Chapter 8: Magnetism

Holstein & Primakoff

March 28, 2017

Contents

1 Introduction 3

1.1 The Relevance of Magnetostatics . . . . . . . . . . . . . . . . . . . . 3

1.2 Non-interacting Magnetic Systems . . . . . . . . . . . . . . . . . . . . 4

2 Coulombic Correlation Effects 6

2.1 Moment Formation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.2 Magnetism and Intersite Correlations . . . . . . . . . . . . . . . . . 9

2.2.1 The Exchange Interaction Between Localized Spins . . . . . . 10

2.2.2 Exchange Interaction for Delocalized Spins . . . . . . . . . . . 12

3 Band Model of Ferromagnetism 15

3.1 Enhancement of χ . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

3.2 Finite T Behavior of a Band Ferromagnet . . . . . . . . . . . . . . . 19

3.2.1 Effect of B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

4 Mean-Field Theory of Magnetism 22

4.1 Ferromagnetism for localized electrons (MFT) . . . . . . . . . . . . . 22

4.2 Mean-Field Theory of Antiferromagnets . . . . . . . . . . . . . . . . 25

1

5 Spin Waves 30

5.1 Second Quantization of Ferromagnetic Spin Waves . . . . . . . . . . . 34

5.2 Antiferromagnetic Spin Waves . . . . . . . . . . . . . . . . . . . . . . 39

6 Criticality and Exponents 44

2

1 Introduction

Magnetism is one of the most interesting subjects in condensed matter physics. Mag-

netic effects are responsible for heavy fermion behavior, ferromagnetism, antiferro-

magnetism, ferrimagnetism and probably high temperature superconductivity.

Unlike our previous studies, most magnetic systems are not well described by

simple models which ignore intersite correlations. The reason is simple: magnetism

is inherently due to electronic correlations of moments on different sites. As we will

J

Figure 1: Both moment formulation and the correlation between these moments (J)

are due to Coulombic effects

see, systems without these inter-spin correlations (or those without well defined mo-

ments to be correlated) have uninteresting and unimportant (energetically) magnetic

properties.

1.1 The Relevance of Magnetostatics

Perhaps the term magnetism is a misnomer, or rather describes only the external

probe (B) which we use to study magnetic behavior. Magnetic effects are mainly due

to electronic correlations, those mediated or due to Coulombic effects, and not due to

magnetic correlations between moments (these are smaller by orders of v/c, ie., they

are relativistic corrections). For example, consider the magnetic correlation between

3

two moments separated by a couple of Angstroms.

U =1

r3[m1 ·m2 − 3(m1 · n)(m2 · n)] (1)

then,

Udipole−dipole ≈m1m2

r3(2)

If we let,

m1 ∼ m2 ∼ gµB ∼eh

m

r ≈ 2A

then,

U ∼ (gµB)2

r3∼

(e2

hc

)2 (a0

r

)3 e2

a0

∼ 10−4eV (3)

Or roughly one degree Kelvin! Magnetic correlations due to magnetic interactions

would be destroyed by thermal fluctuations at very low temperatures.

1.2 Non-interacting Magnetic Systems

We will define non-interacting magnetic systems as those for which the independent

moments do not interact with each other, and only interact with the probing field.

For the moment let’s consider only the magnetic moments due to electrons (as we

will see, since they can interact with each other, they are by far the most important

moments in the system). They have a moment

≈ µB(L + 2S) (4)

The system energy will change by an amount

∆E ∼ gµBB(L+ 2S) ∼ µBB, µB =eh

2m∼ 5.8× 10−5 eV

Te(5)

in an external field. The largest field which can regularly be produced in a lab is

∼ 10Te (100Te or more can be produced at LANL by blowing things up), thus

∆E<∼ 10−3eV or ∼ kB(10oK) (6)

4

This is a very small energy. Thus magnetic effects are wiped out by thermal fluctua-

tion for

kBT > µBB (7)

at about 10 K! Thus experiments which measure the magnetism of non-interacting

systems must be carried out at low temperatures. These experiments typically mea-

sure the susceptibility of the system with a Faraday balance or a magnetometer

(SQUID).

For a collection of isolated moments (spin 1/2), the susceptibility may be calcu-

lated from the moment

↑ s =1

2〈m〉 = gµB

1

2

(e−β

12gµBB − eβ 1

2gµBB

)(e−β

12gµBB + eβ

12gµBB

) (8)

' gµB2

tanh(βµBB) ≈ µBµBB

T(g ∼ 2)

χ =∂ 〈m〉∂B

≈ µ2B

kBT(9)

Once again, the energy of the moment-field interaction is roughly

E ∼ µ2B

TB2 ∼

10−8(eVTe

)2

T(

10−4eVK

)2B2, (k = 1) (10)

When E ∼ T , thermal fluctuations destroy the orientation of the moments with the

external field. If B ∼ 10Te

E ∼ 100

T

K (11)

or E ∼ T at 10K! However, we know that systems such as iron exist for which a

small field can induce a relatively large moment at room temperature. This is sur-

prising since for a metal, or a free electron gas, the susceptibility is much smaller than

the free electron result, since only the spins near the Fermi surface can participate,

χ =µ2BEF

. Note that this is even smaller than the free electron result by a factor of

kBTEF 1!

5

∼ kT/EF

E

D(E)

Figure 2: In a metal, only the electrons near the Fermi surface, which are not paired

into singlets, contribute to the bulk susceptibility χ ∼ kTEF

µ2BkBT∼ µ2

BD(EF )

2 Coulombic Correlation Effects

2.1 Moment Formation

Of course real materials are not composed of free isolated electrons. Nevertheless

some insulators act almost as if they are composed of non-interacting atoms (ions)

with moments given by Hunds rules: maximum S maximum L which leads to large

atomic moments. Hunds rules reflect the atom’s attempt to lower its Coulombic

energy, see Fig. 3. By maximizing the total spin S, the spin part of the wavefunction

Max S Max L

ψ(x)

x

e- e-

Figure 3: Hunds rules, Maximize S and L, both result from minimizing the Coulomb

energy.

becomes symmetric under electron exchange (i.e. for two electrons with s = 12↑ ↓ the

maximum value of total spin is S = 1 with a wavefunction |↑↓〉+ |↓↑〉 ). Then, since

6

the total wavefunction must be antisymmetric under exchange, the spatial part must

be antisymmetric requiring it to have a node. The node keeps the electrons apart,

minimizing their Coulomb energy. The second Hunds rule is also due to Coulombic

interactions. Maximizing L tends to keep the electrons apart, much like a centrifuge.

(alternatively, the radial Schroedinger equation obtains an angular momentum barrier

L(L+ 1)/r2).

ε

µ B U

Figure 4: A simple tight-binding model with a local Coulomb repulsion U . If U = 0,

the rate that electrons hop on and off any site may be approximated using Fermi’s

golden rule ∼ π|B|2D(EF ) ∼ 1∆t

. Then by the uncertainty principal ∆E∆t ∼ h so

each site energy acquires an uncertainty or width ∆E ∼ h∆t∼ π|B|2D(EF ) ≡ Γ.

The sites will form moments (see Fig. 5) if Γ U, |ε|

To illustrate how band formation modifies this scenerio, lets consider a simple

tight binding model (See Fig. 4). By Fermi’s goldon rule, each level acquires a width

(uncertainty in its energy) Γ = πB2D(EF ) and each level can be in one of the four

states shown in Fig. 5. Clearly, the states −−−−©−−− and −−↑−−↓−− do not have a moment,

moment forms

Γ

ε

µ

2ε+U

Figure 5: A moment forms on an orbital provided that Γ U, |ε|. U ∼ e2

ae− arTF

rTF is small for a metal, large for an insulator

and the states −−↑−− and −−↓−− do. If these states mix equally a moment will not form.

The mixing between the states with moments is only through one of the other two

states (−−−−©−−− or −−↑−−↓−−) and may be suppressed, as can the occupancy of the

7

n

m

LI

e

B

ξ

Figure 6: Here ξ = −1c∂φ∂t

, m = −µBL and φ = Bπa2

moment-less states, by increasing the energy of the states without moments. Ie., a

moment will form on each site if −ε Γ and ε + U Γ.

In this limit U B, the system will act more like a system of free moments than

a free electron gas. Thus, one might expect

χinsulator χmetal (12)

for noninteracting systems. However, this is not the case. The reason is that I have

only told you half of the story. A real atom, or a system composed of such atoms, has

a diamagnetic response due to the angular momentum L of the rotating electrons.

This effect is due to Lenz’ Law. So that any introduced magnetic induction will

induce an EMF and hence a current that opposes the electron current which reduces

the moment. ⇒ diamagnetism with χ ∝ a2. In the free electron limit (see J.M.

Ziman, )

χ = µ2BD(EF )

[1− 1

3

( mm∗

)2]

(13)

∆E ∼ 10−8

(eV

T

)2

a10−1eV −1B2 ∼ 10−9eV B2(T ) (14)

For insulators, often with m∗

m< 1 the diamagnetism wins; whereas for metals, gener-

ally with m∗

m≥ 1, the Pauli paramagnetism wins.

8

a

ξ

Figure 7: If the magnetic moments in a small volume are correlated, then the magnetic

susceptibility is strongly enhanced.

2.2 Magnetism and Intersite Correlations

From both Hund’s rules and a simple tight binding picture, we argued that moment

formation in solids results from local Coulomb correlations between electrons. We

also saw that a collection of such isolated moments is rather boring since all magnetic

behavior is washed out by thermal fluctuations at very low temperatures. Consider

once again an isolated moment of magnitude mµB in an external field.

χ ≈ (mµB)2

kBT(15)

E ≈ (mµBB)2

kBT(16)

For magnetism to be significant at room temperature (300K) we must increase the

energy of our system in a field. This may be accomplished by increasing the effective

moment m by correlating adjacent moments. If the range of this correlation is ξ, so

that roughly 4πξ3

3a3moments are correlated, then let ξ

a= 3 so that ∼ 102 moments

are correlated and m ∼ 102. This increases E by about 104, so that E ∼ kbT at

T ∼ 103K. The observed (measured) susceptibility also then increases by about 104,

all by only correlating moments in a range of 3 lattice spacings.

Clearly correlations between adjacent spins can make magnetism in materials rele-

9

1 2

A

r

r rr r

R

ee

eB

AB

2B1B 2A

12

1A

+e+

Figure 8: Geometry of two electrons, 1 and 2, bound to two ions A and B.

vant. Such correlations are due to electronic effects and are hence usually short ranged

due to electronic (Thomas-Fermi) screening. If we consider two s = 12

spins, ↑1 ↓2,

then the correlation is usually parameterized by the Heisenberg exchange Hamilto-

nian, or

H = −2Jσ1 · σ2 (17)

where J is the exchange splitting between the singlet and triplet energies.|↑ ↑〉|↑ ↑〉 + |↓ ↑〉|↓ ↓〉

Et (18)

|↑ ↓〉 − |↓ ↑〉Es (19)

Et − Es = −J (20)

The trick then is to calculate J !

2.2.1 The Exchange Interaction Between Localized Spins

Imagine that we have two hydrogen atoms A and B which localize two electrons 1

and 2. As these two electrons approach, their spins will become correlated.

H = H1 +H2 +H12 (21)

H1 = − h2

2m∇2 − e2

r1A

− e2

r1B

(22)

10

H12 =e2

r12

+e2

RAB

(23)

As we did in Chap. 1 to describe binding, we will use the atomic wave functions to

approximate the molecular wavefunction ψ12.

ψ12 = (φA(1) + φB(1)) (φA(2) + φB(2))⊗ spin part

= (φA(1)φA(2) + φB(1)φB(2) + φA(1)φB(2) + φA(2)φB(1))

⊗spin part (24)

If e2

r12is strong (it is) then the first two states with both electrons on the same ion are

suppressed, especially if the ions are far apart. Thus we neglect them, and make the

Heitler-London approximation; for example

ψ12 ' (φA(1)φB(2) + φB(1)φA(2))⊗ spin singlet (25)

The spatial wave function is symmetric, and thus appropriate for the spin singlet state

since the total electronic wave function must be antisymmetric. For the symmetric

spin triplet states, the electronic wave function is

ψ12 = (φA(1)φB(2)− φB(1)φA(2))⊗ spin triplet (26)

or

ψ12 = φA(1)φB(2)± φB(1)φA(2)⊗ spin part (27)

The energy of these states may then be calculated by evaluating 〈ψ12|H|ψ12〉〈ψ12|ψ12〉 .

E =〈ψ12|H|ψ12〉〈ψ12|ψ12〉

= 2EI +C ± A

1± S, + singlet ,− triplet (28)

where

EI =

∫d2r1φ

∗A(1)

− h2

2m∇2

1 −e2

r1A

φA(1) < 0 (29)

the Coulomb integral

C = e2

∫d3r1d

3r2

1

RAB

+1

r12

− 1

r2A

− 1

r1B

|φA(1)|2 |φB(2)|2 < 0 (30)

11

the exchange integral

A = e2

∫d3r1d

3r2

1

RAB

+1

r12

− 1

r2A

− 1

r1B

φ∗A(1)φA(2)φB(1)φ∗B(2) (31)

and finally, the overlap integral is

S =

∫d3r1d

3r2φ∗A(1)φA(2)φB(1)φ∗B(2) (0 < S < 1) (32)

All EI , C, A, S ∈ <. So

−J = Et − Es = 2EI +C − A1− S

−

2EI +C + A

1 + S

−J =

C − A1− S

− C + A

1 + S> 0

J = 2A− SC1− S2

< 0 (33)

where the inequality follows since the last two terms in the dominate the integral for

A and in the Heitler-London approximation S 1. Or, for the effective Hamiltonian.

H = −2J σ1 · σ2, J < 0 (34)

Clearly this favors an antiparallel or antiferromagnetic alignment of the spins (See

Fig. 9) since then (classically) σ1 · σ2 < 0 and E < 0, so minimizing the energy. This

type of interaction is clearly appropriate for insulators which may be approximate as

a collection of isolated atoms. Indeed antiferromagnets are generally insulators for

this and other reasons.

H = −2J∑〈ij〉

σi · σj, J < 0 (35)

2.2.2 Exchange Interaction for Delocalized Spins

Ferromagnetism, where adjacent spins tend to align forming a bulk magnetic moment,

is most often seen in conducting metals such as Fe. As we will see in this section,

the Pauli principle, the Coulomb interactions, and the itinerancy of free (metallic)

electrons favors a ferromagnetic (J > 0) exchange interaction.

12

Figure 9:

e

eri

jrV

Figure 10: |↑ ↑〉 triplet-symmetric

Consider two like-spin (triplet) free electrons in a volume V (See Fig. 10). If we

describe the spatial part of their wave function with plane waves, then

ψij =1√2V

eiki·rieikj ·rj − eiki·rjeikj ·ri

=

1√2V

eiki·rieikj ·rj

1− ei(ki−kj)·(ri−rj)

(36)

The probability that the electrons are in volumes d3ri and d3rj is

|ψij|2d3rid3rj =

1

V 21− cos [(ki − kj) · (ri − rj)] d3rid

3rj (37)

As required by the Pauli principle, this probability vanishes when ri = rj. This would

not be the case for electrons in the singlet spin state (if the coulomb interaction

continues to be ignored). Thus there is a hole, called the “exchange hole”, in the

probability density for ri ≈ rj for triplet spin electrons, but not singlet spin ones.

13

Now consider the effects of the electron-ion and the electron-electron coulomb

interactions (See Figure 11). If one electron comes near an ion, it will screen the

e

e

Ze Ze

a

b

+ +

Figure 11: Electron a screens the potential seen by electron b, raising its energy. Any-

thing which keeps pairs of electrons apart, but costs no energy like the exchange hole

for the electronic triplet, will lower the energy of the system. Thus, triplet formation

is favored thermodynamically.

potential of that ion seen by other electrons; thereby raising their energy. Thus the

effect of allowing electrons to approach each other, is to increase the electron-ion

coulomb energy, and of course the electron-electron Coulomb energy. Thus, anything

which keeps them apart without an energy cost, like the exchange hole for triplet spin

electrons, will reduce their energy. As a result, like-spin electrons have lower energy

and are thermodynamically favored ⇒ Ferromagnetism.

e

e

r=O rj

i.e. r ≡ 0i

Figure 12: Geometry to calculate the exchange interaction.

To determine the range of this FM exchange interaction, we must average the

effect over the Fermi sea. If one of the electrons is fixed at the origin (See Figure 12),

14

then the probability that a second is located a distance r away, in a volume element

d3r is

P↑↑(r)d3r = n↑d

3r (1− cos [(ki − kj) · r])︸ ︷︷ ︸ (38)

Fermi sea average

n↑ =1

2n =

1

2

# electrons

volume(39)

In terms of an electronic charge density, this is

ρex(r) =en

2(1− cos [(ki − kj) · r])

=en

2

1− 1

(43k3F )2

∫ kF

o

d3kid3kj

1

2

(eı(ki−kj)·r + e−ı(ki−kj)·r

)=

en

2

1− (

4

3k3F )−2

∫ kF

0

d3kieıki·r

∫ kF

0

d3kjeıkj ·r

=en

2

1− 9

(sin kF r − kF r cos kF r)2

(kF r)6

(40)

Note that both of the exponential terms in the second line are the same, since we

integrate over all ki → −ki & kj → −kj. Since we have only been considering Pauli-

principle effects, the electronic density of spin down electrons remains unchanged.

Thus, the total charge density around the up spin electron fixed at the origin is

ρeff (r) = en

1− 9

2

(sin kF r − kF r cos kF r)2

(kF r)6

(41)

The size of the exchange hole, and the range of the corresponding ferromagnetic

exchange potential, is ∼ 1kF∼ a which is rather short.

3 Band Model of Ferromagnetism

Due to the short range of this potential, its Fourier transform is essentially flat in k.

This fact may be used to construct a band theory of FM where the mean effect of a

spin-up electron is to lower the energy of all other band states of spin up electrons

by a small amount, independent of k.

E↑(k) = E(k)− IN↑N

; I <∼ 1eV (42)

15

1/2

2

1

4

ρ en eff

k rF

Figure 13: Electron density near an electron fixed at the origin. Coulomb effects would

reduce the density for small r further, but would not significantly effect the size of the

exchange hole or the range of the corresponding potential, both ∼ 1/kF .

Likewise for spin down

E↓(k) = E(k)− IN↓N

. (43)

Where I, the stoner parameter, quantifies the exchange hole energy. The relative spin

occupation R is related to the bulk moment

R =(N↑ −N↓)

N, M = µB

(N

V

)R (44)

Then

Eσ(k) = E(k)− I(N↑+N↓)

2N− σIR

2, (σ = ±) (45)

≡ E(k)− σIR2. (46)

If R is finite and real, then we have ferromagnetism.

R =1

N

∑k

1

exp

(E(k)− IR/2− EF )/kBT

+ 1

− 1

exp

(E(k) + IR/2− EF )/kBT

+ 1(47)

For small R, we may expand around E(k) = EF .

f(x− a)− f(x+ a) = −2af ′ − 2

3!a3f ′′′ (48)

16

E

E

f

EF

E

f′

EF

E

f′′ f′′′

Figure 14:

All derivatives will be evaluated at E(k) = EF , so f ′ < 0 and f ′′′ > 0. Thus,

R = −2IR

2

1

N

∑k

∂f

∂E(k)

∣∣∣∣∣EF

− 2

6

(IR

2

)31

N

∑k

∂3f

∂E3(k)

∣∣∣∣∣EF

(49)

This is a quadratic equation in R

−1− I

N

∑k

∂f

∂E(k)

∣∣∣∣∣EF

=1

24I3R2 1

N

∑k

∂3f

∂E3(k)

∣∣∣∣∣EF

(50)

which has a real solution iff

−1− I

N

∑k

∂f

∂E(k)

∣∣∣∣∣EF

> 0 (51)

Or, the derivative of the Fermi function summed over the BZ must be enough to

overcome the -1 and produce a positive result. Clearly this is most likely to happen

at T = 0, where ∂f∂E(k)

∣∣∣EF→ −δ(E − EF )

T = 0, − 1

N

∑k

∂f

∂Ek=

∫dE

V

2ND(E)δ(E − EF ) =

V

2ND(EF ) = D(EF ) (52)

17

So, the condition for FM at T = 0 is ID(EF ) > 1. This is known as the Stoner

criterion. I is essentially flat as a function of the atomic number, thus materials such

1.0

50Z

I (eV)

Fe

Co

Ni

NaLi

Z

1.0

D(E ) (eV )-1F

∼

Figure 15:

as Fe, Co, & Ni with a large D(EF ) are favored to be FM.

3.1 Enhancement of χ

Even those systems without a FM ground state have their susceptibility strongly

enhanced by this mechanism. Let us reconsider the effect of an external field (gS = 1)

on the band energies.

Eσ(k) = E(k)− InσN− µBσB (53)

Then

R = − 1N

∑k

∂f

∂Ek(IR + 2µBB)

= D(EF )(IR + 2µBB) (54)

or as M = µBNVR, we get

M = 2µ2B

N

V

D(EF )

1− ID(EF )B (55)

or

χ =∂M

∂B=

χ0

1− ID(EF )(56)

χ0 = 2µ2BNVD(EF ) (57)

= µ2BD(EF ) (58)

18

Thus, when ID(EF ) <∼ 1, the susceptibility can be considerably enhanced over the

non-interacting result χ0. However, this approximation usually overestimates χ since

it neglects diamagnetic contributions, and spin fluctuations (at T 6= 0). As we will

see, the latter especially are important for estimating Tc.

a

ξ

Figure 16: Spin fluctuations can reduce the total moment within the correlated region,

and even reduce ξ itself. Both effects lead to a reduction in the bulk susceptibility

χ ∼ moment2

3.2 Finite T Behavior of a Band Ferromagnet

In principle, one could start from an ab-initio calculation of the electronic band

structure of E(k) and I, such as Ni, and calculate the temperature dependence of R

(and hence the magnetization) using

R =1

N

∑k

f(Ek −IR

2− µBB0 − EF )− f(Ek +

IR

2+ µBB0 − EF ) (59)

with f(x) = 1eβx+1

. However, this would be pointless since all of the approximations

made to this point have destroyed the quantitative validity of the calculation. How-

19

D(E) s electrons (no moments)

d electrons (with moments)

not correlated

correlated

Figure 17: In metallic Ni, the d-orbitals are compact and hybridize weakly due to

low overlap with the s-orbitals (due to symmetry) and with each other (due to low

overlap). Thus, moments tend to form on the d-orbitals and they contribute narrow

features in the electronic density of states. The s-orbitals hybridize strongly and form

a broad metallic band.

ever, it still retains a qualitative use. For Ni, we can do this by approximating the

very narrow d-electron feature in D(E) as a δ function and performing the integral.

However, only the d-electrons have a strong exchange splitting I and hence only they

will tend to contribute to the magnetization. Thus our D(E) should reflect only the

d-electron contribution, we will accommodate this by setting

D(E) ≈ Cδ(E − EF ), (C < 1) (60)

C, an unknown constant, will be determined by the T = 0 behavior. Then

R = C

f(−IR

2− µBB0)− f(

IR

2+ µBB0)

(61)

Let RC≡ R and Tc = IC

4kB, then if B0 = 0

R =1

exp(−2RTcT

)+ 1− 1

exp(

2RTcT

)+ 1

= tanhRTcT

(62)

If T = 0, then R = 1 = RC

= 1C

n↑−n↓N

. For Ni, the measured ground state magnetiza-

tion per Ni atom is µeffµB

= 0.54 =n↑−n↓N

. Therefore, C = 0.54 =µeffµB

.

For small x, tanh x ' x− 13x3, and for large x

tanhx =sinhx

coshx=ex − e−x

ex + e−x=

1− e−2x

1 + e−2x

= (1− e−2x)(1− e−2x) ' 1− 2e−2x (63)

20

Thus,

R = 1− 2e−2TcT , for T Tc (64)

R =√

3(1− TTC

)12 , for small R or T <∼ Tc (65)

However, neither of the formulas is verified by experiment. The critical exponent

R∼

T/Tc

Eq (64)

Eq(65)

Figure 18:

β = 12

in Eq. 65 is found to be reduced to ≈ 13, and Eq. 64 loses its exponential

form, in real systems. Using more realistic D(E) or values of I will not correct these

problems. Clearly something fundamental is missing from this model (spin waves).

Elementary

Excitations

spin flip

B0

Included

spin wave

Not Included

Figure 19: Local spin-flip excitations, left, due to thermal fluctuations are properly

treated by mean-field like theories such as the one discussed in Secs. 3 and 4. However,

non-local spin fluctuations due to intersite correlations between the spins are neglected

in mean-field theories. These low-energy excitations can fundamentally change the

nature of the transition.

21

3.2.1 Effect of B

If there is an external field B0 6= 0, then

R = tanh

RTc + µBB0/2kB

T

(66)

Or for small R and B0, (or rather, large T Tc.)

R =µB

2kTB0 +

TcTR⇒ R =

µB2k

1

T − TcB0 (67)

Thus since M = µBNVR = CµBN

VR⇒ χ = ∂M

∂B0=

Cµ2B2kV

NT−Tc . This form for χ

χ =Const

T − Tc(68)

is called the Curie-Weiss form which is qualitatively satisfied for T Tc; however,

the values of Const and Tc predicted by band structure are inaccurate. Again, this is

due to the neglect of low-energy excitations.

4 Mean-Field Theory of Magnetism

4.1 Ferromagnetism for localized electrons (MFT)

Some of the rare earth metals or ionic materials with valence d or f electrons are both

ferromagnetic and have largely localized electrons for which the band theory of FM is

inappropriate (A good example is CeSi2−x, with x > 0.2). As we have seen, systems

with localized spins are described by the Heisenberg Hamiltonian.

H = −∑iδ

JiδSi · Siδ − gµBB0

∑i

Si (69)

In general, this Hamiltonian has no solution, and we must resort to (further)

approximation. In this case, we will approximate the field (exchange plus external

magnetic) felt by each spin as the average field due to the neighbors of that spin and

the external field. (See Fig. 21.) Then

22

i δ = 1

δ = 2

δ = 3

δ = 4

Figure 20: Terms in the Heisenberg Hamiltonian H = −∑

iδ JiδSi ·Siδ−gµBB0

∑i Si

Here i refers to the sites and δ refers to the neighbors of site i.

iS

JEach site υ nearest neighbors with exchange interaction J

Figure 21: The mean or average field felt by a spin Si at site i, due to both its neighbors

and the external magnetic field, is 1gµB〈∑

δ JiδSiδ〉 +B0 = Bieff . Where 〈∑

δ JiδSiδ〉is the internal field, due to the neighbors of site i.

H ≈ −∑i

gµBBieff · Si = −∑i

Si ·

∑δ

Jiδ 〈Siδ〉 + gµBB0

(70)

If Jiδ = J is a constant (independent of i and δ) describing the exchange between the

spin at site i and its ν nearest neighbors, then

Bieff =J∑

iδ 〈Siδ〉 + gµBB0

gµB=

Jν

gµB〈S〉 +B0 (71)

M = gµBN

V〈S〉 ; ν = #nn . (72)

23

For a homogeneous, ordered system,

Beff =V

Ng2µ2B

νJM +B0 = BMF +B0 (73)

and

H ≈ −gµBBeff ·∑i

Si (74)

ie., a system of independent spins in a field Beff . The probability that a particular

spin is up, is then

P↑ ∝ e−β(−gµBBeff12) (75)

and

P↓ ∝ e−β(+gµBBeff12) (76)

so, on averageN↓N↑

= e−βgµBBeff (77)

and, since N↑ +N↓ = N

M =1

2gµB

N↑ −N↓V

=1

2gµB

N

Vtanh

(β

2gµBBeff

)(78)

Since tanh is odd and Beff ∝M , this will only have nontrivial solutions if J > 0

(if B0 = 0). If we identify

Ms =N

V

gµB2

; Tc =1

4νJ

k(79)

M

Ms

= tanh

(TcT

M

Ms

)(80)

and again for T = 0,M(T = 0) = Ms, and for T <∼ Tc

M

Ms

'√

3

(1− T

Tc

) 12

(81)

Again, we get the same (wrong) exponent β = 12.

When is this approximation good? When each spin really feels an ”average” field.

Suppose we have an ordered solid, so that

BMF =Jν

2gµB= Breal (82)

24

a = tanh (ba)

y = a

y = tanh (ba)

initial slope = b

Figure 22: Equations of the form a = tanh(ba), i.e. Eq. 80, have nontrivial solutions

(a 6= 0) solutions for all b > 1.

Now, consider one spin flip excitation adjacent to site i only, Fig. 23.

If there are an infinite # of spins then BMF remains unchanged but for ν <∞

Breal =Jν

2gµB

ν − 2

ν6= BMF =

Jν

2gµB. (83)

Clearly, for this approximation to remain valid, we need Breal = BMF , which will only

happen if ν −2ν

= 1 or ν 2. The more nearest neighbors to each spin, the better

MFT is! (This remains true even when we consider other lower energy excitations,

other than a local spin flip, such as spin waves).

4.2 Mean-Field Theory of Antiferromagnets

Oxides of Fe Co Ni and of course Cu often display antiferromagnetic coupling between

the transition-metal d orbitals. Lets assume we have such a magnetic system on a

bipartite lattice composed of two inter-penetrating sublattices, like bcc. We consider

the magnetization of each lattice separately: For example, the central site shown

in Fig. 24 feels a mean field from the ν = 8 near-neighbor spins on the “down”

sublattice. so

25

i

Figure 23: The flip of a single spin adjacent to site i makes a significant change in

the effective exchange field, felt by spin Si, if the site has few nearest neighbors.

"up" sublattice

"down" sublattice

J < 0

Figure 24: Antiferromagnetism (the Neel state) on a bcc lattice is composed of two

interpenetrating sc sublattices lattices.

M+ =1

2gµB

N+

Vtanh

gµB2kT

V

N−g2µ2B

νJM−

(84)

M− = (+↔ −) . . . (85)

where M+ is the magnetization of the up sublattice. These equations have the same

form as that for the ferromagnetic case! We can make a closer analogy by realizing

that N+ = N− and M+ = −M−, so that

M+ =1

2gµB

N+

Vtanh

− V νJM+

2kTN+gµB

, J < 0 (86)

M− = −M+ (87)

26

B = V/(⁄ N g µ )νJMMF

−− 2 2

B

Figure 25:

Again, these equations will saturate at

M+s = −M−

s =1

2gµB

N+

V(88)

soM+

M+s

= tanh

TNT

M+

Ms

(89)

where TN = −14νJkB

Now consider the effect of a small external field B0. This will yield a small increase

or decrease in each sublattice’s magnetization ∆M±.

M+ + ∆M+ =1

2gµB

N+

Vtanh

gµB2kT

[B0 +

V νJ

N−g2µ2B

(M− + ∆M−)]

M− + ∆M− = (+↔ −) . . . (90)

Or, since ddx

tanhx = 1cosh2 x

, then

∆M = ∂M∂Beff

∆Beff

.

∆M = ∆M+ + ∆M− =1

2gµB

N+

V

gµB2kT

1

cosh2 x

[B0 +

V νJ

2N−g2µ2B

∆M

](91)

where x = TNT

M+

M+s

. For T > TN ,M+ = 0 and so x = 0, and

∆M = 2g2µ2

BN

8V kBT

[B0 −

4kBTNV

Ng2µ2B

∆M

](92)

T∆M =g2µ2

BN

4V kBB0 −∆MTN (93)

27

∆M =g2µ2

BN

4V kB(T + TN)B0 (94)

χ =g2µ2

BN

4V kB(T + TN)(95)

−T T

χ

N N

T0

Figure 26: Sketch of χ = Const/(T + TN). Unlike the ferromagnetic case, the bulk

susceptibility χ does not diverge at the transition. However, as we will see, this

equation only applies for the paramagnetic state (T > TN), and even here, there are

important corrections.

Below the transition, T < TN , the susceptibility displays different behaviors de-

pending upon the orientation of the applied field. For T TN and a small B0 paral-

lel to the axis of the sublattice magnetization, we can approximate M+(T ) ≈ M+s

and x ≈ TNT

in Eq. 91

B0

Figure 27: When T TN , a weak field applied parallel to the sublattice magnetization

axis only weakly perturbs the spins. Here M+(T ) ≈ M+s and x ≈ TN

T

28

χ ' g2µ2BN

4V kB

1

T cosh2(TNT

)+ TN

(96)

χ ' g2µ2BN

4V kBe−2

TNT (97)

Now consider the case where B0 is perpendicular to the magnetic axis. The

B0

α

B0

BMF

Figure 28: When T TN , a weak field B0 applied perpendicular to the sublattice

magnetization, can still cause a rotation of each spin by an angle proportional to

B0/BMF .

external field will cause each spin to rotate a small angle α. (See Fig. 28) The energy

of each spin in this external field and the mean field ∝ ν JgµB

to the first order in B0

is

E = −1

2gµBB0 sin ∝ +

1

2νJ cosα (98)

Equilibrium is obtained when ∂E∂α

= 0. Since B0 is taken as small, α 1.

E ∼ −1

2gµBB0α +

1

2νJ

(1− 1

2α2

)(99)

or∂E

∂α= 0 =

1

2gµBB0 +

1

2να ⇒ α = −gµBB0

νJ(100)

The induced magnetization is then

∆M =1

2

gµBB0

Vα = −g

2µ2BNB0

2νJV(101)

so

χ⊥ =g2µ2

BN

2ν |J |V= constant (102)

Of course, in general, in a powdered sample, the susceptibility will reflect an average

of the two forms, see for example Fig. 30.

29

T

χ

TN

χ

χ

χ⊥

T

χ powdered sample

Figure 29: Below the Neel transition, the lattice responds very differently to a field

applied parallel or perpendicular to the sublattice magnetization. However, in a pow-

dered sample, or for a field applied in an arbitrary direction, the susceptibility looks

something like the sketch on the right.

5 Spin Waves

We have discussed the failings of our mean-field approaches to magnetism in terms

of their inability to account for low-energy processes, such as the flipping of spins.

(Sα → −Sα) However, we have yet to discuss the lowest energy spin flip processes

which are spin waves.

We will approach spin-waves two ways. First following Ibach and Luth we will

determine a spin wave in a ferromagnet. Second, we will argue that they should be

quantized and then introduce a (canonical) transformation to a Boson representation.

Consider a ferromagnetic Heisenberg Model

H = −J∑iδ

Si · Si+δ (103)

where Si = xSxi + ySyi + zSzi . If we define |α〉 =

1

0

(i.e. |↑〉), β =

0

1

(i.e. |↓〉)

so that

Sz

0

1

= −1

2

0

1

· · · (104)

and

Sx =1

2

0 −ii 0

Sy =1

2

0 1

1 0

Sz =1

2

1 0

0 −1

(105)

30

959085807570656055504540353025201510500.0e+0

1.0e-6

2.0e-6

3.0e-6

4.0e-6

5.0e-6

6.0e-6

7.0e-6

8.0e-6

9.0e-6

1.0e-5

1.1e-5

1.2e-5

T (K)

Figure 30: High temperature superconductor Y123 with 25% Fe substituted on Cu,

courtesy W. Joiner, data from a SQUID magnetometer.

[Sα, Sβ

]= iεαβγS

γ (106)

It is often convenient to introduce spin lowering and raising operators S− and S+.

S+ = Sx + iSy =

0 1

0 0

[Sz, S±

]= ±S+/− (107)

S− = Sx − iSy =

0 0

1 0

[S2, S±

]= 0 (108)

S+

0

1

=

1

0

, S−

0

1

= 0 . . . (109)

They allow us to rewrite H as

H = −J∑iδ

Szi Szi+δ +

1

2

(S+i S−i+δ + S−S

+i+δ

)(110)

Since J > 0, the ground state is composed of all spins oriented, for example

|0〉 = Πi |α〉i (i.e. all up) (111)

31

This is an eigenstate of H, since

S+i S−i+δ |0〉 = 0 (112)

and

Szi Szi+δ |0〉 =

1

4|0〉 (113)

so

H |0〉 = −1

4JνN |0〉 ≡ E0 |0〉 (114)

where N is the number of spins each with ν nearest neighbors.

Now consider a spin-flip excitation. (See Fig. 31)

j

Figure 31: A single local spin-flip excitation of a ferromagnetic system. The resulting

state is not an eigenstate of the Heisenberg Hamiltonian.

|↓j〉 = S−j Πn |α〉n (115)

This is not an eigenstate since the Hamiltonian operator S+j S−j+δ will move the flipped

spin to an adjacent site, and hence create another state.

However, if we delocalize this spin-flip excitation, then we can create a lower

energy excitation (due to the non-linear nature of the inter-spin potential) which is

an eigenstate. Consider the state

j

Figure 32: If we spread out the spin-flip over a wider region then we can create a

lower energy excitation. A spin-wave is the completely delocalized analog of this with

one net spin flip.

32

|k〉 =1√N

∑j

eik·rj |↓j〉 . (116)

It is an eigenstate. Consider:

H |k〉 =1√N

∑j

eik·rj−1

4νJ(N − 2) |↓j〉

+1

2νJ |↓j〉 −

1

2J∑δ

(|↓j+δ〉+ |↓j−δ〉)

(117)

where the sum in the last two terms on the right is over the near-neighbors δ to site

j. The last two terms may be rewritten:

1√N

∑j

eik·rj |↓j+δ〉 = 1√N∑m

eik·(rm−rδ) |↓m〉 (118)

where

rj+δ = rj + rδ = rm (119)

so

H |k〉 =

−1

4νJN + νJ − 1

2J∑δ

(eik·rδ + e−ik·rδ

) 1√N

∑j

eik·rj |↓j〉 (120)

Thus |k〉 is an eigenstate with an eigenvalue

E = E0 + Jν

1− 1

ν

∑δ

cos k · rδ

(121)

Apparently the energy of the excitation described by |k〉 vanishes as k→ 0.

What is |k〉? First, consider

Sz |k〉 =∑i

Szi |k〉 =∑i

Szi1√N

∑j

eik· rj |↓j〉

=1√N

∑j

eik·rj∑i

Szi |↓j〉 = (SN − 1) |k〉 (122)

I.e., it is an excitation of the ground state with one spin flipped. Apparently, since

Ek=0 = E0, the energy to flip a spin in this way vanishes as k → 0.

33

Sz n

32

0

12

1

−12

2

−32

3

Table 1: The correspondence between Sz and the number of spin-wave excitations on

a site with S = 3/2.

5.1 Second Quantization of Ferromagnetic Spin Waves

In the ground state all of the spins are up. If we flip a spin, using a spin-wave

excitation, then

Sz |k〉 = (SN − 1) |k〉 , Sz |0〉 = SN |0〉 (123)

If we add another spin wave, then 〈Sz〉 = (SN − 2). For spin 12, 〈Sz〉 = N

2−n, where

n is the number of the spin waves. Since Sz is quantized, so must be the number of

spin waves in each mode. Thus, we may describe spin waves using Boson creation

and annihilation operators a† and a. By specifying the number nk excitations in each

mode k, the corresponding excited spin state can be described by a Boson state vector

|n1n2 . . . nN〉We can introduce creation and anhialation operators to describe the spin excita-

tions on each site. Suppose, in the ground state, the spin is saturated in the state

Sz = S, then n = 0. If Sz = S − 1, then n = 1, and so on. Apparently

Szi = S − a†iai

S+i ∝ ai

S−i ∝ a†i (124)

If these excitations are Boselike, then[ai, a

†i

]= 1 (125)

34

ai |n〉 =√ni |n− 1〉 (126)

a†i |n〉 =√ni + 1 |n+ 1〉 (127)

This transformation is faithful (canonical) and will maintain the dynamical properties

of the system (given by ∂∂tθ = ih [H, θ]) if it preserves the commutator algebra[

S+i , S

−i

]= 2Szi ,

[S−i , S

zi

]= 2S−i ,

[S+i , S

zi

]= −2S+

i (128)

consider S+S− − S−S+

|n〉 = 2Sz |n〉 = 2(S − n) |n〉 (129)

If S+ = a, and S− = a†, then the left-hand side of the above equation would be

(n+ 1)− n |n〉 = |n〉 6= 2(S − n) |n〉. In order to maintain the commutators, we

need

S+ =√

2S − n a S− = a†√

2S − n (130)

Then

[S+, S−] |n〉 = S+S− |n〉 − S−S+ |n〉

=√

2S − a†aaa†√

2S − a†a |n〉 − a†(2S − a†a)a |n〉

= (2S − n)(n+ 1) |n〉 − n(2S − (n− 1)) |n〉

= (2Sn+ 2S − n2 − n− 2Sn+ n2 − n) |n〉

= 2(S − n) |n〉 (131)

You can check that this transformation preserves the other commutators. Of course,

we need one other constraint, since −S ≤ Sz ≤ S, we also must have

n ≤ 2S (132)

This transformation

S+i =

√2S − a†iaiai S−i = a†i

√2S − a†iai Szi = S − a†iai (133)

is called the Holstein Primakoff transformation.

35

If we Fourier transform these operators,

a†i =1√N

∑k

eik·Ria†k; ai =1√N

∑k

e−ik·Riak (134)

then (since the Fourier transform is unitary) these new operators satisfy the same

commutation relations[ak, a

†k′

]= δkk′

[a†k, a

†k′

]= [ak, ak′ ] = 0 (135)

To convert the Hamiltonian into this form, assume the number of magnons in each

mode is small and expand

S+i =

√2S − niai '

√2S(1− ni

4S)ai (136)

≈

1√N

∑k

eik·Riak −1

4SN32

∑kpq

ei(p+q−k)·Ria†kapaq

Of course this is only exact for ni 2S, i.e. for low T where there are few spin

excitations, and large S (the classical spin limit).

In this limit

S+i '

√2S

N

∑k

eik·Riak (137)

S−i '√

2S

N

∑k

e−ik·Ria†k (138)

Szi = S − 1

N

∑kk′

ei(k−k′)·Ria†kak′ , (139)

the Hamiltonian

H = −J∑iδ

Szi S

zi+δ +

1

2(S+

i S−i+δ + S−i S

+i+δ)

(140)

may be approximated as

H ' −NJνS2 + 2JνS∑k

a†kak

−2JνS∑k

(1

ν

∑δ

eik· Rδ

)a†kak +O(a4

k) (141)

36

H ' E0 +∑k

2JνS(1− γk)a†kak +O(a4k) (142)

where γk = 1ν

∑δ e

ikRδ . This is the Hamiltonian of a collection of harmonic oscillators

k k′

k-qk′+q k k′

Figure 33: The fourth order correction to Eq. 142 corresponds to interactions between

the spin waves, giving them a finite lifetime

plus some other term of order O(a4k) which corresponds to interactions between the

spin waves. These interactions are a result of our definition of a spin-wave as an

itinerant spin flip in an otherwise perfect ferromagnet. Once we have one magnon,

another cannot be created in a “perfect” ferromagnetic background.

Clearly if the number of such excitations is small (T small) and S is large, then our

approximation should be valid. Furthermore, since these are the lowest energy excita-

tions of our spin system, they should dominate the low-T thermodynamic properties

of the system such as the specific heat and the magnetization. Consider

〈E〉 =∑k

hωkeβhωk − 1

. (143)

For small k, hωk = 2JνS(1 − γk) = 2JνSk2 on a cubic lattice. Then let’s assume

that the k-space is isotropic, so that d3k ∼ k2dk, then

γk =2

ν(cos kx + cos ky + · · · ) = 1− k2

ν(144)

and

〈E〉 ≈∑k

2JνSk2

eβ2JνSk2 − 1∝∫ ∞

0

k4dk

eβαk2 − 1(145)

x = βαk2 k =

(x

βα

) 12

dk =1

2

(1

βα

) 12

x−12dx (146)

37

so that

〈E〉 ∝ β−2β−1/2

∫ ∞0

dxx3/2

ex − 1∝ T 5/2 (147)

Thus, the specific heat at constant volume CV ∝ T 3/2, which is in agreement with

experiment.

M/M(0)

T

1 - T 3/2

Figure 34: The magnetization in a ferromagnet versus temperature. At low tempera-

tures, the spin waves reduce the magnetization by a factor proportional to T 3/2, which

dominates the reduction due to local spin fluctuations, derived from our mean-field

theory. This result is also consistent with experiment.

If we increase the temperature from zero, then the change in the magnetization is

proportional to the number of magnons generated

M(0)−M(T ) =

⟨∑k

nk

⟩gµBV

(148)

since each magnon corresponds to spin flip. Thus

M(T )−M(0) ∼ −∫

k2dk

eβαk2 − 1∼ T

32 (149)

which clearly dominates the exponential form found in MFT (1− 2e2TcT ). This is also

consistent with experiment!

38

5.2 Antiferromagnetic Spin Waves

Since the antiferromagnetic ground state is unknown, the spin wave theory will per-

turb around the Neel mean-field state in which there are both a spin up and down

sublattices. Spin operators can then be written in terms of the Boson creation and

up sublattice

down sublattice

Figure 35: To formulate an antiferromagnetic spin-wave theory, we once again con-

sider a bipartite lattice, which may be decomposed into interpenetrating spin up and

spin down sublattices.

annihilation operators as before

“up” sublattice “down” sublattice

Szi = S − ni Szi = −S + ni (150)

S+i =

(S−i)+

=√

2Sfi(S)ai S+i =

(S−i)+

=√

2Sa†ifi(S)

where

fi(S) =

√1− ni

2Sand ni = a†iai (151)

Again this transformation is exact (canonical) within the manifold of allowed states

0 ≤ ni ≤ 2S ⇔ −S ≤ Sz ≤ S . (152)

The Hamiltonian

H = −J∑iδ

Szi Szi+δ =

1

2

(S+i S−i+δ + S−i S

+i+δ

)(153)

39

may be rewritten in terms of Boson operators as

H = +JS2Nν + J∑iδ

a†iaia†i+δai+δ

− JS∑iδ

a†iai + a†i+δai+δ

+ fi(S)aifi+δ(S)ai+δ + a†ifi(S)a†i+δfi(S)

(154)

Once again, we will expand

fi(S) =

√1− ni

2S= 1− ni

4S− n2

i

32S2− · · · (155)

and include terms in H only to O(a2)

H ' JS2Nν − JS∑iδ

a†iai + a†i+δai+δ + aiai+δ + a†ia

†i+δ

(156)

This Hamiltonian may be diagonalized using a Fourier transform

ai =1√N

∑k

e−ik·Riak (157)

and the Bogoliubov transform

ak = αk coshuk − α†−k sinhuk (158)

a†k = α†k coshuk − α−k sinhuk (159)

tanh 2uk = −γk (160)

γk =1

ν

∑δ

eik· Rδ (161)

Here the αk are also Boson operators[αk, α

†k

]= 1. To see if this transform is

canonical, we must ensure that the commutators are preserved.

1 = [ak, a†k′ ] = [αkCk − α†−kSk, α

†k′Ck′ − α−k′Sk′ ]

=C2k [αk, α

†k] + S2

k [α†−k, α−k]

δkk′ (162)

=C2k − S2

k

δkk′ = δkk′

40

where Ck (Sk) is shorthand for coshuk (sinhuk). You should check that the other

relations, [ak, ak′ ] = [a†k, a†k′ ] = 0, are preserved.

After this transformation,

H ≈ JNνS(S + 1) +∑k

hωk

(α†kαk +

1

2

)+O(a4) (163)

where hωk = −2JSν√

1− γ2k. Notice that for small k, hωk ∼

√2JSνk ≡ Ck

(C = −√

2JSν is the spin-wave velocity).

The ground state energy of this system (no magnons), is

E0 = JNνS(S + 1)− JSν∑k

√1− γ2

k (164)

If γk = 0, then each spin decouples from the fluctuations of its neighbors and E0 =

JNνS2 (J < 0) which is the energy of the Neel state. However, since γk 6= 0, the

E = JNνSN2

Figure 36: The Neel state of an antiferromagnetic lattice. Due to zero point motion,

this is not the ground state of the Heisenberg Hamiltonian when J < 0 and S is finite.

ground state energy E0 < EN . Thus the ground state is not the Neel state, and is

thus not composed of perfectly antiparallel aligned spins. Each sublattice has a small

amount of disorder ∝ 〈ni〉+ 1/2 in its spin alignment.

The linear dispersion of the antiferromagnet means that its bulk thermodynamic

41

properties will emulate those of a phonon lattice. For example

〈E〉 '∑k

hωkeβhωk − 1

'∑k

αk

eβαk − 1

∼∫ ∞

0

αk3dk

eβαk − 1

〈E〉 ∼ T 4

∫ ∞0

x3dx

ex − 1(165)

C =∂ 〈E〉∂T

∼ T 3 like phonons! (166)

Which means that a calorimeter experiment cannot distinguish phonon and magnon

excitations of an antiferromagnet.

Ef

θn E i

thermal spin-polarized neutrons

dθdΩdω

∝ S(k, ω) ∝ I F(-i⟨[a(t),a (0)]⟩)†n∼

2θ ∝ k = k − ki j

h ω = E − Ei f

sample

Figure 37: Polarized neutrons are used for two reasons. First if we look at only spin

flip events, then we can discriminate between phonon and magnon contributions to

S(k, ω). Second the dispersion may be anisotropic, so excitations with orthogonal

polarizations may disperse differently.

Therefore, perhaps the most distinctive experiment one may perform on an anti-

ferromagnet is inelastic neutron scattering. If spin-polarized neutrons are scattered

from a sample, then only those with flipped spins have created a magnon. If the

neutron creates a phonon, then its spin remains unchanged. The time of flight of the

neutron allows us to determine the energy loss or gain of the neutron. Thus, if we plot

the differential cross section of neutrons with flipped spins, we learn about magnon

dispersion and lifetime. Notice that the peak in S(k, ω) has a width. This is not just

due to the instrumental resolution of the experiment; rather it also reflects the fact

42

γk

ωk

ω

S(k,ω)

phonon junk subtracted off

magnon

n↑

n↓

2θ

⟨ω ⟩k

Figure 38: Sketch of neutron structure factor from scattering off of a magnetic system.

The spin-wave peak is centered on the magnon dispersion. It has a width due to the

finite lifetime of magnon excitations.

that magnons have a finite life time δt,which broadens their neutron signature by γk.

γkδt ∼ h δt ∼ 1

γk(167)

However in the quadratic spin wave approximation the lifetime of the modes hωk is

infinite. It is the neglected terms in H, of order O(a4) and higher which give the

magnons a finite lifetime.

a a a a ⇒i i i+δ i+δ† †

Figure 39:

43

6 Criticality and Exponents

Before reading this section, please read Ken Wilson’s [ Scientific American, 241, 158

(Aug, 1979)] paper on renormalization group (RG). From the this paper, you have

learned a number of new ideas that serve as the basic paradigms of Fisher scaling

which leads to a better understanding of criticality in terms of critical exponents.

1. Renormalization Group (RG) operates in the space of all possible Hamiltonian

parameters K = (K1, K2, K3 · · · ) which renormalize such that

K ′ = R(K)

where R depends on the system, the RG transform employed (since RG is not

unique) and the renormalization scale b.

2. At the fixed point K = K∗

K∗ = R(K∗)

3. Transitions are indicated by an unstable fixed point.

4. At or near a transition, the parameters K = (K1, K2, K3 · · · ) change very slowly

under repeated RG transformations (critical slowing down).

If R is analytic at K∗, then for small |K − K∗| we may linearlize, so that

K ′α −K∗α =∑β

Tαβ(Kβ −K∗β)

very close to K∗where the T-matrix

Tαβ =dK ′αdKβ

∣∣∣∣K=K∗

has real eigenvalues (symmetric, real). Let λ(i)and φ(i)α be the left eigenvalues and

eigenvectors of T ∑α

φ(i)α Tαβ = λ(i)φ

(i)β .

44

We define the scaling variables ui =∑

α φ(i)α (Kα −K∗α) which transform multiplica-

tively, so that

u′i =∑α

φiα (K ′α −K∗α)

=∑α

φ(i)α

∑β

Tαβ(Kβ −K∗β

)=

∑β

λ(i)φ(i)β

(Kβ −K∗β

)= λ(i)ui

This means that under repeated RG transforms, ui grows or shrinks depending on

the size of λ(i). If

• λ(i) > 1, then the variable u i is relevant since repeated RG transforms will

make it grow movig the system away from the fixed point.

• λ(i) < 1, then it is irrelevant since RG transforms leave it at the fixed point.

• λ(i) = 1, then it is marginal.

We can use this to describe the scaling behavior. The RG transformation is con-

structed to keep the partition function Z the same, and hence to also keep the free

energy the same. So, if f = F/N is the free energy per site, then

f(K) = f0(K) + b−Dfs(K′)

where f0 is the analytic part (no transitions and boring) and fs is the non-analytic part

which describes the phase transformation. So, the singular part scales like fs(K) =

b−Dfs(K′). Or, in terms of the scaling variables (fields)

fs (u1, u2, · · · ) = b−Dfs (λ1u1, λ2u2, )

= b−nDfs (λn1u1, λn2u2, )

where the second line is after n RG steps and includes relevant variables only so that

λ1 > 1, λ2 > 1, etc. so the arguments grow with n. Since n is arbitrary, we may

45

eliminate it by defining A, so that λn1u1 = A 1 (where the inequality ensures that

the linear approximation still holds). Then n ln(λ1) + ln(u1) = ln(A), which we use

to eliminate n, so that

fs(u1, u2 · · · ) = b−D lnA

lnλ1 bD

lnu1lnλ1 fs

(A, , λ

lnAlnλ1

− lnu1lnλ1

2 , · · ·)

Now, to make better contact with conventional notation, we define y1 = lnλ1/ ln b,

y2 = lnλ2/ ln b, etc. so that λ1 = by1 etc. These y are called the scaling exponents.

Generally, we define the relevant variables such that u1 = t = (T −Tc)/Tc, u2 = h, or

u3 = 1/L, which are the reduced temperature, field and inverse system size, which are

all obviously relevant. The corresponding set yj are related to the critical exponents

ν, γ, β · · · .One of the most important applications of these ideas is in a finite-sized scaling

analysis. Suppose we have a simulation (e.g., Monte Carlo or Molecular Dynamics)

of a system of size L and spatial dimension D. Then 1/L must be a relevant field uj

since if L is finite there can be no transitions, since they require that the correlation

length ξ diverge. Suppose, for the sake of an example, that the other relevant fields

are t and h. Under a scale transformation of size b, fs must be scale invariant so that

fs(u1, u2, L

−1)

= b−Dfs(by1u1, b

y2u2, b1L−1

)where L−1 is not only relevant, it must scale with exponent yL = 1 as shown. Let’s

identify, the other relevant variables as discussed, so that

fs(t, h, L−1

)= b−Dfs

(by1t, by2h, bL−L

)This form must be true for any b, L−1, h or t close to the transition (in the scaling

region). So, let’s choose b = L to eliminate a parameter, so that

fs(t, h, L−1

)= b−Dfs (by1t, by2h, 1)

46

Figure 40: Scaled susceptibility

versus t = |T − Tc| /Tc. By

scaling the susceptibility with an

appropriate power of the cluster

size, the curves for different clus-

ter sizes L may be made to cross at

the transition, identifying Tc and

γ/ν. A similar finite size scaling

ansatz exists for the specific heat

and yields another estimate for Tc

and ν

Suppose we want to calculate the susceptibil-

ity χ to the applied field h. It involves two deriva-

tives with respect to h at h = 0)

χ = L−D+2y2g (Ly1t, 0, 1)

where g is the second derivative of fs function.

Now, some of these exponents have well known

identities that we may as well use, such as ξ =

ξ0 |t|−1/y1 so that ν = 1/y1, χ ∝ t−γ with the

hyperscaling relation γ = (2y2 −D)/y1, so that

χ ∝ Lγ/νg(L1/νt, 0, 1)

We may apply a similar analysis to the specific

heat, which is proportional to a second derivative

of fs with respect to t, again for h = 0, so that

C ∝ L2/ν−Dg′(tL1/ν , 0, 1)

Now suppose we calculate χ an C for several different clusters of size L. L must

be large enough so that the system is in linear scaling region of the phase transition.

Then we notice that since χ ∝ Lγ/νg(L1/νt, 0, 1), so that if we plot L−γ/νχ versus t,

then the curves for different L must all have the same value and cross at the transition

where T = Tc so that t = 0, and g(0, 0, 1) is a constant (Fig. 6). This allows us to

obtain an estimate for Tc and γ/ν. Likewise, if I plot LD−2/νC versus t, then again the

curves for different L must cross at the transition where g′(0, 0, 1) is constant. This

produces another estimate for Tc and for ν, which when combined with the scaling

for χ gives γ.

47