R?ssler U. Solid State Theory: An Introduction

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6.6 Itinerant Electron Magnetism 187

(N,ζ )=

)+E

−

)



(1 + ζ)

5/3

+(1− ζ)

5/3



−



− 1



. (6.101)

A minimum of ǫ

(N,ζ )atﬁniteζ would indicate the existence of a stable

ferromagnetic state. This condition leads to the relation

ζ =(1+ζ)

2/3

− (1 − ζ)

2/3

, 0 ≤ ζ ≤ 1. (6.102)

The rhs is a monotonous function, starting for ζ = 0 at zero with a slope of

4/3 and reaching for ζ =1thevalue2

2/3

with inﬁnite slope. With respect to

(

6.102) three situations are possible:

(a) UN/E

< 4/ 3, the relation has no solution for ﬁnite ζ,whichleadstoa

stable paramagnetic state,

(b) 4/3 <UN/E

< 2

2/3

, there exists a solution for 0 <ζ<1representing

a stable ferromagnetic state, with partial spin polarization,

2/3

<UN/E

, there is always a solution with ζ = 1 representing a

ferromagnet with perfect alignment of all spins.

The ratio N/E

is prop ortional to the density of states at the Fermi energy

D(E

), thus the existence of ferromagnetism in a system of itinerant electrons

is ruled by the competition between D(E

) (which is determined by the dis-

persion) and the strength of the exchange energy U. If, for the given U,the

width of the energy band increases, i.e. D(E

) decrease s, the criterio n (c)

or even (b) will b e missed a nd ferromagnetism will not be r ealized. Thus,

the Stoner model provides with the criterion (a), the Stoner condition, a clear

answer with respect to the existence of ferromagnetism for itinerant electrons.

Besides the existence of magnetic order the other basic property o f a spin

system is the excitation sp ectrum out of the ground state. In Chap.

4 it was

the dielectric function which has led us to the excitation spectrum of the free

interacting electrons consisting of single-particle (or electron–hole) and col-

lective excitations (the plasmons). For the latter, it was necessary to employ

the random phase approximation. We remember that the inverse dielectric

function is a density–density correlation function. Here, we want to put the

emphasis on spin-ﬂip excita tions. The corresponding response f u n ction is the

magnetic or spin susceptibility, which is a correlation function between com-

ponents of the magnetic dipole density. The observable of interest here is the

magnetization related to spins S

of electrons at r

M(r)=

m(r)

= gμ



δ(r − r

) (6.103)

as intro duced already in Sect.

2.5. This quantity can be addressed by a mag-

netic ﬁeld with the interaction term V

ext

(t)=−m·B(t). Spin ﬂips are caused

188 6 Spin Waves: Magnons

by the ladder operators S

, which for a spin-1/2 system are expressed in terms

auf Pauli spin matrices as

=(σ

+iσ

)/2=

−

=(σ

− iσ

)/2=

. (6.104)

In the following we assume B

= 0 and choose the appropriate decomposition

of the external perturbation

ext

(t)=−(m

−

−iωt

+ m

−

iωt

) (6.105)

which describes the interaction with a rotating magnetic ﬁeld in the xy plane.

The observable we ar e interested in here is M

= m

/V . It is related to the

susceptibility responsible for spin-ﬂip pro cesses

+−

(q,ω)=

¯h



+∞

−∞

iωτ

Θ(τ)[M

(q,τ),M

−

(−q, 0)]

dτ. (6.106)

This correlation function, which has the same structure as (

2.61), can be eval-

uated in the occupation number representation. In terms of fermion operators

for the Bloch electrons we have

(q)=gμ



†

k+q↑

k↓

−

(−q)=gμ



†

k−q↓

k↑

, (6.107)

which are spin-density ﬂuctuations (multiplied by gμ

). We evaluate the

expectation value under the integral for T = 0 with the eigenstates of the

HF Hamiltonian (

6.96) analogous to the calculation of the inverse dielectric

function in Sect.

4.5 but consider the spin splitting of the HF single-particle

energies ǫ

k±

= ǫ

∓∆/2 and obtain as the HF result for the spin susceptibility

(Problem 6.11)

+−

(q,ω) = lim

δ→0



k↓

− f

k+q↑

¯hω + ǫ

k+q↑

− ǫ

k↓

+iδ

. (6.108)

It resembles the polarization function π

(q,ω) in the HF result for the inverse

dielectric function (see (

4.122)). Its poles (in the lower half of the complex

energy plane) mark the particle–hole excitations with spin ﬂip, the Stoner

continuum depicted in Fig.

6.12.Forq = 0, excitations are possible only for

¯hω = ∆, b ecause the two bands are shifted against each other by the exchange

energy (see Fig. 6.11). With increasing q, spin-ﬂip excitation s become possible

for a continuum with increasing width. In Fig.

6.12, two situations are shown.

If ∆ is smaller than E

the continuum reaches down to vanishing excitation

energies, for whi ch spin-ﬂip excitations are possible in the interval k

−k

−

q<k

−

as can be checked also with Fig.

6.11. This situation is called weak

ferromagnetic case.Inthestr ong ferromagnetic case, E

< ∆, single-particle

excitations with spin ﬂip are possible only for ﬁnite excitation energy.

6.6 Itinerant Electron Magnetism 189

(weak)

Stoner

continuum

∆

Stoner

continuum

(strong)

∆

–k

+–

hω

+–

Fig. 6.12. Spectra of single-particle (Stoner continuum) and collective excitations

with spin-ﬂip for a weak (left)andastrong(right) ferromagnet

These results can be compared with those of the dielectric function for the

free electron system in Sect.

4.5. There we have obtained the single-particle or

electron–hole excitations (without spin ﬂip) in the HF approximation. As we

know from the discussion of the dielectric function, this approximation does

not yield the collective charge-density excitations (or plasmons) for which we

hadtogobeyondHFtotheRPA.Itconsistedinreplacingthefreepolarization

function π

(q,ω)byπ

(q,ω)/(1−v

(q,ω)). Th e corresponding replacement

is possible here by identifying π

(q,ω)with˜χ

+−

(q,ω)=χ

+−

(q,ω)/g

and v

with the exchange interaction U . Thus the RPA result for the spin

susceptibility is easily obtained as

+−

(q,ω)=g

˜χ

+−

(q,ω)

1 − U ˜χ

+−

(q,ω)

. (6.109)

Besides the poles of single-particle excitations of χ

+−

(q,ω), the RPA suscep-

tibility has an additional pole due to the vanishing denominator,

1 − U ˜χ

+−

(q,ω)=0, (6.110)

giving the collective excitations of the spin system, the magnons.

In order to ﬁnd the magnon dispersion, we have, in a ﬁrst step, to evaluate

˜χ

+−

(q,ω) under the conditions of collective excitations outside of the Stoner

continuum, which we expect to occur for ¯hω ≪ ∆ and at small q.Thenwe

have, using the solution of Problem 6.11

˜χ

+−

(q,ω)=

6π

¯hω − ∆+ǫ



¯h

¯hω − ∆+ǫ



−

6π

¯hω − ∆ − ǫ



¯h

¯hω − ∆ − ǫ



(6.111)

190 6 Spin Waves: Magnons

with ǫ

=¯h

/2m. We substitute this result in (

6.110) and ﬁnd

1 −

6π

−

¯hω − ∆+ǫ

−

¯hω − ∆ − ǫ

¯h



(¯hω − ∆+ǫ

)

−

(¯hω − ∆ − ǫ

)

,

=0. (6.112)

Being interested in the leading order term in q, we expand the denominators

of the ﬁrst two terms in the bracket with ∆ ≪ ¯hω ± ǫ

but neglect the small

terms in the dominators of the last terms, which are already prop ortional to

. Making use of the relation

∆



− k

−



6π

(6.113)

the condition for the magnon pole simpliﬁes and can be solved for the magnon

energy

¯hω

≃

¯h

+ k

−

− k

−

2¯h

5m∆

− k

−

− k

−

. (6.114)

With k

= k

(1 ± ζ)and∆=UNζ this reduces to

¯hω

≃

¯h

2mζ



1 −

5UN

(1 + ζ)

5/3

− (1 − ζ)

5/3



. (6.115)

We ﬁnd the characteristic q

dependence giving a dispersion of the ferro-

magnetic magnons (see Sect.

6.3) separated from the Stoner continuum as

indicatedinFig.

6.12. The magnon dispersion of Fe, measured with inelastic

neutron scattering, is shown in Fig.

6.13. It follows more or less that this q

law is almost independent of the direction o f propagation. With increasing q

the magnon dispersion becomes degenerate with the Stoner continuum and

decays into single-particle excitations with spin-ﬂ ip . This situation resembles

that of the collective charge density excitations (plasmons, see Sect.

4.6)when

they become degenerate with the particle-hole continuum of the free electron

gas and decay into particle–hole excitations.

The curvature of the magnon dispersion depends on the relation between

density of states (or Fermi energy over N) and interaction strength U.For,

complete spin polarization ζ =1,(

6.115) simpliﬁes to

¯hω

(ζ =1)=

¯h



1 −

5UN

5/3



(6.116)

while for small spin polarization ζ ≪ 1wehave

¯hω

(ζ ≪ 1) =

¯h

2mζ



1 −

3UN



. (6.117)

Problems 191

Fig. 6.13. Magnon dispersion of ferromagnetic Fe from inelastic neutron scattering

after [

186]. The dashed line represents the quadratic dispersion

In this limit ω

is positive if

. (6.118)

Thus, we arrive at the consistent result that col lective spin excitations are

possible if according to the Stoner condition ferroma gnetism and ﬁnite spin

po larization exist.

Problems

6.1 Derive the expression f or the matrix element of the electron–electron

interaction V

αββ

′

in the general case! Make use of the Four ier expansion

of products of the periodic parts of the Bloch functions. Simplify to the

single-band case and use only the leading term of this Fourier expansion.

6.2 Make use of the orthogonality and normalization of the Bloch functions

nkσ

(r) to show that

nσ

(r − R)=

√



−ik·R

nkσ

(r) (6.119)

192 6 Spin Waves: Magnons

are normalized and that they are orthogonal when centered around diﬀer-

ent sites R (Wannier representation)! Derive the commutation relations

of c

nRσ

†

′

from those of the Bloch states!

6.3 Express the interaction matrix element of Problem 6. 1 in the Wannier

representation with localized atomic orbitals for the single-band case to

obtain (

6.10)!

6.4 Show that the set of operators c

†

↑

↓

†

↓

↑

,andc

†

↑

− c

†

↓

fulﬁlls the

same commutation relations as the components of the spin-1/2 operators.

Make use of the fermion commutation rules for c

†

6.5 Derive the equation of motion (

6.32) for the spin operator S

determined

by the Heisenberg Hamiltonian (

6.23)!

6.6 Given are the commutation rules for the boson operators b

†

,i=1, 2

for the magnons in the two sublattices of an anti-ferromagnet. Verify the

commutation relations (

6.58) for the Boson operators α

,β

obtained by

the Bogoliubov transformation (

6.57)!

6.7 The Hamiltonian of a solid, in which the simultaneous presence of

phonons and magnons shall be studied, can be written as

p−m



¯hω

†

+¯hω

†

+ c



†

+ a

†



, (6.120)

where ω

(ω

) is the phonon(magnon) dispersion and a, a

†

(b, b

†

)are

the boson operators for phonons(magnons). The coupling between b oth

elementary excitations is described by the bilinear term with the cou-

pling constant c

. Make use of the following variant of the Bogoliub ov

transformation to eliminate the coupling (with real Θ

= α

cos Θ

+ β

sin Θ

= β

cos Θ

− α

sin Θ

(6.121)

and express Θ

(or a function of Θ

) by the coupling constant and the

frequencies ω

(ω

)! Discuss the solution, especially for ω

= ω

,and

give the eigenfrequencies o f the new collective magnon–phonon modes

and how their boson op erators are composed of the original phonon and

magnon op erators!

6.8 Replace in (

6.56) the spin wave op erators for the sublattices with the

help of the Bogoliubov transformation to get the Hamiltonian (

6.60)!

6.9 In order to remove the degeneracy of the anti-ferromagnetic magnons,

introduce a weak anisotropy ﬁeld H

in the Hamiltonian (

6.53)acting

with opposite sign on the two sublattices, and an external magnetic ﬁeld.

How does this change the dispersion relation for the anti-ferromagnetic

magnons? Discuss the frequency in the long wavelength limit k → 0and

sketch the result to show the eﬀect of the anisotropy ﬁeld!

6.10 Calculate the temperature dependence o f M (T ) − M(0) for T ≪ T

the spin–wave approximation and verify Bloch’s T

3/2

law!

Problems 193

6.11 Derive the expression (

6.108) of the spin susceptibility in HF approxi-

mation χ

+−

(q,ω)atT = 0 ! The calculation is strictly analogous to that

performed in Sect.

4.5 to obtain the HF result for the inverse dielectric

function. Evaluate the formula by assuming a single-particle dispersion

of the form ǫ

kσ

=¯h

/2m − σ∆/2with∆=U(N

− N

−

)! Make use

of the fact that ∆ is the dominating quantity, i.e. ∆ ≫ ¯hω, ¯h

/2m!

Correlated Electrons

In Sect.

4.7, we have studied in some detail the correlation eﬀects for the homo-

geneous free electron system and ﬁgured out their dependence on the electron

density. Correlation has been considered in the eﬀective single-particle poten-

tial for crystal electrons within the density-functional theory (DFT), which

has led to the independent particle description of the electronic band struc-

ture in Chap.

5. In Chap. 6, we have stressed the imp ortance of the exchange

interaction and of the electron spin for magnetic properties. The aim of this

chapter is to introduce concepts of treating correlation eﬀects for electrons

in a crystalline surrounding beyond the independent particle picture in a

more general sense. One aspect wil l be to describe model systems, which

allow to demonstrate corr elation eﬀects [

122, 193–196]. The motivation comes

from the observation that some group of solids, namely those with the Fermi

energy within narrow bands deriving from d or f electrons, exhibit properti es

which cannot be understood within the single-particle band structure. Among

those are besides the magnetic properties (Chap.

6), the insulating behavior

of some transition metal oxides, and the heavy fermion eﬀects in compounds

of lanthanides and actinides. Another aspect is the quasi-particle concept in

the context of the Fermi liquid theory and the deviations from Fermi liq-

uid behavior in systems with reduced dimension [

197–201]. Finally, we would

like to address also correlation in two-dimensional electron systems in high

magnetic ﬁelds, known as the fractional quantum Hall r egime [

202–204].

As in Chap.

6, we start from the N electron Hamiltonian (see (6.5))

H =



†



αβα

′

αββ

′

†

′

, (7.1)

where α, β, α

′

, and β

′

are complete sets of single-particle quantum numb ers,

α = nkσ,andǫ

the corresponding energy values. The matrix elements of the

electron–electron interaction

αββ

′



′

†

(x)ψ

†

′

)

4πε

|x − x

′

(x)ψ

′

) (7.2)

U. R¨ossler, Solid State Theory,

DOI 10.1007/978-3-540-92762-4

 Springer-Verlag Berlin Heidelberg 2009

196 7 Correlated Electrons

canbeexpressedintermsofthewavefunctionsψ

(x), where x stands for

space and spin coor dinates. We shall separate the Hamiltonian (

7.1) according

to H = H

+ H

int

into the single-particle and the interaction part. In the

context of this chapter, the latter accounts for electrons in partially ﬁlled

bands only, while the interaction of all other electrons is considered as before

in the eﬀective single-particle potential leading to the energy values ǫ

7.1 Retarded Green Function for Electrons

In this section, we introduce the concept of the Green function for electrons,

which is advantageous in describing electronic properties of the interacting

electrons [

64, 205, 206]. In general, a retarded Green function is an object

deﬁned by (compare with (

2.82))

ret

(t, t

′

)=−

¯h

θ(t − t

′

)[

A(t),

B(t

′

)]

, (7.3)

where ... denotes the thermal expectation value as deﬁned in Sect.

2.32.3

and

[

A(t),

B(t

′

)]

A(t)

B(t

′

) ±

B(t

′

)

A(t). (7.4)

The +(−) sign applies if the operators

A(t)and

B(t

′

) are fermion(b oson)

operators. In Sect.

2.6, we have already identiﬁed the response function, which

is a correlation function for observables, as a special type of retarded Green

function. In the context of this chapter, the choice of the operators,

A(t)=c

(t)and

B(t

′

)=c

†

′

), (7.5)

will be a creation and an annihilation operator of an electron in a single-

particle state with quantum numbers α. Thus, the Green function (we use as

before {...,...} for the anti-commutator [...,...]

)

G(α; t, t

′

)=−

¯h

θ(t − t

′

){c

(t),c

†

′

)} (7.6)

is the probability amplitude to ﬁnd the particle created at time t

′

in the state

α at a later time t>t

′

still in the same state. It describes the propagation of

an electron in the state α from t

′

to t and is, therefore, called a propagator.

In order to calculate the Green function (

7.6), we formulate the equation

of motion by taking the derivative with respect to the time argument t.Itcan

be written as

i¯h

∂G(α; t, t

′

)

∂t

∂θ(t − t

′

)

∂t

{c

(t),c

†

′

)}

−

¯h

θ(t − t

′

)



i¯h

∂c

(t)

∂t

†

′

)



. (7.7)

7.1 Retarded Green Function for Electrons 197

With

∂θ(t − t

′

)

∂t

= δ(t − t

′

) , {c

(t),c

†

(t)} =1, (7.8)

and the equation of motion for c

(t)

i¯h

∂c

(t)

∂t

=[c

(t), H] (7.9)

(here [...,...] denotes the commutator [...,...]

−

) this can be cast into the

form

i¯h

∂G(α; t, t

′

)

∂t

= δ(t − t

′

) −

¯h

θ(t − t

′

){[c

(t), H],c

†

′

)}. (7.10)

The second term on the rhs has again the form (

7.3) of a retarded Green

function, but it has a more complex form. As it will turn out, it contains in

general more than two fermion operators due to the electron–electron inter-

action. One could also formulate, for this higher order Green function, an

equation of motion with a similar structure but with an even more complex

Green function, and so forth. This is a generic hierarchy problem typical for

interacting systems. Concepts of many-particle physics focus on ﬁnding for

this problem approximate solutions, essentially by truncating the hierarchy

at some level.

To become acquainted with the concept of Green functions, let us ﬁrst look

for the simplest solution of (

7.10) which can be given for the noninteracting

electrons, i.e., by considering the Hamiltonian



′

†

′

. (7.11)

Evaluation of the commutator yields

(t), H

]=e

¯h



′

†

′

−

¯h

= ǫ

(t). (7.12)

Here, we have used the fact that the time argument of all operators is the same

and determined by H

and that the commutation relation for the fermion

operators yields [c

†

′

]=c

′

α,α

′

. The equation of motion for the Green

function G

(0)

(α; t, t

′

) of the noninteracting electron

i¯h

∂

∂t

− ǫ

(0)

(α; t, t

′

)=δ(t − t

′

) (7.13)

can be easily integrated to obtain

(0)

(α; t, t

′

)=−

¯h

θ(t − t

′

−

¯h

(t−t

′

)

. (7.14)

After carrying out the Fourier transformation (note the time dependence

t − t

′

= τ)