunction

extends over the whole crystal, this quantity is of order N À1 , with N the

number of unit cells in the crystal. The only nonvanishing matrix element

of sþ is h" jsþ j #i ¼ 0, and for sÀ only h# jsÀ j "i ¼ 0 survives, while sz has

elements h" jsz j "i ¼ 0=2 and h# jsz j #i ¼ À0=2. Thus

1X y y y y

Vkk 0 ½ck 0 " ck# SÀ þ ck 0 # ck" Sþ þ °c k 0 " ck" À c k 0 # ck# ÞSz :

H1 ¼ °11:2:1Þ

20 k;k 0

Diagrams illustrating the type of scattering caused by each of these terms

would then be of the form shown in Fig. 11.2.1. The scattering probability is

proportional to the square of the modulus of the elements of the T-matrix for

this perturbation, as indicated in Section 4.7. In the Born approximation the

T-matrix is replaced by H1 itself and then the scattering probability is found

from terms with precisely two annihilation operators and two creation opera-

tors, and turns out to be composed of terms of the form

Q°k; k 0 Þnk# °1 À nk 0 " Þ

corresponding to the process of Fig. 11.2.1(a) and other terms corresponding

to the other processes. The scattering is still elastic, since we are not assuming

the energy of the impurity to depend on its spin direction after the Bloch

electron has scattered and departed. The various occupation numbers nk can

then be averaged to give the fk that enter the Boltzmann equation as in

Section 8.2, and one ¬nds the resistivity still to be independent of the tem-

perature.

The interesting e¬ects occur when we consider the second-order terms in

the T-matrix. Let us for simplicity look at those processes in which the net

result is that an electron in the state k " is scattered into the state k 0 ". While

Figure 11.2.1. In the Born approximation a magnetic impurity can scatter an elec-

tron in these four di¬erent ways.

388 The Kondo effect and heavy fermions

in ¬rst order only the process shown in Fig. 11.2.1(c) contributes to this, there

are other possibilities in second order. We recall that

1 1

T ¼ H1 þ H1 T % H1 þ H 1 H

E À H0 1

E À H0

to second order. Of the sixteen types of second-order term that we ¬nd when

we substitute expression (11.2.1) into this, we examine only those two invol-

ving products of S À with Sþ . These are sums of the form

2 X

1 1

y y

Vk1 k2 Vk3 k4 ck1 " ck2 # SÀ ck3 # ck4 " Sþ °11:2:2Þ

E À H0

20 k1 k2 k3 k4

and

2 X

1 1

y y

Vk1 k2 Vk3 k4 ck3 # ck4 " S þ c k1 " ck2 # S À : °11:2:3Þ

E À H0

20 k1 k2 k3 k4

For these to have the net e¬ect only of scattering k " into k 0 " we must

always have k2 ¼ k3 , k4 ¼ k, and k1 ¼ k 0 . In terms of diagrams we can

picture these processes as in Fig. 11.2.2. The diagram (a) represents expres-

sion (11.2.2), in which an electron is ¬rst scattered from k " to the virtual

state k2 #, and then ¬nally to the state k 0 ". In diagram (b), however, the ¬rst

thing that happens is the creation of an electron“hole pair. That is, an elec-

tron already in state k2 # is scattered into k 0 ". The incoming electron in state

k " then drops down into this vacancy in a process that we can depict as the

annihilation of an electron“hole pair. The energy of the intermediate state

di¬ers from that of the initial state by E k 0 À E k2 , which is just the negative of

the energy di¬erence, E k2 À E k , of the process of Fig. 11.2.2(a). We can thus

Figure 11.2.2. These two second-order processes both contribute to the scattering

amplitude for a conduction electron. Because of the change of spin, the occupancy

nk2 of the intermediate state does not cancel from the total scattering amplitude.

389

11.2 The resistance minimum and the Kondo effect

add the contributions of expressions (11.2.2) and (11.2.3) and use the anti-

commutation relations of the c™s and cy ™s to ¬nd

2 X

1 1

y

½°1 À nk2 # ÞS À Sþ þ nk2 # S þ SÀ :

Vk 0 k2 Vk2 k c k 0 " ck"

E k À E k2

20 k2

°11:2:4Þ

If S þ and S À were not operators but simply numbers they would commute,

and the terms in nk2 would cancel. We would then be back in the situation of

having the type of scattering that leads to a temperature-independent resis-

tivity. However, as we may verify from the de¬nitions in Eqs. (3.10.12),

S À Sþ À Sþ S À ¼ À20Sz :

There is thus a contribution to the scattering matrix that is proportional to

X hnk2 # i

y

: °11:2:5Þ

ck 0 " ck" Sz Vk 0 k2 Vk2 k

E k À E k2

k2

The presence of the term in nk2 has the consequence that the scattering

probability becomes strongly temperature-dependent. When we form jTkk 0 j2

we shall ¬nd contributions of the form

PK °k"; k 0 "Þ / hnk" °1 À nk 0 " Þig°k 0 ; kÞ

where g°k 0 ; kÞ is the sum over k2 in expression (11.2.5), and the subscript K

refers to the contribution responsible for the Kondo e¬ect. Because the

thermal average of the expectation value of nk2 # is given by the Fermi“Dirac

function, we see that g°k 0 ; kÞ depends on the temperature.

The total probability P°k"; k 0 "Þ of a scattering event occurring in which

the net e¬ect is that an electron is transferred from k" to k 0 " can be written

as

P°k"; k 0 "Þ ¼ fk" °1 À fk 0 " ÞQ°k"; k 0 "Þ:

Here Q°k"; k 0 "Þ is composed of two parts. One is independent of the

temperature and is due to ¬rst-order processes plus those contributions

from higher-order processes that do not involve the occupation numbers of

the intermediate states. The second part contains contributions from the

occupation numbers of the intermediate states, and in second order has

the temperature dependence of g°k; k 0 Þ. The qualitative nature of this

390 The Kondo effect and heavy fermions

temperature dependence may be seen by making a few approximations. We

¬rst assume that the matrix elements Vk 0 k2 and Vk2 k vary slowly over the

range of energies over which we have to integrate k2 , so that we can replace

each of them by a constant V0 . We then change from a sum over k2 to an

integration over energies E 2 by introducing the density of states D°EÞ, and

then approximate this by its value when E k2 is equal to the chemical potential

. We specialize to the value of g°k; k 0 Þ when E k is also equal to to ¬nd

°W ^^

f °E Þd E

g°k; k 0 Þ % V0 D°Þ

2

;

^

E

ÀW

where W is an energy characteristic of the width of the band. After inserting

^

the form of the Fermi“Dirac function f and de¬ning x E =2kT we have

° W=2kT

tanh x

0

g°k; k Þ % ÀV0 D°Þ

2

dx:

x

0

The integral is one we have met before in deriving Eq. (7.5.10) for the critical

temperature of a BCS superconductor, and so we ¬nd the result

g°k; k 0 Þ % ÀV0 D°Þ ln °1:14 W=kTÞ:

2

The low-temperature resistivity then takes the form

5

kT

T

°TÞ ¼ 0 þ 1 À 2 ln : °11:2:6Þ

‚ W

The resistance has its minimum when the derivative d=dT vanishes. Thus

4

Tmin

À 2 ¼0

51

‚5 Tmin

and

2 ‚5

5

¼ :

Tmin

51

For low impurity concentrations, the scattering events on di¬erent impurities

are independent and their contributions add incoherently. The sum over

scattering events on all impurities is then proportional to the concentration

c of impurities, so we have thus shown that

Tmin / c1=5 :

391

11.3 Low-temperature limit of the Kondo problem

The fact that Eq. (11.2.6) erroneously predicts an in¬nite resistance as T

approaches zero is a consequence of the inadequacy of considering only

second-order terms in the T-matrix, and of our implicit high-temperature

assumption that the energy of the local spin is independent of its orientation.

At lower temperatures the local spins form bound collective states with the

conduction-band electrons. In order to ¬‚ip the local spin, this binding must

be broken by the thermal energy of excited conduction electrons. As the

temperature is reduced this becomes more and more unlikely, and the local

spin becomes ˜˜frozen out.™™ In fact, the singularity we discovered signals this

formation of a bound state.

11.3 Low-temperature limit of the Kondo problem

It is clear that perturbation theory, especially when limited to second-order

calculations, is not going to be useful if we want to learn the nature of the

low-temperature behavior of a system in the Kondo limit. Instead, we will try

to use our intuition to guess a reasonable variational wavefunction that

describes a collective bound state, and see whether we can make this state

have a lower energy than a state with an independent localized spin and a

Fermi sea. We start with an impurity atom embedded in a host metal. In

the low-concentration limit, we can ignore interactions between impurities,

and so for simplicity consider a single local spin- 1 state coupled antiferro-

2

magnetically to the sea of conduction electrons. We would like to construct

some kind of trial wavefunction that describes the formation of a bound state

of conduction electrons and the local moment. One natural possibility for the

case of antiferromagnetic coupling would be to have excitations above the

Fermi surface combine with the local moment to form a spin singlet, which

would be an antisymmetric combination of spin-up and spin-down states of

the local spin and states above the Fermi surface. Another possibility would

be to form spin singlets of states below the Fermi surface “ holes “ and local

spins. Simple variational wavefunctions that describe these possibilities were

¬rst written down by Yosida. We will here follow Mahan™s treatment of the

problem.

The trial wavefunctions we consider are of two types. The ¬rst is

X yy yy

jÉa i ¼ ak ½ c k# À ck" jFi:

jkj>kF

Here, y and y create up- and down-spin states of the local spin, jFi is the

¬lled Fermi sea, and the ak are coe¬cients that we will have to determine. We

392 The Kondo effect and heavy fermions

see that this state is a spin singlet: it is an antisymmetric combination of a

local up-spin plus a delocalized down-spin above the Fermi surface with

a local down-spin and a delocalized up-spin that is also above the Fermi

surface. The second type of state is formed with annihilation operators for

conduction electrons below the Fermi surface, and is written as

X

bk ½y ck" þ y ck# jFi:

jÉb i ¼

jkj<kF

This state combines local spins with holes. It too is an antisymmetric combi-

nation of antiparallel local spin and electron spin, and thus a spin singlet, but

now the electrons are below the Fermi surface. The two states jÉa i and jÉb i

are orthogonal to one another, since they contain di¬erent numbers of elec-

trons.

Next we have to evaluate the expectation value of the Kondo Hamiltonian

in the states jÉa i and jÉb i, and minimize these expressions with respect to

the coe¬cients ak and bk . Note that the states are not normalized, so we

have to divide the expectation values by the respective norms, which also

depend on ak and bk , and minimize these entire expressions. This is a rather

lengthy operation, and so we instead take a short cut by making another

approximation. The states jÉa i and jÉb i are the simplest spin-singlet states

we can think of that consist of electrons above the Fermi sea, or holes

below it, combined with local spins. However, when the interaction term in

the Kondo Hamiltonian acts on either of these states, other, more compli-

cated terms are created. Let us look at this in some detail. The interaction

term is

JX y y y y

f°c k" cp" À ck# cp# ÞSz þ c k" cp# S À þ c k# cp" S þ g:

H1 ¼ À

0 k;p

Here we have taken the coupling J to be independent of momentum transfer

and to be negative, which means that we are considering antiferromagnetic

coupling. We then let this interaction act on our state jÉa i.

JX y

f½cp" cq" À cy cq# Sz þ cy cq# S À þ cy cq" S þ g

H1 jÉa i ¼ À

0 p;q p# p" p#

X yy yy

‚ ak ½ c k# À ck" jFi:

jkj>kF

393

11.3 Low-temperature limit of the Kondo problem

The simplest terms in this expression are the ones for which q ¼ k, and are

JX y y yy yy

À ak f½c p" ck" À cp# ck# Sz ½ ck# À c k" jFi

0 p;k

y yy yy y yy yy

þ cp" ck# S À ½ c k# À ck" jFi þ c p# ck" S þ ½ ck# À ck" jFig:

The operators Sz , SÀ and S þ all act only on the local spin, and Sz y ¼ 0y=2,

Sz y ¼ À0y=2, S À y ¼ 0y , S À y ¼ 0, S þ y ¼ 0, and S þ y ¼ 0y , so we

obtain

& '

X 1y y yy yy y yy y yy

ÀJ ½c c À c p# ck# ½ ck# þ ck" þ c p" ck# c k# À c p# ck" ck" jFi:

ak

2 p" k"

p;k

y

Since jkj > kF , the operator combination ck" ck" gives unity when it acts on

the ¬lled Fermi sphere, and the presence of the operator cy requires jpj > kF .

p

Applying the same argument to the other terms, and collecting up the surviv-

ing components, we see that we are left with something similar to what we

started with, namely singlet states made up of conduction electrons and local-

moment electrons. This is good news, as we would like Éa to be an eigenstate

of H1 , and the terms we have just examined satisfy that wish. The bad news is

that there are also many other terms bearing less resemblance to jÉa i. For

example, terms with p; q 6¼ k will lead to expressions like

y yy

c p" cq# c k# jFi:

This is a term which, in addition to the electron at momentum 0k above the

Fermi surface, has a particle“hole pair consisting of an electron with momen-

tum 0p above the Fermi surface and a hole of momentum 0q below the Fermi

surface. This is inconvenient, since it will lead to some cumbersome algebra

when we vary ak to ¬nd a minimum in energy. Our Ansatz state started out

just with electrons above the Fermi sea, but the action of the Hamiltonian on

this state generates various kinds of other electrons and holes. Another way

to say this is that we would like to keep our state in a restricted part of the

Hilbert space consisting only of electron-plus-local-moment singlets, but act-

ing with the Hamiltonian on this state takes us to a bigger part of the Hilbert

space. We avoid this di¬culty with a simple remedy: we insist on staying in

our restricted Hilbert space by throwing out all the parts of HjÉa i that do not

yy yy

consist of terms of the form ° ck# À c k" ÞjFi. Technically, we can do this

394 The Kondo effect and heavy fermions

by using the projection operator

X yy X

yy

½ c k# À c k" jFihFj ½ck# À ck" :

jkj>kF jkj>kF

The eigenvalue equation we wish to solve is

°H0 þ H1 À E 0 À E a ÞjÉa i ¼ 0:

Here E 0 is the ground-state energy of the unperturbed Hamiltonian,

P y

H0 ¼ k; E k c k; ck; , and E a is the shift in energy due to the perturbation

H1 . We wish to determine ak such that this shift is as negative as possible. Let

us ¬rst look at the simplest terms:

X yy yy

°H0 À E 0 À E a ÞjÉa i ¼ °H0 À E 0 À E a Þ ak ½ ck# À c k" jFi

jkj>kF

X yy yy

¼ ak ½E k À E a ½ c k# À ck" jFi:

jkj>kF

This term does not contain any parts outside our restricted Hilbert space and

so we need not worry about projections, and the contribution to the equation

for ak is

X

ak ½E k À E a :

jkj>kF

We have to do a little more work for the terms generated by H1 jÉa i. First, we

look at the piece of H1 that involves Sz . This is

X X

y y yy yy

Sz ½c p" cq" À cp# cq# ak ½ c k# À c k" jFi

jkj>kF

p;q

°11:3:1Þ

0X y X

y yy yy

¼ ½c p" cq" À cp# cq# ak ½ c k# þ ck" jFi;

2 p;q jkj>k F

y y y y

where we have used Sz ¼ 0 =2, and Sz ¼ À0 =2. Let us look at the

term in y in Eq. (11.3.1). Terms that survive the projection and give nonzero

contribution must contain only one electron excitation above the ¬lled Fermi

sea jFi. This means that the product of three electron operators in (11.3.1)

must reduce to one by having the other two form a simple number operator.

This can happen if p ¼ q or if q ¼ k. For the former case, we have

Xy X

y yy

½cp" cp" À c p# cp# ak ck# jFi;

jkj>kF

p

395

11.3 Low-temperature limit of the Kondo problem

Py y

and p ½c p" cp" À cp# cp# just counts the di¬erence between the number of up-

y

and down-spins after we have applied c k# to the ¬lled Fermi sea, and is thus

y

equal to À1. We then obtain for the terms in in Eq. (11.3.1)

0X 0X

yy yy

À ak ck# jFi À ak c p# jFi: °11:3:2Þ

2 jkj>k 2 p;jkj>k

F F

The terms in y in Eq. (11.3.1) are

X X

0y

y y yy y yy

ÀSz ½cp" cq" À c p# cq# ak ck" jFi ¼ ½c p" cq" À cp# cq# ak c k" jFi:

2

jkj>kF jkj>k F

Again we must demand that either p ¼ q or q ¼ k, and we are left with only

0X 0X

yy yy

ak cp" jFi þ ak c p" jFi: °11:3:3Þ

2 jkj>k 2 p;jkj>k

F F

Continuing with the terms generated by the operators S À and S þ , we ¬nd

that

X X X X

y yy y yy

cp" cq# SÀ c p# cq" Sþ

ak c k# jFi À ak c k" jFi

jkj>kF jkj>kF

p;q p;q

reduces to

X X

y y y y

0 ak c p" jFi À 0 ak cp# jFi: °11:3:4Þ

p;jkj>kF p;jkj>kF

We now combine expressions (11.3.2), (11.3.3), and (11.3.4). In (11.3.2) and

(11.3.3) there are single summations over k, while all the other terms

are double summations over p and k. We drop the single summations,

since they will be smaller by a factor of order N than the double ones. The

result is

3J X X yy yy

½ cp# À cp" jFi:

ak

2 jkj>k jpj>k

F F

The eigenvalue equation for E a then becomes

3J X

À Á

ak E k À E a þ a ¼ 0:

2 jpj>k p

F

396 The Kondo effect and heavy fermions

We avoid having to solve for the eigenvectors ak by the trick of dividing by

E k À E a and summing over jkj > kF . The sum of the ak then cancels to leave

us with

3J X 1

1¼À :

2 jkj>k E k À E a

F

^

We change from a sum over k to an integration in which the energy E

measured relative to the Fermi energy runs from zero to a value W related

to the bandwidth, and approximate the density of states by its value D at the

Fermi surface. We then have

°W ^ 3JD E a

dE

3JD

ln

1¼À ¼ W À E :

^ À E a

E

2 2

0 a

Remembering that J is negative, we then ¬nd the solution with E a < 0 to be

W

E a ¼ À : °11:3:5Þ

e2=°3jJjDÞ À 1

This expression reminds us of the condensation energy of a BCS supercon-

ductor given in Eq. (7.4.2), which was proportional to eÀ2=VD , with V the

attractive electron“electron interaction. When jJjD is small, E a is similarly

proportional to eÀ2=°3jJjDÞ , showing that the coupling constant jJj in this case

enters nonperturbatively into the problem. The function eÀ2=°3jJjDÞ cannot be

expanded in a power series in jJj, and is said to have an essential singularity

at jJj ¼ 0. For bands more than half-¬lled, the state jÉa i, which consists of

bound spin-singlet pairs of conduction electrons and local spins, has a lower

energy than E 0 , and is a better candidate for the ground state. The state jÉb i

may also be examined by means of a similar approach, the main di¬erence in

the analysis being that jkj < kF . One ¬nds that the state jÉb i should be the

ground state for bands that are less than half full.

The de¬nition of the Kondo temperature TK is

1

kTK W exp À :

2DjJj

This energy is similar to the small-jJj limit of our expression (11.3.5) for the

energy reduction ÀE a , except for the factor of 1 instead of 2 in the exponent.

2 3

This di¬erence comes from the fact that those who originally de¬ned it did so

in terms of a triplet state, which is obtained if the coupling J is ferromagnetic

397

11.4 Heavy fermions

(J > 0), rather than the singlet state that we have considered. We again notice

an analogy with the BCS theory of superconductivity, in which we ¬nd a

similar expression for the critical temperature Tc . This is not an accident.

Both TK and Tc de¬ne temperatures below which perturbation theory fails.

In the BCS case, Tc signals the onset of the formation of bound Cooper

pairs and a new ground state with an energy gap. The Kondo e¬ect is a little

more subtle. Here TK de¬nes a temperature at which the energy contributions

from second-order perturbation theory become important. This happens

when the local spin on a single impurity starts to become frozen out at an

energy set by the Kondo coupling J and the density of states at the Fermi

energy.

In summary, we have seen that the internal dynamics of the local spins

interacting with the sea of conduction electrons become important at low

temperatures. The net e¬ect of this interaction is very much like a resonant

state appearing at temperatures $ TK . In fact, Wilkins has noted that the

Kondo e¬ect is very well described by a density-of-states expression that adds

a resonant state at the Fermi energy for each impurity with a local moment.

He writes this expression as

c

DK °EÞ ¼ D°EÞ þ ¼ D°EÞ þ D°EÞ; °11:3:6Þ

°E À E F Þ2 þ

2

where

¼ 1:6 kTK . This expression adds a Lorentzian peak of weight unity

at the Fermi energy for each impurity atom with a local moment. From this

expression one can calculate, for example, the change in speci¬c heat and the

change in electrical resistance. All the many-body physics has then resulted in

a simple change in the density of states at the Fermi surface consisting of the

addition of a resonant state for each impurity. At high temperatures, the

sharp resonances are unimportant in the presence of thermal smearing at

the Fermi surface. As the temperature is decreased, the sharp resonances

become more important in the scattering of the conduction electrons. At

still lower temperatures, there is insu¬cient energy to ¬‚ip the local spins,

which become frozen in ¬xed orientations.

11.4 Heavy fermions

We treated the Kondo problem by ¬rst considering the e¬ects of a single

impurity, and then simply multiplying the expected e¬ect by the number of

impurities present. We were thus assuming that the magnetic impurities were

su¬ciently dilutely dispersed in the metallic host material that we did not

398 The Kondo effect and heavy fermions

have to consider interactions between them. The starting point for the theory

of heavy fermions, in contrast, is a regular lattice, typically consisting of a

basis of a rare earth or actinide and a metal. Examples are UPt3 and CeAl3 .

In this type of compound, it is possible for the electrons to form Bloch states,

and display metallic behavior in the sense that the resistance diminishes to

quite a small value as the temperature approaches absolute zero. This is quite

distinct from the e¬ects of the dilute magnetic impurities in the Kondo

problem, which lead to a minimum in the resistance. On the other hand,

the magnetic elements experience RKKY interactions, which can lead to

the formation of nontrivial magnetic ground states. In addition, some of

the heavy-fermion materials, UPt3 being an example, have superconducting

ground states that are not the usual BCS type of superconductor. In view of

this rich diversity of interesting properties, it is perhaps useful to start by

pointing out what heavy-fermion materials do have in common, and why the

fermions are said to be ˜˜heavy.™™ Let us ¬rst consider the electrical resistivity.

In typical transition metals, this has a temperature dependence given by

°TÞ ¼ °0Þ þ AT 2 at low temperatures, where A is proportional to the e¬ec-

tive mass of the electrons. In metallic heavy-fermion systems at su¬ciently

low temperatures, the resistivity still has this simple behavior, but the con-

stant A can be as much as seven orders of magnitude larger than for transi-

tion metals! Similarly, the speci¬c heat of normal metals at low temperatures

is of the form C°TÞ ¼

T þ BT 3 , where

T is the electronic contribution,

and the term in T 3 is due to phonons. For heavy fermions, the electronic

contribution

T is perhaps two to three orders of magnitude greater than in

normal metals, and is so large that the phonon contribution can often be

ignored. Finally, there is the magnetic susceptibility , which for heavy fer-

mions at low temperatures is enhanced by several orders of magnitude over

that of conventional metals. All these quantities, A,

, and , are propor-

tional to the e¬ective mass m* of the electrons for conventional metals. Their

enhanced values lead us to the interpretation that we are indeed dealing with

˜˜heavy fermions.™™ That the concept of a large e¬ective mass is useful for

these compounds is demonstrated by the Wilson ratio, R. This is the ratio of

the zero-temperature limit of the magnetic susceptibility (in units of the

moment per atom, g2 J°J þ 1Þ2 ) to the zero-temperature derivative of the

J B

speci¬c heat (in units of k ), and is

22

°0Þ=g2 J°J þ 1Þ2

J B

R¼ :

°0Þ= k

22

The fact that the Wilson ratio is approximately unity for all the nonmagnetic

399

11.4 Heavy fermions

heavy fermions suggests that, whatever the mechanism responsible for the

anomalous behavior, there is a consistent pattern that can be interpreted in

terms of a large e¬ective mass. We take this argument back one further step

by recalling that the electronic density of states is itself proportional to the

e¬ective mass in the case of a single band, and decide that our task should be

to examine the likely magnitude of D°E F Þ in these systems.

Our starting point will be the Anderson Hamiltonian, which is constructed

from the following ingredients. First, there is a sea of conduction electrons

formed from the conduction-band states of the host material (Al, Pt, or Zn,

for example) and from the s and p electrons of a dopant such as U or Ce.

These delocalized conduction electrons will in general have some dispersion

relation E k;n for Bloch states labeled by a wavevector k in the ¬rst Brillouin

zone and a band index n. This is a distracting complication, and so we assume

that the bands are of free-electron form. Secondly, there are the localized d or

f states of the dopant atoms. To be speci¬c we assume that we are dealing

with f electrons, and name their dispersion relation E f °k; nÞ. However, since d

and f electrons are very closely bound to the core of the dopant atoms, these

states do not overlap signi¬cantly. The band formed by them is consequently

very ¬‚at, and their energies can be taken to be a constant, E f (not to be

confused with the Fermi energy E F ). The next ingredient is one that leads

to strong correlations. It is the on-site repulsion term. This is due to the

localized nature of the d and f orbitals. If there are several d or f states ¬lled

on the same atom (which is possible because of their relatively high spin

degeneracy), the Coulomb interaction between them contributes strongly to

the energy. This interaction can then be written

X

ni; ni; 0 ;

U

i;; 0

where the summation runs over all dopant sites i, and is an enumeration of

the degenerate multiplet of local states. The on-site repulsion is given by U.

This kind of localized Coulomb repulsion is sometimes called a Hubbard

term, because it is a central piece in another model of strongly correlated

electrons, the Hubbard Hamiltonian. Lastly, there is an interaction between

the delocalized conduction electrons and the local states. The idea here is that

a conduction electron with a certain spin can hop onto a local site, or vice

versa. This is in contrast to the Kondo Hamiltonian, where local states and

conduction electrons could exchange spin, but were not allowed to transform

into one another. We take this local interaction to be a very short-ranged

potential centered at the sites i of the dopants.

400 The Kondo effect and heavy fermions

Putting it all together, we then have the periodic Anderson model,

X X X

y

f y fi; þ U f y fi; f y 0 fi; 0

HA ¼ E k c k; ck; þ E f i; i; i;

i; 0 ;

i;

k;

X y

V°c k; fi; þ f y ck; Þ:

þ °11:4:1Þ

i;

k;i;

Here fi; annihilates an f electron of quantum number at site i. In the last

term, which is the hybridization term, we have e¬ectively taken the inter-

action V to be a delta-function in real space, making the Fourier transform

independent of k. This is a reasonable approximation provided the range of

the interaction is much shorter than the Fermi wavelength. We have also

taken the confusing step of writing the hybridization term as coupling the

local states with conduction electrons labeled by a quantum number rather

than by a simple spin . The reason for this is that usually denotes an

enumeration of the symmetries of the degenerate local states, which

depend on angular momentum, spin, and spin“orbit coupling. The conduc-

tion electron states have to be decomposed into the same symmetries by

combining the Bloch states into angular momentum states. This is a tedious

element of general practical calculations, and we here acknowledge that it

may be necessary by using the notation instead of . Fortunately, for local

states having only spin-up and spin-down degeneracies, no decomposition is

required, and the coupling to the conduction electrons is just through the spin

channels .

Equation (11.4.1) is a very rich Hamiltonian, which is capable of describing

a variety of physical situations, depending on the values of U and V, and of

the degeneracy of the local states. In particular, this Hamiltonian contains

the Kondo Hamiltonian. By this we mean that by making a clever transfor-

mation, one can show that there are Kondo-like interactions included in the

periodic Anderson model. This procedure is called the Schrie¬er“Wol¬ trans-

formation. One starts with the single-impurity Anderson Hamiltonian, in

which there is just one impurity site with its degenerate d or f orbitals.

One then seeks a canonical transformation that eliminates the hybridization

terms between the conduction electrons and the impurity states. This is very

similar in spirit to the canonical transformation we applied when studying the

Frohlich and Nakajima Hamiltonians in Sections 6.5 and 6.6. There we

¨

eliminated the term linear in electron“phonon coupling, while in the present

case we eliminate the terms linear in the coupling between the conduction

electrons and the impurity states.

401

11.4 Heavy fermions

We start by considering the single-impurity Hamiltonian

H ¼ H0 þ H 1 °11:4:2Þ

where

X X

y

f y f þ Un" n#

H0 ¼ E k ck; ck; þ Ef

k;

and

X y

Vk °ck; f þ f y ck; Þ:

H1 ¼

k;

For simplicity we are considering only spin-up spin-down degeneracy, but are

allowing the hybridization term Vk to depend on k. Were it not for the

Hubbard term, Un" n# , the whole Hamiltonian H could easily be diagona-

lized, since it would then be just a quadratic form in annihilation and creation

operators. We might thus be tempted to perform the diagonalization and

then treat the Hubbard term as a perturbation. However, we have seen in

Chapter 2 that the strength of the Coulomb interaction makes this a di¬cult

task. Instead, we follow the procedure of Section 6.5, and look for a unitary

transformation to eliminate the interaction terms Vk to ¬rst order. Just as in

Eq. (6.5.3), we seek a unitary operator s that will transform H into a new

Hamiltonian H 0 ¼ eÀs Hes from which ¬rst-order terms in H1 have been

eliminated. For this to happen, we again require

H1 þ ½H0 ; s ¼ 0;

which then to second order in Vk leaves us with

H 0 ¼ H0 þ 1 ½H1 ; s:

2

From our experience with the electron“phonon interaction, where the elim-

ination of the ¬rst-order terms led to the appearance of an e¬ective electron“

electron interaction term, we are prepared for some interesting consequences

in the present case.

To proceed, we try an operator s of the form

X y

ak Vk ° f y ck À ck f Þ;

s¼ °11:4:3Þ

k;

with the coe¬cients ak; to be determined. With this form of s, we ¬nd that

X X

y y

ak Vk °E f À E k Þ° f y ck þ c k f Þ þ U ak Vk °c k f þ f y ck ÞnÀ :

½H0 ; s ¼

k; k;

402 The Kondo effect and heavy fermions

This poses a little problem, since the presence of four operators in the last

term means that this expression cannot be equated with ÀH1 , which contains

only two. We solve this by making the mean-¬eld approximation of replacing

nÀ by its expectation value hnÀ i. The solution for the coe¬cients is then

ak ¼ °E k À E f À UhnÀ iÞÀ1 :

Because hnÀ i can only take on the values 0 or 1, we can rewrite this solution as

1 À nÀ nÀ

ak ¼ þ : °11:4:4Þ

Ek À Ef Ek À Ef À U

Note that we have quietly removed the angular brackets from hnÀ i and

restored it to the status of being an operator. This is important in the next

step we take, which is to calculate the e¬ective interaction term in our trans-

formed Hamiltonian H 0 . It is

1X

1 y y

½ak Vk ° f y ck À ck f Þ; Vk 0 °c k 0 0 f 0 þ f y 0 ck 0 0 Þ:

½H1 ; s ¼ À °11:4:5Þ

2 2 k;k 0 ; 0

The presence of the operator nÀ in the expression for ak means that we are

commuting a product of four operators with a product of two operators.

Two operators disappear in the process, and this leaves us with a product of

y

four operators, the most important of which are of the form f y ck c k 0 À fÀ

and their Hermitian conjugates. These terms have the form of an interaction

between a conduction electron and a local state, in which the spins of both are

reversed and in which the conduction electron is scattered: this is precisely

the form of the Kondo interaction. In addition to these terms, the unitary

transformation also yields other terms not included in the Kondo

Hamiltonian. The Anderson Hamiltonian is therefore richer in the sense that

it contains more physics. This also makes it more di¬cult to solve. While the

Kondo problem can be solved exactly using so-called Bethe Ansatz techniques,

no exact solutions are known to the Anderson Hamiltonian.

Given the fact that the single-impurity Anderson Hamiltonian contains a

Kondo-like term, it was to be expected that in the dilute limit, where one

can consider just a single dopant atom, there would appear in the

Anderson Hamiltonian a Kondo resonance at the Fermi energy. What is

more surprising is that spectroscopic measurements ¬nd such a resonance in

heavy-fermion materials such as CeAl3 , in spite of the fact that the Ce atoms are

not at all dilute in the Al host. The reason for this is that the measurements are

typically performed at temperatures too high for the coherence expected for a

regular lattice of Ce and Al atoms to develop. As in the Kondo model, the

403

Problems

appearance of this resonance depends nonperturbatively on an e¬ective

coupling constant.

One approach to studying the single-impurity Anderson Hamiltonian is

very similar to the way in which we analyzed the low-temperature behavior

of the Kondo Hamiltonian. One builds up approximate eigenstates by includ-

ing several collective states. Each collective state consists of one or more

electron“hole pair excitations from the Fermi sea, plus possible combinations

of occupations of the local states. Because more states with greater com-

plexities are included than in the low-temperature treatment of the Kondo

problem, the algebra is more complicated, and so we do not present the

details here. The picture that emerges, and which is fairly typical of heavy

fermions, is that the density of states contains a number of interesting fea-

tures. The ¬rst of these is a sharp Kondo-like peak at the Fermi energy,

which is expected because the Kondo physics is contained in the Anderson

Hamiltonian. In addition, there are two broader peaks, one below the Fermi

surface at E f , and the other above the Fermi surface at approximately

E f þ U. The Kondo peak is very sharp with a small spectral weight, while

the spectral weights of the broader peaks depend on the degeneracy of the

local state. Experimental measurements of the density of states by means of

spectroscopies that probe both the ¬lled states below the Fermi surface and

the empty states above it con¬rm the accuracy of this generic picture.

Problems

11.1 Calculate the energy E b of the state jÉb i for spin- 1 antiferromagnetic

2

coupling. Follow the steps of the calculation of E a , but note the dif-

ference that the state jÉb i involves electron states below the Fermi

surface, which will change some signs and the limits of the ¬nal integral

over energies. Show that jÉb i has a lower energy than jÉa i if the con-

duction band is less than half full.

11.2 A commonly encountered Kondo system consists of Fe impurities in

Cu. Assume a concentration of 1% and antiferromagnetic coupling of

the order of 1 eV. Estimate the Kondo temperature (you will need to

guess the density of states and conduction band width for Cu). How

large a change in the speci¬c heat (cf. Eq. (11.3.6)) would you expect

due to the presence of the Fe impurities at low temperatures?

11.3 In a traditional BCS s-wave superconductor, electrons at the Fermi sur-

face with opposite momenta and spin are bound in singlets. Suppose

404 The Kondo effect and heavy fermions

that such a superconductor, which in its normal state has a conduction

band that is more than half-¬lled, is now doped with magnetic impurities

with spin-1/2 local moments. Form an expression for the ratio of the

Kondo temperature TK to the BCS critical temperature Tc . How do you

think this system will behave as it is cooled down if Tc > TK ? What if

Tc < TK ?

11.4 In our attempt to diagonalize the Hamiltonian (11.4.2) we de¬ned in

Eq. (11.4.3) an operator s in terms of coe¬cients ak . These coe¬cients

were assumed to be numbers and not operators, and so they commuted

with H. However, in Eq. (11.4.4) we rather inconsistently gave them the

character of operators, and this led to the Kondo interaction when we

formed ½H1 ; s. What would we have found if we had assumed from the

start that ak was linear in n ?

y

11.5 What is the coe¬cient of the Kondo operator f y ck ck 0 À fÀ found from

the commutator ½H1 ; s in Eq. (11.4.5)?

11.6 The Kondo e¬ect is the result of the scattering impurity having an

internal degree of freedom, namely its spin. Something similar happens

when electrons are scattered from a moving impurity, the internal

degree of freedom in this case being the vibrational motion of the

impurity. For the purposes of this exercise we consider a substitutional

impurity of mass closely equal to that of the host material, so that its

motion can be described in terms of phonon operators. The electron

scattering matrix element VK , with K ¼ k 0 À k, is phase shifted to

VK eiKÁy by the displacement y of the impurity. To ¬rst order in y the

P

scattering perturbation is then VK °1 þ iK Á q yq N À1=2 Þ. Show that in

second order, processes like those shown in Fig. P11.1 do not exactly

cancel each other because of the energy of the virtual phonons involved.

(This makes an observable contribution to the Peltier coe¬cient in dilute

alloys at low temperatures, and is known as the Nielsen“Taylor e¬ect.)

Figure P11.1. These two scattering processes do not exactly cancel.

Bibliography

Chapter 1. Semiclassical introduction

Standard introductory texts on quantum mechanics and solid state physics are

1. Quantum Physics, 2nd edition, by S. Gasiorowicz (Wiley, New York, 1996)

2. Introduction to Solid State Physics, 7th edition, by C. Kittel (Wiley, New

York, 1996).

Some classic references for the topic of elementary excitations are

3. Concepts in Solids, by P. W. Anderson (reprinted by World Scienti¬c,

Singapore, 1998)

4. Elementary Excitations in Solids, by D. Pines (reprinted by Perseus, Reading,

Mass., 1999).

The concept of the soliton is described in

5. Solitons: An Introduction, by P. G. Drazin and R. S. Johnson (Cambridge

University Press, Cambridge, 1989).

Chapter 2. Second quantization and the electron gas

A detailed but accessible discussion of the electron gas and Hartree“Fock theory

is given in

1. Many-Particle Theory, by E. K. U. Gross, E. Runge and O. Heinonen (IOP

Publishing, Bristol, 1991)

This book also introduces the concepts of Feynman diagrams and Fermi liquid

theory. The primary reference for the nuts and bolts of many-particle physics is

2. Many-Particle Physics, 3rd edition, by G. D. Mahan (Plenum, New York,

2000).

This work is particularly useful for learning how to do calculations using

Feynman diagrams. Its ¬rst chapter treats second quantization with many examples

from condensed matter physics, while its Chapter 5 treats electron interactions.

A classic reference for the interacting electron gas is

`

3. The Theory of Quantum Liquids, by D. Pines and P. Nozieres (reprinted by

Perseus Books, Cambridge, Mass., 1999).

405

406 Bibliography

Chapter 3. Boson systems

Lattice vibrations are thoroughly discussed in

1. Dynamics of Perfect Crystals, by G. Venkataraman, L. A. Feldkamp, and

V. C. Sahni (M.I.T. Press, Cambridge, Mass., 1975)

and are also included in the reprinted classic

2. Electrons and Phonons, by J. M. Ziman (Oxford University Press, Oxford,

2000).

One of the most comprehensive references on the subject of liquid helium is

3. The Physics of Liquid and Solid Helium, edited by J. B. Ketterson and

K. Benneman (Wiley, New York, 1978).

For the subject of magnons, and a clear introduction to magnetism in general, we

recommend

4. The Theory of Magnetism I, Statics and Dynamics, by D. C. Mattis

(Springer-Verlag, New York, 1988).

The reduction of the Dirac equation by means of the Foldy“Wouthuysen transfor-

mation may be found in

5. Relativistic Quantum Mechanics, by J. D. Bjorken and S. D. Drell

(McGraw-Hill, New York, 1964)

Chapter 4. One-electron theory

A reference textbook on solid state physics that contains good pedagogical chapters

on one-electron theory is

1. Solid State Physics, by N. W. Ashcroft and N. D. Mermin (Holt, Reinhart

and Winston, New York, 1976).

More detail is to be found in

2. Electronic Structure and the Properties of Solids, by W. A. Harrison (Dover,

New York, 1989)

and

3. Elementary Electronic Structure, by W. A. Harrison (World Scienti¬c,

Singapore, 1999)

with specialized descriptions of particular methods in

4. The LMTO Method: Mu¬n-Tin Orbitals and Electronic Structure, by H. L.

Skriver (Springer-Verlag, New York, 1984)

5. Planewaves, Pseudopotentials and the LAPW Method, by D. J. Singh

(Kluwer Academic, Boston, 1994)

6. Electronic Structure and Optical Properties of Semiconductors, 2nd edition,

by M. L. Cohen and J. R. Chelikowsky (Springer-Verlag, New York, 1989).

A very readable text on quasicrystals is

7. Quasicrystals: A Primer, by C. Janot (Clarendon Press, Oxford, 1992).

407

Bibliography

Chapter 5. Density functional theory

The standard references are

1. Density Functional Theory. An Approach to the Quantum Many-Body

Problem, by R. M. Dreizler and E. K. U. Gross (Springer-Verlag, New

York, 1990)

2. Density-Functional Theory of Atoms and Molecules by R. G. Parr and W.

Yang (Oxford University Press, Oxford, 1989).

The book by Dreizler and Gross is couched in the language of condensed matter

physics, while the one by Parr and Yang is more suitable for quantum chemists.

A classic review article is

3. General density functional theory, by W. Kohn and P. Vashishta, in Theory

of the Inhomogeneous Electron Gas, edited by S. Lundqvist and N. H. March

(Plenum Press, New York, 1983).

The volume

4. Electronic Density Functional Theory. Recent Progress and New Directions,

edited by J. F. Dobson, G. Vignale, and M. P. Das (Plenum Press, New

York 1998)

has a clear exposition of GGA approximations by K. Burke, J. P. Perdew and Y.

Wang, as well as a good article on TDDFT by M. Petersilka, U. J. Gossmann

and E. K. U. Gross, and on time-dependent current DFT by G. Vignale and W.

Kohn. In addition, this volume also contains an article on ensemble DFT by O.

Heinonen, M. I. Lubin, and M. D. Johnson.

There is also a review article on TDDFT by E. K. U. Gross, J. F. Dobson, and

M. Petersilka in

5. Density Functional Theory II, Vol. 181 of Topics in Current Chemistry, edited

by R. F. Nalewajski (Springer-Verlag, New York, 1996), p. 81.

A very understandable review article on TDDFT is

6. A guided tour of time-dependent density functional theory, by K. Burke and

E. K. U. Gross, in Density Functionals: Theory and Applications, edited by

D. Joubert (Springer-Verlag, New York, 1998).

The original papers laying out the foundation for density functional theory and

the Kohn“Sham formalism are

7. P. Hohenberg and W. Kohn, Phys. Rev. 136, B864 (1964)

8. W. Kohn and L. J. Sham, Phys. Rev. 140, A1133 (1965).

Chapter 6. Electron“phonon interactions

A good source on this topic is Reference 2 of Chapter 3, which is further developed

in

1. The electron“phonon interaction, by L. J. Sham and J. M. Ziman, in Solid

State Physics, edited by F. Seitz and D. Turnbull, Vol. 15, p. 221 (Academic

Press, New York, 1963)

and

2. The Electron“Phonon Interaction in Metals, by G. Grimvall (Elsevier,

New York, 1981).

408 Bibliography

There is a wealth of material on electron“phonon interactions in

3. Polarons in Ionic Crystals and Polar Semiconductors, edited by J. T. Devreese

(North-Holland, Amsterdam, 1972)

while a good source for discussion of Peierls distortions is

4. Density Waves in Solids, by G. Gruner (Addison-Wesley, Reading, Mass.,

¨

1994).

Chapter 7. Superconductivity

The classic reference is

1. Theory of Superconductivity, by J. R. Schrie¬er (reprinted by Perseus,

Reading, Mass., 1983).

An excellent introductory volume that treats both type I and type II super-

conductors is

2. Introduction to Superconductivity, 2nd edition, by M. Tinkham

(McGraw-Hill, New York, 1996).

We can also recommend

3. Superconductivity of Metals and Alloys, by P. G. de Gennes (reprinted by

Perseus Books, Cambridge, Mass., 1999).

A very complete review of BCS superconductivity is contained in the two volumes

of

4. Superconductivity, edited by R. D. Parks (Marcel Dekker, 1969).

The original BCS paper, which is truly a landmark in physics, is

5. J. Bardeen, L. N. Cooper, and J. R. Schrie¬er, Phys. Rev. 108, 1175 (1957).

Chapter 8. Semiclassical theory of conductivity in metals

A classic text on the subject is again Reference 2 of Chapter 3. A more recent

treatment is

1. Quantum Kinetics in Transport and Optics of Semiconductors, by H. Haug

and A.-P. Jauho (Springer-Verlag, New York, 1998).

A reference for two-dimensional systems is

2. Transport in Nanostructures, by D. K. Ferry and S. M. Goodnick

(Cambridge University Press, Cambridge, 1997).

A review article that focuses on Monte Carlo solutions of Boltzmann equations in

semiconductors is

3. C. Jacoboni and L. Reggiani, Rev. Mod. Phys. 55, 645 (1983).

An introduction to thermoelectricity is

4. Thermoelectricity in Metals and Alloys, by R. D. Barnard (Taylor and

Francis, London, 1972).

409

Bibliography

Chapter 9. Mesoscopic physics

Introductory-level discussions of the Landauer“Buttiker approach are given in

¨

1. Electronic Transport in Mesoscopic Systems, by S. Datta (Cambridge

University Press, Cambridge, 1995).

A more advanced book is Reference 2 of Chapter 8. One of the best review arti-

cles on the Landauer“Buttiker approach is

¨

2. The quantum Hall e¬ect in open conductors, by M. Buttiker in

¨

Nanostructured Systems, edited by M. Reed (Semiconductors and

Semimetals, Vol. 35) (Academic Press, Boston, 1992).

In addition to discussing the multi-terminal Landauer“Buttiker approach, this

¨

article also describes in great detail its applications to the integer quantum Hall

e¬ect.

A good review of weak localization can be found in

3. Theory of coherent quantum transport, by A. D. Stone in Physics of

Nanostructures, edited by J. H. Davies and A. R. Long (IOP Publishing,

Bristol, 1992).

For more discussion of noise in mesoscopic systems, we recommend in particular

4. R. Landauer and Th. Martin, Physica 175, 167 (1991)

5. M. Buttiker, Physica 175, 199 (1991).

¨

We also recommend the original literature on the Landauer“Buttiker formalism,

¨

namely

6. R. Landauer, IBM J. Res. Dev. 1, 223 (1957)

7. R. Landauer, Phil. Mag. 21, 863 (1970)

8. M. Buttiker, Phys. Rev. Lett. 57, 1761 (1986).

¨

The classic work on weak localization is

9. E. Abrahams, P. W. Anderson, D. C. Licciardello, and T. V. Ramakrishnan,

Phys. Rev. Lett. 42, 673 (1979).

Chapter 10. The quantum Hall effect

The standard reference is

1. The Quantum Hall E¬ect, 2nd edition, edited by R. E. Prange and S. M.

Girvin (Springer-Verlag, New York, 1990)

while the book

2. The Quantum Hall E¬ects, Integral and Fractional, 2nd edition, by

T. Chakraborty and P. Pietilainen (Springer-Verlag, New York, 1995)

¨

gives a particularly detailed discussion of the fractional quantum Hall e¬ect and

gives technical descriptions of numerical calculations. For composite fermions

there is the volume

3. Composite Fermions: A Uni¬ed View of the Quantum Hall Regime, edited by

O. Heinonen (World Scienti¬c, Singapore, 1998).

For skyrmions we recommend

4. H. A. Fertig, L. Brey, R. Cote, and A. H. MacDonald, Phys. Rev. B 50,

11018 (1994).

410 Bibliography

A volume that contains many useful articles, and particularly the one by S. M.

Girvin and A. H. MacDonald, is

5. Perspectives in Quantum Hall E¬ects: Novel Quantum Liquids in Low-

Dimensional Semiconductor Structures, edited by S. Das Sarma and A.

Pinczuk (Wiley, New York, 1996).

The original literature reporting the discoveries of the integer quantum Hall e¬ect

and the fractional quantum Hall e¬ect, and Laughlin™s explanation of the frac-

tional quantum Hall e¬ect are

6. K. von Klitzing, G. Dorda, and M. Pepper, Phys. Rev. Lett. 45, 494 (1980).

7. D. C. Tsui, H. L. Stormer, and A. C. Gossard, Phys. Rev. Lett. 48, 1559

¨

(1982).

8. R. B. Laughlin, Phys. Rev. Lett. 50, 1395 (1983).

Chapter 11. The Kondo effect and heavy fermions

Reference 2 of Chapter 2 has a good chapter on spin ¬‚uctuations. It includes the

Kondo problem and heavy fermions. A very complete account of these topics is

1. The Kondo Problem to Heavy Fermions, by A. C. Hewson (Cambridge

University Press, Cambridge, 1997).

Some review articles on this topic are

2. D. M. Newns and N. Read, Adv. Phys. 36, 799 (1987).

3. P. Fulde, J. Keller, and G. Zwicknagl, in Solid State Physics, Vol. 41, p. 1,

edited by H. Ehrenreich and D. Turnbull (Academic Press, San Diego, 1988).

4. N. Grewe and F. Steglich in Handbook on the Physics and Chemistry of the

Rare Earths, Vol. 14 (Elsevier, Amsterdam, 1990).

Index

absolute thermoelectric power, 311 chemical potential, 85, 204, 272