凝聚态物理导论

有关固体磁性的基本概念和规律在上个世纪电磁学的发展史中就开始建立了。19世纪中期：分子电流为基础——最初关于磁性介质的理论；19世纪后半期：发展了铁磁磁化现象的试验方法——确立磁化规律的基本要素——分子场初步假说和顺磁磁化的Curie定律；20世纪初期：发展了顺磁性的Langevin理论和铁磁性的Weiss理论；20世纪前半期和中期：量子力学的提出和整个物理学的发展——铁磁性，反铁磁性及 铁磁性理论的发展，并发展了许多新的物理实验技术，如电子磁性 ，核磁 及铁磁

等。§6.1

Magnetism

of

atoms(1)

Electronic

states

in

atoms

or

ionsSingle

electron

Hamiltonian(2li+1)-fold

degeneracySpin-orbit

couplingInclusion

of

Coulombinteraction

(CI)(2L+1)(2S+1)-fold

degeneracy(2J+1)-fold

degeneracyL-S

coupling;

(CI>LS)J-J

coupling.

(CI<LS)(2)

Hund’s

rulesThe

rules

that

determine

the

ground

state

of

an

atom.Under

the

condition

that

satisfies

Pauli

exclusion

principle,

Stakes

its

um;Under

the

condition

that

satisfies

Pauli

exclusion

principle,

Stakes

its um

where

L

is

the

largest;If

the

number

ofelectrons

inthe

outer

s is

less

than

the

half-filling,

thenJ=|L-S|;

if

the

number

of

electrons

in

the

outer

sis

larger

than

the

half-filling,

then

J=|L+S|.Cr+3

(3d3):

the

ground

state(S=3/2,

L=3,

J=3/2)2S+1LJ(3)

Atom

in

a

magnetic

fieldp→p+eAIntroduce

a

magnetic

field

B0

along

z

direction,Take

gauge:satisfyingLz(without

inclusion

of

spin)Presume

B0

is

weak,

the

perturbation

energy

up

to

theorder

isE=E0B0=0E=E0+ΔEB0≠0Zeeman

splitting321ML=0-1-2-3Magnetic

moment:Intrinsic

orbitmagnetic

moment(indep.

B0)Induced

magnetic

moment:(dep.

B0)Origin

of

diamagnetismThe

case

including

spin:(Spin

moment-magnetic

fieldinteraction)Total

magnetic

moment:The

perturbation

energy

up

to

theorder

is(Landé

g-factor)Intrinsic

magnetic

moment:§6.2

Magnetism

in

solidsFive

basic

types

of

magnetism

have

been

observed

and

classified

onthe

basis

of

the

magnetic

behavior

of

materials

in

response

to

magneticfields

at

different

temperatures.

These

types

of

magnetism

are:ferromagnetism,

ferrimagnetism,

antiferromagnetism,paramagnetism,

and

diamagnetism.(1)

Diamagnetism

of

saturated

electronic

structuresThe

ionic

and

covalent

solids,

similar

to

noble

gases, have

filledelectron

structures.Paramagnetic

moment:(2)

Paramagnetism

ofbandcarriersElectrons

in

the

conduction

band

of

semiconductors

are

less,

and

onemay

presume

they

satisfy

Boltzmann

statistics.The

average

moment

of

band

electrons:At

room

temperature

T,(3)

Paramagnetism

of

impurities

and

defects(ESR,

EPR)Measuring

gfactorFrom

web(4)

Pauli

Paramagnetism

and

Landau

diamagnetismEFEFH=0H≠0,nonequilibriumH≠0In

Chapter

3,

we

gotPauli

paramagnetismPauliEnergy

levels

of

an

electron

in

a

magnetic

field

are

called

Landau

levels:Landau(5)

Knight

shiftThe

Knight

shift

is

a

shift

in

the

nuclear

magnetic

resonance

frequencyof

a

paramagnetic

substance

published

in

1949

by

the

Americanphysicist.The

Knight

shift

is

due

to

the

conductionelectrons

in

metals.

Theyintroduce

an

"extra"

effective

field

at

the

nuclear

site,

due

to

the

spinorientations

of

the

conduction

electrons

in

the

presence

of

an

externalfield.

This

is

responsible

for

the

shift

observed

in

the

nuclear

magneticone

is

the

Pauliponent

waveresonance.

The

shift

comes

from

two

sources,paramagnetic

spin

susceptibility,

the

other

is

thefunctions

at

the

nucleus.Depending

on

the

electronic

structure,

Knight

shift

may

be

temperaturedependent.

However,

in

metals

which

normally

have

a

broad

featurelesselectronic

density

of

states,

Knight

shifts

are

temperature

independent.Knight

shift:§6.3

Theory

of

paramagnetismCurie

lawCurie-Weiss

lawAtomicmomentLangevin

theory:PierreCurieAverage

moment:(Brillouin

function)Paul

LangevinMarcel

Louis

Brillouin1854-1948Curie

lawCurie

constantVan

Vleck

paramagnetismQuench

of

orbital

angular

momentumBy

Patrik

Fazekas§6.4

Theory

of

ferromagnetismMagneticsMagneticwallsTfθ

θp1/χCurie

lawFMmagnetic

hysteresis

loopMsMrHsHc(1)

Weiss

molecular

field

theory

on

spontaneous

magnetizationTwo

assumptions

for

ferromagnets:The

existence

of

an

internal

field---molecular

field;The

existence

of

magnetic

s.Pierre-Ernest

Weiss(1865-1940)x0T1BJ(x)T2MT3(T3>

T2>

T1)(2)

Paramagnetism

at

high

temperatures(3)

Localized

electron

model

for

spontaneous

magnetizationGeneralization:Werner

Heisenberg#

of

nearestneighborsmagnetizationWeiss分子场的实质来源于原子间的交换作用，而交换作用来源于Pauli不相容原理。(4)

Spin

waves1930年，Bloch基于Heisenberg

model提出了自旋波的概念，用于

在低温下自发磁化强度与温度的关系，得到了M(T)随T3/2变化的规律，这就是著名的Bloch

T3/2定律。Felix

Bloch根据Heisenberg

model，铁磁体的基态是所有自旋沿同一方向排列。在低温下，有一部分自旋将处于激发态，最低的激发态对应于一个自旋反转。由于同近邻的自旋间有耦合，一个自旋的反转必定引起整个系统自旋的不同程度的反转，产生集体激发，这种自旋的集体激发被称为自旋波(spin

wave)，其对应的准粒子为磁振子(magnon).A

boson经典的自旋波理论是利用自旋角动量S在磁场中的进动关系，可以求得一维单原子链的自旋波的色散关系:自旋波的量子理论是利用Holstein-Primakoff变换关系，将自旋算符变换成磁振子算符，即可求出上述色散关系。和声子类似，自旋波得能量是量子化的：在低温时，波数为k的自旋波的平均粒子数:注意到每激发一个磁振子相当于一个自旋反向，则有(D:

spin-wave

stiffness)At

low

TRandy

S.

Fishman

et

al,

Phys.

Rev.

Lett.

99,

157201

(2007).Spin

Wave

Excitations

in

a

Frustrated

Magnet

CuFeO2Due

toantiferromagneticinteractions

betweennearest-neighbor

Fe3+spins

in

each

hexagonalplane,

CuFeO2

is

ageometrically-frustratedantiferromagnet.(5)

Itineran ectron

magnetism

(band

magnetism)金属磁性材料中原子磁矩并不是整数，例如铁是2.21μB

，钴是1.70

μB

，镍是0.6μB

，它们与 原子磁矩的大小相差甚远。局域电子模型不能说明金属磁性材料的磁性，而能带模型却比较成功地说明了金属磁性材料的磁性及原子词句的非整数性。能带理论认为，过渡金属中3d与4s带是交叠在一起的，3d电子虽然存在能带结构，但它们又相域，电子间的交换作用使自旋简并的电子能带发生。考虑电子间交换作用后，能带 成不对称形式，可以看出自旋向上的电子比自旋向下的电子数目多，在3d能带中形成未被抵消的自发磁矩，因而可以发生自发磁化。54.420.583d10According

to

Stoner

model,

the

conditionshould

be

satisfied,

where

U

is

the

on-site

Coulomb

interaction

betweenelectrons.金属磁矩的非整数性可以这样解释：一般认为S带的电子对铁磁性没有贡献，d带贡献的大小依赖于能带的性质。如镍，有10个价电子，饱和磁化说明每个原子只有0.58个电子磁矩，能带模型认为，它是9.42个价电子处在d带，0.58个电子处在s带，9.42个电子中，5个电子自旋向上，4.42个电子自旋向下。这即解释了为什么没个原子只有0.58个电子磁矩。§6.5

Antiferromagnetism

and

ferrimagnetism(1)

Antiferromagnetism相邻磁矩反平行排列，大小相等，方向相反，互相抵消，对外呈现出总磁矩为零。在温度TN时，自发的反平行排列

了，成为Neel温度。在Neel温度以上，

顺磁性。T>TN:Louis

Néel(1904-2000)NT

反铁磁性是靠什么机制产生的呢？Cramer和Anderson先后用超交换模型即使了MnO晶体的铁磁性，超交换作用有时也称为间接交换作用。TχparamagnetismχTferromagnetismTccomplexTχantiferromagnetismTN-θCurieCurie-WeissNeelPhilip

Warren

Anderson(2)

Ferrimagnetism铁磁性实际上是一种特殊的反铁磁性，在研究其自发磁化时，需要将晶格分为两个子晶格，然后按照铁磁性的理论在每个子格子上进行，铁磁体具有

温度。四氧化三铁是典型的

铁磁体，以及其它的铁氧体：Fe(A-Fe)O4型，A=Mn2+、Fe2+、Ni2+、Zn2+。铁磁体具有两个主要特点：(1)有相当大的磁化强度，但比铁磁体里的磁化强度小；(2)这类材料的电阻率都相当大，具有半导体的性质，可用铁氧体材料来制作微波元件等。§6.6

Low-dimensional

magnetismLow

spatial

dimensionality:

D

<

3One-Dimensional

(1D)Systemschainswireszigzagladdersalternating

chainsrandom

chainsTwo-Dimensional

(2D)

Systemssquaretrianglebrick-wallKagomébherringboneTwo-Dimensional

(2D)

Systems¼

depleted

square

latticeRectangular

latticeInterpenetratedb

latticeTurtle

back

lattice?+=

10.8-11.2

AstronMolecular

magnetsMn12-acFe8(1980)V15

:

low

spin

molecule

with

spin

1/2S=1/2J~-800KJ'~J1~-150

KJ''~

J2

~

-300KAFM

couplingNi12-WheelS=12[Ni12(chp)12(O2CMe)12(THF)6(H2O)6]Cr8:

S=0(8

AFMcoupledS=3/2

Cr

centers)[Cr8F8(O2CCMe3]1612

FM

coupled

S

=

1nickel

centresNi24-wheel:AFM,

but

notdiamagneticMn6Cr4Cr8Laboratoire

Louis

Néel,

Wolfgang

Wernsdorfer1D

&

2D

Ising

model

can

be

exactly

solved.(Si

:

up

or

down)1D

exact

solution

(Ising1925):Magnetization:When

H➔0,

M➔0.

No

spontaneous

magnetization

for

T>0.Specific

heat:Susceptibility:Ising

ModelErnst

IsingZero-field

susceptibility

for

Ising

chain

with

spin

½

(Fisher

1963)Specific

heatMeasure

parallel

to

determinethe

sign

of

J.2D

exact

solution

(Onsager

1944)Specific

heat

on

a

square

lattice:T/Tc1It

is

found

that

CH=0(T)

is

logarithmic

divergent

at

T=Tc:M/NgμBMagnetization

(1952):for

T<TcCritical

pointLarsOnsager2D

Ising

model

exhibits

a

phase

transition

at

T>0:OnsagerBlote

et

al

(SC,1969)Critical

exponentsExact

solutions

of

2D

Ising

model

establish

the

foundationof

modern

theory

of

the

critical

phenomenon.Critical

temperatureHeisenberg

ModelExact

resultsHeisenberg

S=1/2

AFM

chain

by

Bethe

ansatz(Bethe

1931,

Hulthen

1938)Energy

of

the

ground

state:For

anisotropic

exchange

integrals

(Orbach

1958,

Walker

1959)Werner

HeisenbergHans

Albrecht

BetheExcitation

energy

(des

Cloizeaux

&

Pearson

1962)xyFor

S=1/2

AFM

chain,the

excitationfrom

the

ground

state

(singlet)

to

theexcited

state

(triplet)

is

gapless:-0Susceptibility

at

T=0

(Griffiths

1964,&1966)Spin-spin

correlation

function

(Shanker

et

al

1990)Power-law

decayLieb,

Schultz

and

Mattis

TheoremConsider

a

1D

AFM

chain

with

the

Hamiltonian,satisfying

the

periodic

boundary

condition

SN+1=S1,

N=even.

ForS=odd

integer/2,

e.g.,

S=1/2,

3/2,

…,

the

excitation

from

theground

state

to

the

excited

state

is

gapless.Ground

state

for

Heisenberg

antiferromagnet(Anderson

1951)Ground

state

for

Heisenberg

ferromagnetFor

the

Heisenberg

ferromagnet

(J<0),

the

fully

ferromagnetic

stateФFM

is

one

of

the

ground

state

multiplet,

whereMagnetization

plateaus

(Oshikawa

et

al

1997)Consider

zero-temperature

quantum

spin

chains

in

a

uniformmagnetic

field,

with

axial

symmetry,For

integer

or

half-integer

spin,

S,the

magnetization

curve

can

haveplateaus,

and

the

magnetizationper

site

m

is

topologicallyzed

as

n(S-m)=

integer

atthe

plateaus,

where

n

is

the

period

1/6of

the

ground

state

determined

bythe

explicit

spatial

structure

ofHamiltonian.mH/JConfirmed

both

theoretically

and

experimentallyHida

1994plateaun=3,

S=1/2(3-site

translation

invariant)Mermin-Wagner

TheoremFor

the

quantum

Heisenberg

model,

with

theshort-range

interactions

satisfying,there

cannotexist

any

magnetic

(including

FM

and

AFM)long-range

order

at

any

nonzero

temperature

in

one

andtwodimensions.CorollaryIf

there

exists

a

gap

from

the

ground

state

to

the

excitedstate,

there

will

be

no

magnetic

LRO

in

1D

and

2D

atzero

temperature.Goldstone

TheoremIf

there

exists

a

magnetic

LRO,

then

the

excitation

fromthe

ground

state

to

the

excited

state

will

be

gapless.Jeffrey

GoldstoneN.

David

MerminMagnetic

Long-Range

OrderHeisenberg

AFModelMagneticLong-RangeOrder

in

theGround

StateMagneticLong-RangeOrder

at

T>0D=1No(owing

to

quantumfluctuations)No(owing

tothermalfluctuations)D=2S=1/2,

Yes(numerical)S≥1,

Yes(rigorous)No(owing

tothermalfluctuations)D=3S=1/2,

YesS

≥1,

Yes(rigorous)S=1/2,

Yes(?)S

≥1,

Yes(rigorous)Heisenberg

Model

on

Square

LatticeModified

Spin-wave

theory

(Takahashi

1989)Uniform

susceptibilityFor

quantum

Heisenberg

chains

with

spin

integer,there

will

be:the

ground

state

is

unique;there

exists

a

gap

between

the

singletground

state

and

the

triplet

excited

state;the

ground-state

spin

correlation

functiondecays

exponentially.Within

the

continum

limit,

Haldane

observedSROCorrelation

lengthΔ

c-1Haldane

gap:c:

spin-wave

velocity~JSHaldane

Scenario

(1983)F.DuncanM.HaldaneAKLT

Model

(1987)S=1

Heisenberg

AFM

spin

chain

with

biquadratic

interactions(Affleck,

Kennedy,

Lieb,

Tasaki)The

ground

state

is

a

valence

bond

state

(VBS)(<i,j>:

nearest

neighbors)It

can

be

proven

thatAs

the

eigenvalues

of

H

0,

the

VBSis

the

unique

singlet

ground

state.They

proved

it

exists

an

excited

gap

from

theground

state:

=0.75JConfirmation

of

Haldane

conjecture!Bilinear-Biquadratic

ModelS=1

Heisenberg

AFM

chain

with

bilinear

and

biquadratic

interactionsCritical

points

separatingHaldane

phase

fromother

phasesΔ=0Haldane

phase:-

/4<

θ

<

/4Δ

=0.411J

at

θ=

0for

S=1,

=6Δ

=0.085J

at

θ=

0for

S=2,

=49Δ

=0Can

half-integer

spin

chains

have

a

gap?(Oshikawa

et

al

1997)Translationally

invariant

spin

chains

in

anapplied

field

can

be

gapful

without

breakingtranslation

symmetry,

onlywhen

themagnetization

per

site,

m,

obeys

S-m=integer.Such

gapped

phases

correspond

to

plateaus

atthese zed

values

of

m.Half-integer

S

spinchainscanhave

“Haldane

gap

phase”

undersome

conditions.S=3/2,

m=1/2Quantum

Phase

TransitionsA

quantum

phasetransition

(QPT)

is

aphase

transition

betweendifferent

quantum

phases(phases

of

matter

at

zerotemperature).QPT