Fundamentals of the Physics of Solids / 15-Elementary Excitations in Magnetic Systems

15.2 Quantum Mechanical Treatment of Spin Waves

525

a† =		1	eik·Rl S+
k	√			(15.2.17)
			l
		2SN	l

from the ground state. Taking its adjoint as the annihilation operator, their commutator is

a

, a†

=

1

e−ik·Rl eik ·Rl

S−

, S+

(15.2.18)

k

2SN

l

l

k

l,l

1

)

z

= −

e−i(k−k

·Rl 2Sl .

2SN

l

At low temperatures, where the magnetization is only slightly di erent from the saturation value, the z component of the spins can be well approximated by −S, therefore

		1			)
ak , ak†		≈		e−i(k−k		·Rl = δkk .	(15.2.19)
			N

l

Thus, in this approximation, the creation and annihilation operators of spin waves satisfy commutation relations characteristic of bosons.

Transforming the formula (15.2.17) for a†k back into lattice representation using inverse Fourier transforms, we introduce the operators

aj† =	√	1	Sj+,	aj	=	√	1	Sj− .	(15.2.20)
		2S					2S

In terms of these, the spin operators are

S+ =

√

−

= √

(15.2.21)

2Sa†

,

S

2Sa

j

.

j

j

j

This pair has to be complemented by a third expression, for Sjz . Acting on the state |Mj , in which the z component of the spin at site j is Mj ,

a†a M	=	1		S+S−	|	M
j j | j			2S j j				j
		1
							1
	=		2S	S(S + 1)				− Mj (Mj − 1) |Mj			(15.2.22)
	=	S + Mj + 2S							(S + Mj ) − 2S (S + Mj )2		|Mj .
									1

If S is large enough and the quantum number Mj of the states of interest (i.e., the expectation value of Sjz ) di ers little from −S then the number operator

a†j aj measures this deviation. Thus in the space of these states

Sjz = −S + aj†aj .

(15.2.23)

Inserting this representation of the spin operators into the Heisenberg Hamiltonian, in the present approximation

526 15 Elementary Excitations in Magnetic Systems

H = − ij	Jij S2	+ S	ij	Jij ai†ai + aj†aj − ai†aj − aj†ai
				(15.2.24)
− gμBμ0H
		−S + ai†ai .

i

Introducing the Fourier transforms of the operators a†j and aj via

a†	=	1			eik·Rj a† ,	a =	1		e−ik·Rj a ,
		√					√			(15.2.25)
k					j	k			j
			N	j				N	j

which is equivalent to approximating the spin operator by the bosonic operator in (15.2.17), the Hamiltonian takes the simple form

H = E0 + ωkak† ak ,	(15.2.26)
k

where the expression for ωk is identical to (15.2.9) obtained for a single spin wave, and E0 is the known energy of the ferromagnetic ground state. H has the same form as the Hamiltonian of a gas of bosons in which the energy of a particle with wave vector k (momentum k) is ωk. The elementary quanta of the latter are called magnons. Since the ferromagnetic ground state is exactly known, and – contrary to the case of phonons – there are no quantum fluctuations in it, the term 12 corresponding to zero-point vibrations does not appear.

According to our previous considerations, this Hamiltonian reproduces correctly the ground state and the excited states with one raised spin. Finding further eigenstates is not so easy. Since the interaction of spins in the Heisenberg model – even in the case of uniaxial anisotropy – is such that raising the z component of one spin is always accompanied by the lowering another, thus

the operator	Sz	=	i	Sz commutes with the Hamiltonian,
	tot			i	[Stotz , H] = 0 ,	(15.2.27)

Stotz is conserved. The projection of the total spin along the quantization axis can therefore be used as a quantum number to label the states. In the ground

state Stotz = −N S, while in one-magnon states Stotz = −N S + 1, as one spin has been raised. Higher excited states are expected to have two, three, etc.

raised spins. Therefore further excited states are sought in the Hilbert space with two raised spins, in the form

|ψ = cij Si+Sj+|0 .	(15.2.28)

ij

If the two raised spins are not very close to each other, then these states could be considered – as suggested by the approximate form obtained for the Hamiltonian – as if two magnons were propagating in them independently.

15.2 Quantum Mechanical Treatment of Spin Waves

527

Such states can be constructed as the products of two one-magnon states. Likewise, states containing several raised spins lend themselves to interpretation in terms of several magnons propagating independently. We shall return to the justification of this point later; now it will be assumed that an arbitrary excited state of the ferromagnet may be regarded as if it were composed of a number of independently propagating magnons. To put it otherwise: the thermodynamic behavior of ferromagnets is well approximated by that of a gas of magnons.

15.2.3 Thermodynamics of the Gas of Magnons

Since the creation of each magnon corresponds to raising a spin, in the independent spin-wave approximation the deviation of the thermodynamic average of the z component of the total spin Stot from the ground-state saturation value is given by the number

Nsw = ak† ak		(15.2.29)
k
of thermally excited magnons at temperature T , as
Stotz = −N S + ai†ai = −N S +	ak† ak .	(15.2.30)
i	k

Replacing summation over wave vectors by integration in the usual way, and using the Bose–Einstein distribution function for the number of magnons, one finds that the number of thermally excited magnons is

Nsw = V

dk

1

,

(15.2.31)

(2π)3

eβ ωk − 1

and so the magnetization is

− 1

.

M = V

gμB Stotz = V |g|μBS 1 − N S

(2π)3

eβ ωk

(15.2.32)

1

N

V

dk

1

In general, the temperature dependence of magnetization may be determined numerically from this formula. At low temperatures, however, one can proceed analogously to the Debye approximation used earlier for phonons. Approximating the dispersion relation by its asymptotic form in the longwavelength limit, the integral in the square brackets may be written as

1	kmax	1
	0
		4πk2 dk		,	(15.2.33)
(2π)3			eβDk2 − 1

where kmax, the equivalent of the Debye wave number, is a cuto related to the size of the Brillouin zone.

528 15 Elementary Excitations in Magnetic Systems

Unlike for phonons, the dispersion relation is now quadratic in the wave number, therefore the thermodynamics of a gas of magnons is di erent from that of a gas of phonons. Due to the rapid fall-o of the Bose–Einstein distribution, the cuto can be neglected at low temperatures. Extending the upper limit of integration to infinity and introducing the new variable βDk2 = x, the following value is found for the previous expression:

1

kBT

3/2

∞ x1/2 dx

1

kBT

3/2 √

π

0

=

ζ(3/2)

4π2

D

ex − 1

4π2

D

2

(15.2.34)

=

3/2

4πD

ζ(3/2) ,

kBT

where we have made use of (C.2.8). Inserting this expression and D = 2JSa2 for the sti ness constant, as implied by (15.2.13), into (15.2.32),

M = V

|g|μBS &1 − N S

ζ(3/2) 4πD

'

N

V

kBT

3/2

= V

|g|μBS &1 − N Sa3 8π3/2

2JS

(15.2.35)

' .

N

V

ζ(3/2)

kBT

3/2

The deviation of the magnetization from its saturation value is due to thermally excited magnons. At low temperatures, the leading correction is proportional to the 3/2th power of the temperature. This result, known as the Bloch T 3/2 law,2 is well confirmed by various experiments. Fitting the above function to the temperature dependence of the magnetization of nickel (shown in Fig. 14.12(a)) provides a much better agreement in the low-temperature region than the mean-field theory.

In simple cubic lattices V = N a3. Inserting the numerical value ζ(3/2) = 2.612,

M = a3		|g|S &1 − 2S	0.117		2JS	' .	(15.2.36)
	μB	1			kBT	3/2

For body-centered cubic lattices V = N a3/2, and for face-centered ones V = N a3/4. Introducing a multiplicative factor α that takes the values 1/2 and 1/4 in the two cases,

M = α a3 |g|S		&1 − 2S		0.117		2JS	' .	(15.2.37)
	1 μB		α			kBT	3/2

If more realistic dispersion relations are used, corrections to the leading term appear. Assuming only nearest-neighbor exchange interactions in simple cubic lattices, the dispersion relation is

2 F. Bloch, 1930.

15.2 Quantum Mechanical Treatment of Spin Waves		529
ωk = 4SJ 3 − cos kxa − cos ky a − cos kz a .	(15.2.38)

The temperature dependence of magnetization can be determined in this case, too. In the low-temperature regime

M = M0

− A3/2

3/2

− A5/2

5/2

− A7/2

2JS

7/2

2JS

2JS

− . . . ,

kBT

kBT

kBT

(15.2.39)

where the coe cients can be expressed in terms of Bessel functions.

To calculate the magnetic energy and the specific heat of the system, the thermal average of magnon excitation energies has to be evaluated. If the ground-state energy is neglected, the internal energy of the gas of free magnons (considered as bosonic particles) becomes

E =	k	ωk ak† ak	= (2π)3			e ωk /kBT		1 dk .		(15.2.40)
				V		ωk
							−

In the regime of low temperatures, this expression can be evaluated similarly to the temperature dependence of magnetization above. The result is

E =

V 1 (k T )5/2

∞ x3/2 dx

(15.2.41)

a3 4π2 (2JS)3/2

0

ex − 1

B

1 (kBT )5/2 3π1/2

= αN

ζ(5/2) .

4π2

(2JS)3/2

4

Due to the extra factor of ωk in (15.2.40) compared to the formula for magnetization, the energy will go with the 5/2th power of temperature. Thus the contribution of magnons to the specific heat is proportional to T 3/2:

	C = N kBA			kBT		3/2
						,	(15.2.42)
				2JS
	where	15
	A = α		ζ(5/3) = 0.113 α .				(15.2.43)
		32π3/2

This contribution to the low-temperature specific heat of ferromagnetic materials can indeed be easily observed in experiments, once the contribution of phonons, proportional to T 3, has been separated.

It is instructive to examine the results for the di erential susceptibility of ferromagnets below the Curie temperature. To this end, we shall determine the change in the magnetization and the number of thermally excited magnons

due to a magnetic field,					− eβ ωk (H=0)			.
Nsw =	k	eβ ωk (H)		1			1		(15.2.44)
		1			1
			−			−

530 15 Elementary Excitations in Magnetic Systems

The excitation spectrum in the presence of a magnetic field will be written asωk(H) = Dk2 + γH. Replacing summation by integration,

Nsw = V

(2π)3

eβ(Dk2 +γH) − 1

− eβDk2

− 1

dk

1

1

dk

eβDk2

1

eβγH

(15.2.45)

= V

eβDk2

1

eβ−(Dk2 +γH)

1

.

(2π)3

−

−

For weak fields, the factor 1 − eβγH in the numerator of the integrand is proportional to H. If this were factored out, susceptibility would be proportional to the remaining integral. The latter, evaluated at H = 0, would yield a divergent result, since close to the lower limit the integrand is proportional to 1/k4. Thus the term γ H is also retained in the expansion valid in the vicinity of the lower limit, so we have

N

sw

= V

dk

−βγ H

.

(15.2.46)

(2π)3 βDk2 (βDk2 + βγH)

With the new variable

x =

,

1/2

Nsw = −βD

D/γH k

d3x x2

D

(2π)3

(x2

+ 1) .

(15.2.47)

1

γH

V

1

Now the integral is convergent at the lower√limit. The most important point is

that the magnetization is proportional to H, hence, below T , the suscepti-

√ C

bility exhibits a strong field dependence proportional to 1/ H. In contrast to the finite value obtained from the mean-field theory, the contribution of spin waves render the susceptibility divergent in the H → 0 limit. In experiments large but finite initial susceptibilities are measured instead. This is because macroscopic samples always contain domains with di erent directions of magnetization, and the field dependence of magnetization is in fact governed by these.

15.2.4 Rigorous Representations of Spin Operators

If the operators ak and a†k were boson operators, the operators in the lattice representation would also behave as boson operators. Obviously this cannot be so, since the z component of the spin at a given lattice site has only (2S + 1) allowed values. Formulas (15.2.21) and (15.2.23) can be only approximately true. If, nevertheless, the spin operators are to be represented by boson operators, more complicated expressions are needed to ensure the right commutation relations among the spin operators. Several such representations exist. Below, we shall see three commonly used forms.

In the Holstein–Primako representation3 the spin operators are expressed as

3 T. Holstein and H. Primakoff, 1940.

15.2 Quantum Mechanical Treatment of Spin Waves

531

+	√				1/2
					1
Sj = 2Saj† 1 − aj†aj /2S ,
Sj− = √				1 − aj†aj /2S	/2	(15.2.48)
		2S			aj ,

Sjz = −S + a†j aj .

With the help of the commutation relations

ai , aj† = δij ,

ni , ai = −ai ,

ni , ai† = ai†

(15.2.49)

for boson operators, it is straightforward to show that the spin operators indeed satisfy the correct commutation relations. For example, for the operators

Sj+ and Sj−,

Sj+, Sj−

	j		−	j		1/2		1/2
= 2S a†		1		a†aj		/2S aj
= 2S	− 1 − aj†aj /2S						aj aj† 1 − aj†aj /2S
	aj aj			−		j aj − 1)	1/2		1/2
	†		1		(a†		/2S
	− 1 − aj†aj /2S 1 + aj†aj 1 − aj†aj /2S
= 2S aj†aj 1 − aj†aj /2S + aj†aj /2S
	− 1 + aj†aj 1 − aj†aj /2S
= 2S −1 + aj†aj /S = 2Sjz .									(15.2.50)

If we wish to go beyond the leading-order terms (which are identical to the expressions used previously), and take into account the corrections due to the factor under the square root in the Holstein–Primako transformation, the calculations will run into di culties because of the square root. That is why it is often more convenient to use the Dyson–Maleev representation,4 in which

+	=	√
Sj		√	2Saj† ,
Sj− =					1 − aj†aj /2S aj ,	(15.2.51)
			2S

Sjz = −S + a†j aj .

Again, it is easy to demonstrate that the usual commutation relations hold among the spin operators, however, Sj+ and Sj− are not each other’s adjoints, as they should be. Nonetheless, the Hamiltonian of the Heisenberg model proves to be Hermitian.

4 F. J. Dyson, 1956 and S. V. Maleev, 1957.

532 15 Elementary Excitations in Magnetic Systems

Whether one form is used or the other, the corrections are small for large values of S, and we get back to the same expression that was used earlier. Therefore it is straightforward to assume that the gas of free magnons provides a good approximation for large spins. In reality, however, the obtained results are su ciently precise even for S = 1/2 in the low-temperature regime.

In the foregoing, we gave two representations of the spin operators in terms of operators satisfying bosonic commutation relations. Further representations of the spin operators are equally possible. In the Schwinger representation,5 two bosons are associated with each lattice site, and – dropping lattice indices

– the vectors		a1
	= a1† a2† ,
a†		a = a2		(15.2.52)
are constructed. Then the operator
	S = 21 a† · σ · a			(15.2.53)
is defined with the help of the Pauli matrices. Its components,
	Sz = 21 (a1†a1 − a2†a2) ,
	S+ = Sx + iSy = a1†a2 ,			(15.2.54)
	S− = Sx − iSy = a2†a1

satisfy the commutation relations of the dimensionless angular momenta. The properties of bosons imply

	S2 = S(S + 1) ,								(15.2.55)
where			a = 1	(a†a		+ a†a		) .
S = 1 a†		·			1				(15.2.56)
2			2	1		2	2

The eigenvalues of S are integers or half-integers (0, 1/2, 1, 3/2, 2, . . . ). To describe the states of spin S in terms of Schwinger bosons, we have to restrict the allowed states to the subspace of the Hilbert space where the condition

S = 1	(a†a	1	+ a†a		)	(15.2.57)
2	1		2	2

is satisfied. If this relation is used to eliminate the boson labeled 2, and the expressions

a2 =	√2S		#1 − 2S1		$	1/2
						,
				a1†a		(15.2.58)
a2† =	√2S		#1 − 2S1		$	1/2
				a1†a

5J. Schwinger, 1952. Julian Schwinger (1918–1994) shared the Nobel price in 1965 with S. Tomonaga and R. P. Feynman “for their fundamental work in quantum electrodynamics, with deep-ploughing consequences for the physics of elementary particles”.

15.2 Quantum Mechanical Treatment of Spin Waves

533

are substituted into (15.2.54), we obtain the Holstein–Primako representation of the spin operators by a single bosonic variable. If, instead, the boson labeled 1 is eliminated, then the spin operators will be expressed as

+	√							1/2
					−	j			,
Sj								1/2
	= 2S 1					a†aj	/2S	aj
Sj−	= √					− aj†aj /2S				(15.2.59)
		2Saj† 1							,
Sjz = S − aj†aj .

This representation is just as good as that given in (15.2.48), but using it is practical only when the spin at the jth lattice site points upward in the ground state. We shall use this form for studying antiferromagnetic excitations.

Alternatively, in the S = 1/2 case spin operators can be represented by anticommuting fermion operators as

Sj =		cjσ† σσσ cjσ ,		(15.2.60)
	σσ
or, using the explicit form of the Pauli operators
Sj+ = cj†↑cj↓,
S−	= c† c	j↑	,	(15.2.61)
j	j↓

Sjz = 12 (c†j↑cj↑ − c†j↓cj↓) .

Finally, it should be mentioned that in one dimension the Jordan–Wigner transformation6 can also be used to obtain a representation of spin-1/2 spin operators in terms of spinless fermions:

Sj+ = cj† exp iπ		cl†cl ,
Sj− = exp	l<j	cl†cl cj ,	(15.2.62)
	− iπ l<j

Sjz = c†j cj − 12 .

15.2.5 Interactions Between Magnons

The expressions for spin operators in terms of magnon creation and annihilation operators are fairly complicated. As a consequence, besides bilinear terms, others containing more operators will also be included in the Heisenberg Hamiltonian, whether the Holstein–Primako or the Dyson–Maleev representation is used. The explicit forms of these terms are, however, di erent in

6 P. Jordan and E. P. Wigner, 1928.

534 15 Elementary Excitations in Magnetic Systems

the two representations. Expanding the square root in the Holstein–Primako transformation,

Sj+ = √2Saj† 1 − aj†aj /4S + . . . ,
Sj− = √2S 1 − aj†aj /4S + . . . aj ,	(15.2.63)

Sjz = −S + a†j aj .

When this is substituted into the Heisenberg Hamiltonian, and the terms quadratic and quartic in the boson operators are separated, we get

				H = H0 + H1 ,	(15.2.64)
where H0 is the same as in (15.2.24),
H0	= − ij	Jij S2 + S	ij	Jij ai†ai + aj†aj − ai†aj − aj†ai	(15.2.65)

− gμBμ0H	−S + ai†ai ,
i

while, making use of the relation Jij = Jji , the quartic part is

H1	= − ij	Jij ai†ai aj†aj − 21 ai aj†aj†aj − 21 ai†ai ai aj† .	(15.2.66)

As we have already seen, when expressed in terms of the Fourier transforms of the operators, H0 can be diagonalized. This is the Hamiltonian of the free magnon gas. On the other hand,

H1 =	J z	(γk1 + γk3 − 2γk1−k3 ) ak†	1 ak† 2 ak3 ak4 δ(k1 + k2 − k3 − k4 + G)
	2N
		k1k2
		k3k4

(15.2.67) contains magnon interactions (scattering processes), so-called dynamical interactions. They have to be taken into account if the corrections to thermodynamic quantities, etc. due to magnon–magnon interactions are to be determined.

If the Dyson–Maleev transformation is applied, only such four-operator terms appear – while if the Holstein–Primako transformation is used instead, the quartic terms do not describe the interaction in full, the expansion of the square root brings in further terms, corresponding to multi-particle scattering.

The interactions lead to a temperature-dependent change in the magnon energy. To demonstrate this, consider the diagonal terms of the interactions that do not change the state of the magnons. For long-wavelength magnons, when umklapp processes can be neglected (only the G = 0 term survives),