R?ssler U. Solid State Theory: An Introduction
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218 7 Correlated Electrons
γT
T*
T
C
V
LaAl
3
CeAl
3
Fig. 7.10. Low-temperature behavior of the specific heat for a heavy fermion system
(CeAl
3
)comparedwithanormalmetal(LaAl
3
). The characteristic temperature T
∗
indicates the transition from heavy fermion to normal fermi liquid behavior (after
[122])
In general, heavy fermion systems are compounds with ions containing
partially filled 4f (lanthanides) of 5f shells (actinides). For further charac-
terization of their properties, one looks at the r atio R = π
2
k
2
B
χ
Pauli
/3μ
2
eff
γ,
where μ
eff
is the effective magnetic moment of the quasi-particles. For non-
interacting quasi-particles, R = 1, because the effective mass drops out. In
contrast, γ and χ
Pauli
are modified differently by quasi-particle interactions
which is reflected in R = 1. Th e enhancement of γ and χ
Pauli
occurs only
in a small temperature range below a characteristic temperature T
∗
of a few
Kelvin. Above T
∗
, it disappears and these systems b ehave like normal Fermi
liquids. This can be seen by comparing the temperature dependence of the spe-
cific heat of CeAl
3
and LaAl
3
(Fig.
7.10). The Ce compound differs from the
otherwise identical system with La by the partial occupation of the 4f shell.
The appropriate Hamiltonian for modeling the electronic prop erties of
heavy fermions is[
122]
H =
kσ
ǫ
k
c
†
kσ
c
kσ
+
im
ǫ
fm
f
†
im
f
im
+
U
2
i,m=m
′
n
f
im
n
f
im
′
+
1
√
N
k,σ,i,m
V
kσm
c
†
kσ
f
im
e
−ik·R
i
+ c
kσ
f
†
im
e
ik·R
i
. (7.100)
The first term describes th e disp ersive energy band ǫ
k
of the delocalized
electrons with wave vector k and spin σ. The second term represents the single-
particle energies of the f electrons, localized at lattice sites R
i
,wherem =
1 ...ν
f
counts the orbital degeneracy, while the third term is
Hubbard’s on-site Coulomb interaction of these electrons (with number oper-
ator n
f
im
). The fourth term accounts for the hybridization of the f electrons
with the delocalized Bloch electrons. Equation (
7.100)isavariantofthe
Anderson
6
Hamiltonian, which was originally formulated for only one site
6
Philip W. Anderson, *1923, Noble prize in Physics 1977 together with Sir Neville
Mott and John H. Van Vleck.
7.5 Heavy Fermion Systems 219
occupied with an ion carrying f electrons (Anderson impurity model). This
Hamiltonian combines two aspects of electronic structure: The hybridization
(see Sect.
5.4) (which co nverts the localized f level into a band with finite
width) with the correlation of the Hubbard model (which prevents double
occupancy of f states at a given site).
Without the correlation term, the problem can be diagonalized as in
Sect.
5.4. The correlation can be taken into account by a proper modifica-
tion of the hybridization or hopping matrix element, which ensures that a
conduction band electron does not hop to an already singly occupied f state.
In fact, according to Fermi’s golden rule, the rate of hopping between the con-
duction band and an f state is proportional to this matrix element squared.
If it is renormalized by a fa ctor 1 − n
f
,wheren
f
is the occupancy of the f
state, the rate will be reduced as desired. For the hoppi ng matrix element,
this means the replacement
V
kσm
→ rV
kσm
=
˜
V
kσm
, (7.101)
where r
2
=1−n
f
. In principle, this modification has to be done site dependent
and with the number op erator of the localized f electrons. By using r in the
following as a parameter, a mean-field approach has been adopted, by which
these site-dependent number operators are replaced by their mean value n
f
.
The relation between r and n
f
is to be included as a subsidi ary condition with
the Lagrange parameter Λ in the Hamiltonian
H
MF
=
kσ
ǫ
k
c
†
kσ
c
kσ
+
im
˜ǫ
fm
f
†
km
f
km
+
k,σ,m
rV
kσm
c
†
kσ
f
km
+ f
†
km
c
km
+ ΛN(r
2
− 1). (7.102)
Here, the fermion operators of the f electrons are in the Bloch representation
and ˜ǫ
fm
= ǫ
fm
+ Λ. This mean-field version of the Anderson Hamiltonian
depends on the two parameters r andΛwhichhavetobedeterminedatthe
end. It is advantageous to simplify the model by considering only a nondegen-
erate conduction band and an f orbital with degeneracy ν
f
= 2 by dropping
the indices m and σ.
The diagonal form of the simplified mean-field operator
H
MF
=
kl
E
l
(k)b
†
kl
b
kl
+ΛN (r
2
− 1) (7.103)
is obtained by replacing the fermion operators in (
7.102) according to
f
†
k
=
l
y
kl
b
†
kl
,c
†
k
=
l
x
kl
b
†
kl
, (7.104)
where the index l refers to the two branches of the hybridized bands. The
coefficients are determined by the normalization condition
220 7 Correlated Electrons
|y
kl
|
2
+ |x
kl
|
2
= 1 (7.105)
and the coupled linear homogeneous equations
˜ǫ
f
− E
l
(k)
y
kl
+
˜
V
k
x
kl
=0
˜
V
∗
k
y
kl
+
ǫ
k
− E
l
(k)
x
kl
=0. (7.106)
These equations signalize a second order perturbation calculation with
respect to th e hybridization a nd by eliminating e.g., x
kml
we have
˜ǫ
f
− E
l
(k)+
˜
V
k
˜
V
∗
k
E
l
(k) − ǫ
k
y
kl
=0. (7.107)
The coefficient y
kl
is the probability amplitude for the quasi-particle in the
band l to be in the f state. From (
7.107), one obtains the two quasi-particle
bands
E
±
(k)=
1
2
((ǫ
k
+˜ǫ
f
) ± W (ǫ
k
)) , with W (ǫ
k
)=
(ǫ
k
− ˜ǫ
f
)
2
+4
˜
V
2
.
(7.108)
They are depicted in Fig.
7.11, where a l inear dispersion is assumed for the
conduction band ǫ
k
. Th e requirement that the lower branch crosses the chem-
ical potential for k = k
F
, E
−
(k
F
)=μ, allows one to determine from (
7.107)
the renormalized energy of the f level
˜ǫ
f
= μ +
˜
V
2
ǫ
k
F
− μ
, (7.109)
which fo r weak hybridization is slightly above μ.
E
ε
~
f
µ
ε
f
k/k
F
01
Fig. 7.11. Quasi-particle dispersion for heavy fermion behavior. The linear disper-
sion of the conduction band hybridizes with the renormalized level at ˜ǫ
f
of the f
state (after [122])
7.5 Heavy Fermion Systems 221
The next task is to determine the two parameters r and Λ of the
model. One obvious condition is that the ground state expectation value of
the mean-field Hamiltonian takes a minimum with respect to the f state
occupation n
f
or the parameter r. This condition can be formulated as
∂/∂rΨ
0
(r)|H
MF
|Ψ
0
(r) = 0, which reduces because of the normalization of
the ground state, to Ψ
0
|∂H
MF
/∂r|Ψ
0
=0orwith(
7.102)
1
N
k
˜
V
k
Ψ
0
|c
†
k
f
k
+ f
†
k
c
k
|Ψ
0
+2Λr
2
=0. (7.110)
The secon d relation is n
f
=1− r
2
,reformulatedas
1
N
k
Ψ
0
|f
†
k
f
k
|Ψ
0
+(r
2
− 1) = 0. (7.111)
The first terms in these relations with the expectation values can be expressed
with the help of the expansions (
7.104) and the possibility to create the ground
state |Ψ
0
by applying b
†
kl
for |k|≤k
F
to the fermion vacuum
1
N
k
Ψ
0
|f
†
k
f
k
|Ψ
0
=
ν
f
2
D(E
F
)(ǫ
k
F
+ W (ǫ
k
F
) − W (0)) = n
f
(7.112)
1
N
k
˜
V Ψ
0
|c
†
k
f
k
|Ψ
0
= −ν
f
D(E
F
)
˜
V
2
ln
ǫ
k
F
− ˜ǫ
f
+ W (ǫ
k
F
)
−˜ǫ
f
+ W (0)
!
. (7.113)
This leads in the leading logarithmic approximation to
r
2
=1− ν
f
D(E
F
)
(ǫ
k
F
− μ) −
˜
V
2
μ
(7.114)
and
Λ=ν
f
D(E
F
)V
2
ln
(ǫ
k
F
− μ)μ
˜
V
2
!
. (7.115)
It is useful to define a characteristic temperature T
∗
by
k
B
T
∗
= μ exp
−
Λ
ν
f
D(E
F
)V
2
!
, (7.116)
which allows one to express the renor malized energy of the f level as
˜ǫ
f
= μ + k
B
T
∗
(7.117)
and by eliminating ǫ
k
F
to write
n
f
=1−
k
B
T
∗
ν
f
D(E
F
)V
2
. (7.118)
222 7 Correlated Electrons
Inspection of Fig.
7.11 sheds some light on the physics: Without hybridization
the f level at ǫ
f
would be occupied, n
f
= 1, and also the conduction band
states up to k
F
. With hybridization, the f level is shifted above μ,therebythe
occupation is reduced, and the conduction band is flattened out close to the
chemical potential. The latter is connected with an increase of the density of
states and a gain in energ y, because the occupied conduction band states are
now at lower energy. This gain in total energy can be quantified by comparing
the ground state energies without (E
(0)
0
)andwith(E
0
)hybridization:
E
0
− E
(0)
0
= −k
B
T
∗
. (7.119)
Without giving further details of the calculation [
122], one obtains for the
Sommerfeld coefficient in the specific heat
γ =
1
3
π
2
k
2
B
n
f
˜ǫ
f
− μ
(2J + 1) (7.120)
and for the spin susceptibility
χ
spin
=(g
J
μ
B
)
2
n
f
ν
f
(˜ǫ
f
− μ)
2J +1
3
. (7.121)
Both quantities show a strong increase with ˜ǫ
f
bei n g close to the chemical
potential μ. This explains the h eavy fermion behavior.
7.6 Fractional Quantum Hall States
In this chapter, we have seen so far that the influence of electron correlation
increases with the localization of the electrons due to their o rbit al motion.
We found that systems with electrons at the Fermi energy in atomic d and
f orbitals are destined to exhibit effects of the electron–electron interaction.
There is quite a different system, where the localization is not a genuine prop-
erty of an atomic orbital, but where it is due the orbital motion enforced by
applying a mag netic field. Remember the two-dimensional electron systems
of Sect.
5.6 and the discussion of the quantum Hall effects. A magnetic field
applied perpendicul ar to the semiconductor hetero-interface forces the elec-
trons into cyclotron orbits whose radius decreases with increasing magnetic
field. This is the localization mechanism which eventually leads to a regime
where correlation effects become significant. It is the regime of the fractional
quantum Hall effect. This effect has been discovered only a few years after the
integer QHE.
In 1982 Tsui, Sto ermer, and Gossard [
226] reported the observation of a
QHE not at integer filling factors ν but at the fractional filling ν =1/3ofthe
lowest spin-split Landau level in a high-mobility GaAs/AlGaAs heterostruc-
ture. In the following years, with increasing perfections of the heterostructure
7.6 Fractional Quantum Hall States 223
MAGNETIC FIELD [T]
510152025300
0
0
0.5
1.0
1.5
2.0
2.5
3 2 N=1
n = 4 3 2
1
3 /2
1 /2
4/3
4/5
5/7
3/4
2/3
3/5
4/7
5/9
6/11
7/13
6/13
5/11
4/9
3/7
2/5
5/3
10/7
9/7
8/5
7/5
11/9
N=0
P
xx
P
xy
[h /e
2
]
Fig. 7.12. Low-temperature data of the longitudinal (ρ
xx
) and Hall resistance
(ρ
xy
) of a high-mobility two-dimensional electron system in a GaAs/AlGaAs het-
erostructure. N indicates the Landau level quantum number and ν the filling factor.
After [
227]
samples, a whole family of fractional quantum Hall states was discovered
preferentially in the s ame materia l system (see Fig.
7.12) but also in semicon-
ductor heterostructures with other material combinations. The characteristic
features of the QHE, the plateaus of the Hall conductance, and the vanishing
of the longitudinal magneto-resistance (in the first observation just a dip at
ν =1/3), appear systematically at filling factors
ν =
p
2p ± 1
(7.122)
with integer p. Remarkable is here the o dd denominator and the convergence
of the two series of fractions for the ± sign from above and below toward the
even fractional filling with ν =1/2, for which no plateau is detected and the
longitudinal magneto-resistance does not decrease to zero.
Shortly after the discovery of the effect (and before most of the other
fractions were discovered), Laughlin came up with the fundamental idea of an
incompressible quantum liquid caused by electron correlation and designed the
corresponding N electron wave function [
228]. Tsui, St¨ormer, and Laughlin
received jointly in 1998 the Nobel prize in physics for their discovery, which
together with other contributions to the subject is well documented in several
monographs [
202–204].
224 7 Correlated Electrons
In order to present Laughlin’s idea, we have to briefly go through the single-
particle description of two-dimensional electrons in a perpendicular magnetic
field.
Supplement: Two-dimensional electrons in a magnetic field
The system Hamiltonian (without Zeeman term)
H =
1
2m
∗
(p + eA )
2
(7.123)
can be written with the symmetric gauge of the vector potential A = B(y,−x, 0)/2
in the form
H =
1
2m
∗
p
2
x
+ p
2
y
+
m
∗
2
ω
∗
c
2
!
2
x
2
+ y
2
−
ω
∗
c
2
L
z
. (7.124)
The effective mass m
∗
(and the cyclotron frequency ω
∗
c
= eB/m
∗
) accounts for
the fact that the electrons are in a subband deriving from the conduction band
of a semiconductor. The symmetric gauge restores the cylindrical symmetry of the
system according to which the z component of the angular momentum L
z
commutes
with the system Hamiltonian, which immediately can be recognized as that of two
harmonic oscillators in the x, y planes, respectively. Accordingly, two sets of oscillator
operators with standard commutation rules are introduced
a
†
x
=
1
√
2
x
l
−
il
¯h
p
x
!
,a
x
=
1
√
2
x
l
+
il
¯h
p
x
!
a
†
y
=
1
√
2
y
l
−
il
¯h
p
y
!
,a
y
=
1
√
2
y
l
+
il
¯h
p
y
!
, (7.125)
where l =
#
(¯h/m
∗
ω
∗
c
)=
#
(¯h/eB) is the magnetic length (see Sect. 4.2). The
single-particle Hamiltonian
H =
¯hω
∗
c
2
&
a
†
x
a
x
+ a
†
y
a
y
+1
− i
a
y
a
†
x
− a
x
a
†
y
'
(7.126)
can be diagonalized by oscillator operators
a
†
+
=
1
√
2
a
†
x
+ia
†
y
,a
+
=
1
√
2
(a
x
− ia
y
)
a
†
−
=
1
√
2
a
†
x
− ia
†
y
,a
−
=
1
√
2
(a
x
+ia
y
) , (7.127)
which obey the commutation relations [a
±
,a
†
±
] = 1. They can be understood as
operators for right/left circular rotations around the direction of the magnetic field.
With these operators, one finds
H =
¯hω
∗
c
2
a
†
+
a
+
+ a
†
−
a
−
+1
+
¯hω
∗
c
2
a
†
+
a
+
− a
†
−
a
−
(7.128)
=¯hω
∗
c
ˆn
+
+
1
2
!
(7.129)
7.6 Fractional Quantum Hall States 225
with eigenvalues
ǫ
n
+
n
−
=¯hω
∗
c
n
+
+
1
2
!
. (7.130)
Note that the eigenvalues depend only on the quantum number n
+
while n
−
counts
the level degeneracy. The eigenstates of H can be created by multiple application
of the raising operators onto the oscillator vacuum
|n
+
n
−
=
%
1
n
+
!n
−
!
a
†
+
n
+
a
†
−
n
−
|00. (7.131)
Instead of n
±
, it is advantageous to use the quantum numbers n = min(n
+
,n
−
)
and the angular momentum quantum number m = n
+
−n
−
, which are related with
the Landau level quantum number n
L
= n +(m + |m|)/2. (Note: in Fig.
7.12 n
L
is
denoted N.)
In combining (
7.125)and(7.127), it is suggestive to introduce the complex
dimensionless variable z =(x − iy)/l and the corresponding derivative ∂/∂z =
(∂/∂x +i∂/∂y)l/2 along with the conjugate definitions. This gives the following
convenient properties
∂
∂z
z − z
∂
∂z
=1,
∂
∂z
z
∗
− z
∗
∂
∂z
= 0 (7.132)
and similarly for the complex conjugates. The oscillator operators can now be written
as
a
†
+
=
1
2
z
∗
− 2
∂
∂z
!
,a
+
=
1
2
z +2
∂
∂z
∗
!
a
†
−
=
1
2
z − 2
∂
∂z
∗
!
,a
−
=
1
2
z
∗
+2
∂
∂z
!
(7.133)
which lead to the Hamiltonian in dimensionless coordinate representation
H =
1
2
¯hω
∗
c
−4
∂
∂z
∂
∂z
∗
+ zz
∗
!
−
1
2
¯hω
∗
c
z
∂
∂z
− z
∗
∂
∂z
∗
!
. (7.134)
The lowest energy eigenfunction can be obtained from the conditions a
±
|00 =0,
which easily lead to the normalized w ave function of the lowest Landau level.
Making use of the complex notation in the (x, y) plane, the normalized
wave function for the lowest Landau level (n
L
=0)isgivenby
ϕ
00
(r)=
1
√
2πl
exp
−
r
2
4l
2
!
=
1
√
2πl
e
−|z|
2
/4
. (7.135)
This wave function with angular momentum m = 0 is degenerate with finite
angular momentum wave functions
ϕ
0m
(z)=
1
√
2π2
m
m!l
z
m
e
−|z|
2
/4
, (7.136)
which have a maximum probability |ϕ
0m
(r)|
2
on a circle with radius
√
2ml
and a spread of the order of the magnetic length l. Thus, a sample of circular
226 7 Correlated Electrons
shape with radius R can accommodate only states with m values fulfilling the
condition 2ml
2
<R
2
. This allows one to count the degeneracy of the Landau
level. The maximum value of m is determined by the number of elementary
flux quanta threading the sample area. This is the same result as the one
obtained in Sect.
4.2 assuming the asymmetric Landau gauge.
Considering fr actional quantum Hall states, Laughlin constructed a N-
electron wave function from the single-particle wave functions of the lowest
Landau level. Their general form, a linear combination of the ϕ
0m
,isϕ(z)=
f(z)exp(−|z|
2
/4) with a polynomial f(z). The N -electron wave function,
expressed as a linear combination of Slater determinants composed of these
single-particle wave functions, has the general form
Ψ(z
1
,...,z
N
)=f(z
1
,...,z
N
)exp
−
N
i=1
|z
i
|
2
/4
. (7.137)
Here, f (z
1
,...,z
N
) is a polynomial in every variable z
i
and its individual
terms are products of z
m
i
i
indicating that the electron i is in an angular
momentum eigenstate with m
i
. The symmetry of the problem requires that the
total angular momentum ¯hM =
i
¯hm
i
is conserved and the wave function
Ψ(z
1
,...,z
N
) should contain only terms with the same M. Thus, the polyno-
mial f has to b e homogeneous. The antisymmetry of the Slater determinants
makes this polynomial also antisymmetric in the particle coordinates.
On top of these symmetry requirements, the N-particlewavefunction,in
order to describe a ground state, should by construction take into account
that the electrons try to avoid each other due to the repulsive Coulomb
interaction. This can be achieved by writing the polynomial as a product
of functions g(z
i
− z
j
) depending on the inter-particle separation. This form,
which accounts for two-particle correlations, is known as a Jastrow-type wave
function and was used before in atomic physics. Together with the general
symmetry considerations g(z) has to be an odd power polynomial. Moreover,
Ψ is an eigenfunction of the total angular momentum ¯hM where M counts
the powers of the z
i
, which are all the same and M = N(N − 1)m/2. Thus,
one arrives at the N electron wave function
Ψ
m
(z
1
,...,z
N
)=
*
i<j
(z
i
− z
j
)
m
exp
−
N
i=1
|z
i
|
2
/4
. (7.138)
This is Laughlin’s wave function.
As we have seen, the maximum possible angular momentum of a single
particle state is determined by the degeneracy of the Landau level which is
the sample area divided by 2πl
2
or the numb er Φ/Φ
0
of flux quanta threading
the sample. On the other hand, the maximum power (or angular momentum
quantum number) of each z is given by m(N − 1) and we can equate (for
N ≫ 1)
mN =
Φ
Φ
0
or m =
Φ
NΦ
0
=
1
ν
. (7.139)
7.6 Fractional Quantum Hall States 227
The last relation connects the angular momentum with the filling factor ν.
For fractional filling ν =1/m with odd m, the Laughlin wave function is an
antisymmetric many-body wave function, where each electron p osition is an
m-fold zero with respect to the dependence on all other electron positions. At
thesametime,m is the number of flux quanta (or vortices) attached to each
electron.
An interpretation of this many-body wave function is possible by looking at
|Ψ
m
(z
1
,...,z
N
)|
2
=e
−Φ
m
(z
1
,...,z
N
)
(7.140)
where
Φ
m
(z
1
,...,z
N
)=−2m
i<j
ln |z
i
− z
j
| +
1
2l
2
N
i=1
|z
i
|
2
(7.141)
is formally identical with the electrostatic energy of a charge-neutral two-
dimensional plasma, where the first term accounts for the electron–electron
interaction and the second for the interaction with the neutralizing back-
ground. The corresponding expression would be obtained by replacing 2m →
e
2
and 1/2mπl
2
by the particle density. Thus, |Ψ
m
(z
1
,...,z
N
)|
2
is the classical
probability distribution of N electrons in a plane with logarithmic interaction.
The quality of Laughlin’s wave function has been tested by projection
onto ground state wave functions, which for small N were calculated numeri-
cally by exact diagonalization. These projections were found to be very close
to 1. Another result, that characterizes Laughlin’s wave function is the radial
distribution function
g
m
(|z
1
− z
2
|)=
N(N − 1)
ρ
2
m
...
Ψ
m
(z
1
,...,z
N
)
"
"
2
d
2
z
3
...d
2
z
N
.
(7.142)
1.0
0.5
0
1234
g(x)
x0
m = 3
liquid
crystal
Fig. 7.13. Radial distribution function for the Laughlin state with m =3(solid
line) and the Wigner crystal state in the HF approximation (dashed line)forthe
same density. Here, x is the inter-particle distance in units of
√
2ml (after [203])