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10.8 Composite fermions
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 xފ;
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 Š 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 Š:
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 Š:
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" Š 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 ½ y ck" þ y ck# Š 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# ŠSz 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" Š 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# ŠSz ½ ck# À c k" Š jFi
0 p;k
y yy yy y yy yy
þ cp" ck# S À ½ c k# À ck" Š jFi þ c p# ck" S þ ½ ck# À ck" Š jFig:

The operators Sz , SÀ and S þ all act only on the local spin, and Sz y ¼ 0 y=2,
Sz y ¼ À0 y=2, S À y ¼ 0 y , S À y ¼ 0, S þ y ¼ 0, and S þ y ¼ 0 y , so we
obtain
& '
X 1y y yy yy y yy y yy
ÀJ ½c c À c p# ck# Š½ ck# þ ck" Š 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" ŠjFihFj ½ ck# À ck" Š:
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" ŠjFi
jkj>kF
X yy yy
¼ ak ½E k À E a Š½ c k# À ck" ŠjFi:
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 Š:
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# Š ak ½ c k# À c k" ŠjFi
jkj>kF
p;q
°11:3:1Þ
0X y X
y yy yy
¼ ½c p" cq" À cp# cq# Š ak ½ c k# þ ck" ŠjFi;
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# Š 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# Š 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# Š ak ck" jFi ¼ ½c p" cq" À cp# cq# Š 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" ŠjFi:
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 ; sŠ ¼ 0;

which then to second order in Vk leaves us with

H 0 ¼ H0 þ 1 ½H1 ; sŠ:
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 ; sŠ 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 ފ:
½H1 ; sŠ 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 ; sŠ. 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 ; sŠ 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

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