Paradigm of Condensed Matter Theory
Theory of Quantum Magnetism
Tao Xiang http://www.itp.ac.cn/~txiang/
11 September 2006

Acknowledgement
In this lecture, I have used many pictures downloaded from Internet. I am very grateful to the authors of these pictures, although I do not even known their names in many cases.

Magnetism Is an Evergreen Tree of Science
The study of magnetism as a cooperative phenomena has been responsible for the most significant advances in the theory of thermodynamic phase transitions. This has transformed statistical mechanics into one of the sharpest and most significant tools for the study of condensed matter.

Han Dynasty
Chinese Compass
Magnets: Ancient Gift
China
• 4000 BC magnetite 磁铁矿
• 3000-2500 BC meteoric iron
Greek
• 800 BC lodestone 磁石
Fe
3
O
4
Magnetic memory

Outset of Modern Theory of Magnetism
• Pierre Curie discovered the Curie law of paramagnetic materials and Curie transition temperature
 
C
T
FeBr(C
44
H
28
N
4
)
Pierre Curie
1903 Nobel Prize
1859-1906

Classical Theory of Paramagnetism
Energy of a dipole
E
  

H
Probability of a dipole in energy E
Average of dipole orientation p ( E )
 e

E / k
B
T dn
 d
 sin
 e

H cos

/ k
B
T
Magnetization:
M

N


 coth

H k
B
T
 k
B
T

H


Paul Langevin
1872-1946
Susceptibility
 
M
H
1
~
T

paramagnetic minerals
Olivine (Fe,Mg)
2
橄榄石
SiO
4
Montmorillonite (clay)
蒙脱石（粘土）
Siderite (FeCO
3
菱铁矿 , 陨铁
)
Serpentinite
Mg
3
Si
蛇纹岩
2
O
5
(OH)
4
Chromite (FeCr
2
铬铁矿
O
4
) diamagnetic minerals
Quartz (SiO
2
石英
)
Calcite (CaCO
3
)
方解石
Graphite (C)
石墨
Halite (NaCl)
岩盐
Sphalerite (ZnS)
闪锌矿

Theory of Molecular Field
• 1907 Pierre Weiss formulated the first modern theory of magnetism: molecular field
--- first self-consistent mean-field theory
 
T
C

T c 
EuO
 
1
Pierre Weiss
1864-1940

Bohr-van Leeuwen Theorem
Niels Bohr
1885-1962
At any finite temperature, and in all finite applied electrical or magnetic fields, the net magnetization of a collection of electrons
(orbital currents) in thermal equilibrium vanishes identically.
Classic Theory of magnetism is irrelevant and Quantum Theory is needed!
Van Vleck
1977 Nobel

Spin: Origin of Magnetic Moment
• Electron spin
• Ion spins: Hund’s Rule
Paul Dirac
↑ ↑ ↑
George Uhlenbeck
Cr3+: 3d 3
1st rule: S=3/2
2nd rule: L=3
S-O coupling: J = 3/2
S.A. Goudsmit
↑ ↓ ↑ ↑ ↑ ↑ Fe: 3d6
S = 2, L = 2, J = L+S = 4

79 elements are magnetic in atomic state
H
Li Be
Na Mg
He
B C N O F Ne
Al Si P S Cl Ar
K Ca Sc Ti V Cr Mn Fe Co Ni Cu Zn Ga Ge As Se Br Kr
Rb Sr Y Zr Nb Mo Tc Ru Rh Pd Ag Cd In Sn Sb Te I Xe
Cs Ba La Hf Ta W Re Os Ir Pt Au Hg Tl Pb Bi Po At Rn
Fr Ra Ac
Ce Pr Nd Pm Sm Eu Gd Tb Dy Ho Er Tm Yb Lu
Th Pa U Np Pu Am Cm Bk Cf Es Fm Md No Lw

15 elements are magnetically ordered in the solid state
H
Li Be
Na Mg
K Ca Sc Ti V
Rb Sr Y Zr Nb Mo Tc Ru Rh Pd Ag Cd In Sn Sb Te I
Cs Ba La Hf Ta W Re Os Ir Pt Au Hg Tl Pb Bi Po At
Xe
Rn
Fr Ra Ac
B C
Al Si P
He
N O F Ne
S Cl Ar
Cr Mn Fe Co Ni Cu Zn Ga Ge As Se Br Kr
Ce Pr Nd Pm Sm Eu Gd Tb Dy Ho Er Tm Yb Lu
Th Pa U Np Pu Am Cm Bk Cf Es Fm Md No Lw

Quantized Langevin Theory
M

N
J z
J 
 
J
 g

B
J exp

 g

B
JH
J z
J 
 
J exp(
 g

B
JH
/ k
B
T

/ k
B
T )

N

B gJ f

 g

B
HJ k
B
T


J
0
-1
-2
J z
2
1
Curie Law:
 m
( T )

M
H

1
T

Pauli Paramagnetism of Metal
M
 
B
 n
 
B
2
  
F
H

( T )

M
H
 
0

B
2
  
F
Density of states
1
2
  
F

E

2

B
H
Wolfgang Pauli

Different types of collective magnetism
BCC Iron
Ferromagnet
MnF
2
Antiferromagnet
Paramagnet
GdCo
5
Ferrimagnet
MnO
Antiferromagnet
Er
6
Mn
23

Different types of collective magnetism

Susceptibility

H

J
 ij i
 j
Heisenberg
Nobel 1932
• J < 0 Ferromagnetic coupling (metal or insulator)
• J > 0 Antiferromagnetic coupling (insulator, freeze charge degrees of freedom)
 

S c c i i
2 i

Hint from Hydrogen molecule
• Direct Coulomb prevents two electrons to form a chemical bond
• H
2 or chemical bond is formed by the exchange interaction of electrons energy
H
2
分子
J triplet singlet

Exchange Interaction of Electrons


( 1 , 2 )



1
   
1 2 2
 
1
   
2 2 r
1


S
( 1 , 2 ) E
0 ground state, spin singlet


( 1 , 2 )



1
   
1 2 2
 
1
   
2 2 r
1


T
( 1 , 2 )
E
1
1st excitation, spin triplet
W Heitler F London
Z. Physik, 44 ， 455 (1927)

S
( 1 , 2 )

1
2


1

2
 
1

2


P
( 1 , 2 )




1
2


1

1

2


2

1

2


1

2

Exchange interactions
S=1 wave function antisymmetric S=0 wave function symmetric

Effective description of low energy states of H
2
• Heisenberg exchange interaction

J S
1


S
2






4
3
1
4
J
J Singlet
Triplet energy
H
2
分子
J triplet singlet

Exchange interactions
In solids: direct exchange is present but small because d and f orbitals are localized:
J
12
  dr
1 dr
2

1

( r
1
)

2
( r
2
) V ( r
12
)

1

( r
2
)

2
( r
1
)
Indirect mecanisms are usually larger:
• Superexchange (short range, ferro or AF)
• RKKY (long range, oscillating sign)
• Double exchange (ferro)
• Itinerant magnetic systems

Superexchange

More Examples of Superexchange interactions
Antiferromagnetic
Ferromagnetic
Strong:
2 different orbitals weak:
90 ° coupling

Double exchange
La1-xCaxMnO3
Colossal Magnetoresistance
Mn 4+ (S=3/2) and Mn 3+ (S=2)
Ferro: possible hopping
AF: no hopping
Mn 3+ Mn 4+
Mn 3+ Mn 4+

Mermin-Wagner Theorem
No long range magnetic ordering for Heisenberg spins with short range interactions at finite temperature in 1-D and 2-D
 
1
2 m p 2
   d
2

1
N
 
 

 
T
 d


3d
2d
1d
C
D

Ferromagnetism magnetic moments are spontaneously aligned
M
T 3/2
Paramagnetic
Ferromagnetic
T c
Curie temperature
T

Holstein-Primakoff Transformation
    
S b 2 S b b i i i i
S i
 
2
 
S b b b i i i
S iz
  b b i i

S
 b b 2 S i i


Spin Wave Expansion
Low temperature excitations are dominated by spin waves b i
 b i

S
2 S
 b i
 b i

2 S

1
 b i
 b i
4 S

  i i
2
32 S
2
...
Spin wave:
Harmonic motion of
Holstein-Primakoff bosons

Ferromagnetic Spin Wave
H
 
J
 ij i
 j
 
J
 ij


1
2

 
S S i j
  
S S i j


S S iz jz


    
S b 2 S b b i i i i
S i
 
2
 
S b b b i i i
S iz
  b b i i

S
   ij

  
J S b b b b i j j i


2

Sb b i i

S
2

Ferromagnetic Spin Wave
   ij

  
H J S b b b b i j j i


2

Sb b i i

S
2
  ij
  b b k k k

JNdS
2
 k

2 SdJ


1

1 d
 d 

1 cos k



~ k
2 dimension

Comparison with Experimental Result

Bloch Law of Magnetization
M
 
S iz
  †
S b b i i
1 
N k e
 k
1

1
~
S
 d
 e


T
3/ 2

1

~ k
2 d

~ kdk
 
~
2 k dk ~
 
1/ 2 d

Experiment vs Self-Consistent Spin Wave

Antiferromagnetism
Neel
Nobel 1970
U t cuprate high temperature superconductors

Typical Antiferromagnetic Susceptibility
 
1

KNiF
3

Low Dimensional Magnetic Materials
Li
2
VO(Si,Ge)O
4
NaV
2
O
5

Two Sublattices HP Transformation
A B A sublattice
B sublattice
S i

S i
S iz


2 S b i


2 S b i
 b i
 b i

S
S j
 
2 S b
 j
S j

S jz

2 S b j

S
 b j
 b j

Antiferromagnetic Spin Wave
H

J
 ij


1
2

S i

S
 j

S i

S
 j


S iz
S jz



J
 ij

S
 b i
 b
 j
 b j b i


2 Sb i
 b i

S
2

  ij
 k
 k
  k

1
2

JNdS ( S

1 )
Bogoliubov transformation
 k

SdJ 1



1 d
 d 

1 cos k



2
~ v k

Spin Wave in La
2
CuO
4

S=1/2 Heisenberg antiferromagnet on square lattice
Magnon excitations spin-wave theory
Linear
Data points for
Cu(DCOO)
2
4D
2
0

Schwinger Boson Representation
S i

 b
 i
 b i


 
2 b b i
 i
 i
 b b i

2 S



 b b i i






• SU(2) symmetric
• Commonly used in the mean-field treatment
• Magnetic long range order corresponding to the condensation of bosons

Jordan-Wigner Transformation
S=1/2 spin operators in 1D can be represented using purely
Fermion operators
S i
  a i
 exp
 i
 
 
S a a i z i i
 a a i j





1
2 ij
 a a j j

1D X-Y model can be readily diagonalized with this transformation
H

1
2
 i
 i
 
S S i

1

S S i
  i

1

H
  k cos
 k a a k k q (

)

Two Spinon contribution to S(Q,w) q (

) Q (

)

Two Spinon Excitation in S=1/2 Spin Chain
Two spinon continuum in uniform spin ½ chain
I (meV -1 )

Paradigm of Quantum Magnetism
Phenomena
Theory
Crisis
New Theory