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

15.2 Quantum Mechanical Treatment of Spin Waves

535

there are two possibilities: either k1 = k3 and k2 = k4, or k1 = k4 and k2 = k3. Rearranging the interaction part leads to

	Jz		(γk1 + γk2 − γ0 − γk1−k2 ) ak†	1 ak1 ak†
H1 =	N	k1k2			2 ak2 .	(15.2.68)

Assuming that the number of excited magnons fluctuates little, i.e., the terms

ak†	1 ak1	− ak†	1 ak1	ak†	2 ak2	− ak†	2 ak2		(15.2.69)

are negligible, the diagonal part of the interaction Hamiltonian can be written in the form

H1	=	Jz				(γk1 + γk2 − γ0 − γk1−k2 ) ak† 1 ak1 ak† 2 ak2
		N
				k1k2
			J z
	+						(γk1 + γk2 − γ0 − γk1 −k2 ) ak†	1 ak1	ak†	2 ak2	(15.2.70)
			N			k1k2
			Jz				(γk1 + γk2 − γ0 − γk1 −k2 ) ak†	1 ak1	ak† 2 ak2 .
	− N
						k1k2

With a change of variables the total Hamiltonian can be written as

H = E(T ) +

ωk(T )ak† ak ,

(15.2.71)

k

where

ω

(T ) = ω

+

2Jz

(γ

+ γ

q −

γ

0 −

γ

) a† a

q

N

k−q

k

k

q

k

q

(15.2.72)

1

= ωk −

( ωk + ωq − ω0 − ωk−q ) aq† aq .

N S

q

This is the energy needed to excite a magnon when other magnons are already present in the system. The pairwise interaction between the magnons renormalizes the energy of both of them, but this energy correction has to be taken into account in the total energy only once. This is taken care of by the term

E(T ) = −JNz (γk1 + γk2 − γ0 − γk1−k2 ) a†k1 ak1 a†k2 ak2 . (15.2.73)

k1k2

The renormalized energy of the magnons can be written in a simple form for a simple cubic ferromagnet with nearest-neighbor interactions. Making use of the cubic symmetry, it can be proved that

536 15 Elementary Excitations in Magnetic Systems

ωk(T ) = ωk	&1 − 2JzS2N	q	ωq aq† aq ' .	(15.2.74)
	1

The energy of all magnons is renormalized by the same temperature-dependent factor. The correction appearing in the brackets is proportional to the thermal energy of the system due to magnetic excitations. It was shown in (15.2.41) that this quantity is proportional to T 5/2 at low temperatures. Even more generally, the correction term in the dispersion relation is proportional to k2 in the long-wavelength regime, while its strength depends on the 5/2th power of temperature.

Over and above such processes yielding temperature-dependent energy corrections, there exist others that scatter the magnon out of its initial state. This may be interpreted as magnons having a finite lifetime. It can be shown that the inverse lifetime is proportional to k4 in ferromagnets, that is longwavelength magnons decay very slowly.

In equation (15.2.39) for the magnetization we saw that the corrections to the Bloch T 3/2 law involve half-integer powers of the temperature. On the other hand, the lowest-order correction due to magnon–magnon interactions is proportional to the fourth power of T (since the energy of magnons goes with T 5/2). Aside from dynamical interactions, we must take into consideration that, strictly speaking, magnons are not bosons. At each lattice site, the spin can be raised up to 2S times. The resulting correction is the so-called kinematical interaction. A rigorous treatment was presented by F. J. Dyson (1956), who showed that the first-order correction to the temperature dependence of magnetization due to kinematical interactions – similarly to that due to dynamical interactions – is proportional to T 4.

15.2.6 Two-Magnon Bound States

A simple manifestation of magnon–magnon interactions is the existence of bound states between two magnons under suitable conditions. To see how they arise, let us recall that in the ground state of the isotropic Heisenberg model with ferromagnetic (J > 0) exchange interaction, spins are lined up parallel, irrespective of the dimensionality of the lattice. The low-lying onemagnon excitations were obtained as linear combinations of states in which one spin is raised or lowered by one unit relative to the ground state. If spin waves were propagating independently in the lattice, then the wavefunction of the excitation of two spin waves, of wave vectors k and k , could be chosen as

|ψ =	1	eik·Ri Si+	eik ·Rj Sj+|0 .	(15.2.75)
	2SN
		i	j

This corresponds to the assumption – mentioned in connection with (15.2.28)

– that the coe cients cij can be chosen in a product form. The above state is, however, not an exact eigenstate of the Hamiltonian because it contains terms

15.2 Quantum Mechanical Treatment of Spin Waves

537

in which the two raised spins are located at adjacent sites, or, for S > 1/2, in which the spin at one site has been raised twice – and the result of the action of the Hamiltonian on such configurations is di erent from the results obtained when the two reversed spins are far apart. Since the weight of these configurations is on the order 1/N in the wavefunction, the excitation energy of such states is expected to be approximately

E = ωk + ωk .

(15.2.76)

It can be shown that for the overwhelming majority of states with two reversed spins the energy is quite close to such a value. However, in addition to states with two nearly free spin waves, there exist other states – although with a small thermodynamic weight – that can be considered as bound states of two spin waves. For simplicity, we shall present the calculation in one dimension.

Assuming that only nearest neighbors interact, the Hamiltonian of the isotropic spin chain is written as

H = −J Sl · Sl+1 ,	(15.2.77)

l

which di ers from the previously used form in a factor of 2. In contrast to the foregoing, the state |F in which the projection of each spin is maximal along the z direction will be chosen as the ferromagnetic ground state. Writing the wavefunction of the state with one lowered spin as

		1		√1
|ψk =	√				eikxl Sl−|F ,	(15.2.78)
		2S		N	l

the energy of one-magnon excitations is

ωk = 2SJ(1 − cos ka) .

(15.2.79)

Two-magnon excitations can be obtained by reducing the z component of the total spin of the ground state by two. In state

|		l l	|		(15.2.80)
ψl,l		= S−S−	F

the projection of the spins at sites l and l have been reduced by one unit each. If S = 1/2, the operator S− reverses the spin, and the two sites are necessarily di erent. If, however, S > 1/2, then it is possible to reduce the z component of the spin twice at the same site. These states are not eigenstates of the Heisenberg Hamiltonian, since, depending on the relative location of the two lattice sites – whether they are coincident, adjacent, or separated at a larger distance –

H|ψl,l = (E0 + 4JS)|ψl,l − J [S(2S − 1)]1/2 (|ψl−1,l + |ψl,l+1 ) ,

H|ψl,l+1 = (E0	+ J(4S − 1))|ψl,l+1 − J [S(2S − 1)]1/2 (|ψl,l
+|ψl+1,l+1 ) − JS (|ψl−1,l+1 + |ψl,l+2 ) ,		(15.2.81)
H|ψl,l = (E0	+ 4JS)|ψl,l −JS (|ψl−1,l +|ψl+1,l +|ψl,l −1 +|ψl,l +1 ) .

538 15 Elementary Excitations in Magnetic Systems

We shall look for eigenstates expressed as linear combinations of these,

NN

|		=					≤		(15.2.82)
			|		,	xl		xl .
Ψ			c(xl, xl ) ψl,l

l=1 l =l

Applying the Hamiltonian on |Ψ , and making use of relations (15.2.81) for the states |ψl,l , equations are obtained for the coe cients c(xl, xl ). If the energy of the excited state is written as E = E0 + ω, we have

[ ω − 4JS]c(xl, xl) + J [S(2S − 1)]1/2 [c(xl−1, xl) + c(xl, xl+1)] = 0 , [ ω − J(4S − 1)]c(xl, xl+1) + J [S(2S − 1)]1/2 [c(xl, xl) + c(xl+1, xl+1)]

+ J S [c(xl−1, xl+1) + c(xl, xl+2)] = 0 ,

(15.2.83)

[ ω − 4J S]c(xl, xl ) + JS [c(xl−1, xl ) + c(xl+1, xl ) + c(xl , xl −1) + c(xl, xl +1)] = 0

for identical, adjacent, and further separated sites.

As a generalization of the expression exp(ikxl ) used for one-magnon states and leading to plane-wave-like solutions, we shall seek the coe cient c(xl, xl ) in the form

c(xl, xl ) = A12ei(k1xl +k2 xl ) + A21ei(k2xl +k1 xl ) .

(15.2.84)

Both terms are necessary if we wish to obtain a symmetrized form for xl ≤ xl . As an immediate consequence, if periodic boundary conditions are imposed, we have

c(xl, xl ) = c(xl , xl + N a) ,		(15.2.85)
or
A12 = A21eik1 N a = A21e−ik2N a .		(15.2.86)
The choice A12 = Aeiφ/2, A21 = Ae−iφ/2 leads to
eik1 N a = eiφ ,	eik2N a = e−iφ ,	(15.2.87)
or alternatively
N ak1 = 2πI1 + φ ,	N ak2 = 2πI2 − φ ,	(15.2.88)

where I1 and I2 are integers. Physically di erent solutions are obtained only when I1 and I2 are both in the interval [0, N ).

Inserting this into the last equation of (15.2.83), the energy eigenvalue is found to be

ω = 2JS [2 − (cos k1a + cos k2a)] .

(15.2.89)

This is apparently the same as the sum of the energies of two free magnons,

15.2 Quantum Mechanical Treatment of Spin Waves		539
ω = ωk1 + ωk2 .	(15.2.90)

Care must be taken however, since due to the interaction between the magnons, the two wave numbers are shifted with respect to the free-magnon values. Their sum is nonetheless a good quantum number:

	k = k1 + k2 =	2π		(I1 + I2) .	(15.2.91)
		N a

In terms of k and the variable q = (k1 − k2)/2, the excitation energy reads

ω = 4JS 1 − cos			21 ka cos(qa) .		(15.2.92)
If q is real, excitation energies are in the interval between				21 ka ,
ω+ = 4J S 1 + cos	21 ka	and ω− = 4J S 1 − cos			(15.2.93)
as shown in Fig. 15.3.

!

/a	a	k

Fig. 15.3. The spectrum of excitations due to lowering two spins in a chain of S = 1/2 spins

The possible values of k1 and k2, along with the phase φ are determined from the condition that the first two equations of (15.2.83) (for c(xl, xl) and c(xl, xl+1)) should both hold. Expressing c(xl, xl) from the first equation and substituting it into the formula for c(xl, xl+1), one gets

[ ω − J(4S − 1)]c(xl, xl+1) + JS (c(xl−1, xl+1) + c(xl , xl+2))

=J2S(2S − 1) (2c(xl, xl+1) + c(xl−1 , xl) + c(xl+1, xl+2)) . (15.2.94)ω − 4JS

Making use of the expression for ω and the assumption for c(xl, xl ), after a tedious calculation one arrives at

cot(φ/2) =	21	[cot(k1a/2) − cot(k2a/2)] 1 + (2S − 1)	cos[(k1	+ k2)a/2]	.
			cos[(k1	− k2)a/2]

(15.2.95)

540 15 Elementary Excitations in Magnetic Systems

It is readily seen that φ changes sign upon the interchange of k1 and k2, so the wavefunction is una ected by the same interchange. For k1 = k2, φ = ±π, and the wave function vanishes. It is therefore su cient to consider the case k1 < k2 (I1 < I2).

If k2 and the corresponding I2 are treated as known, and k1 is varied from zero to k2, the phase φ changes from 0 to π. Each wave number becomes slightly larger than the free-magnon value 2πI1/N a. The shift in energy is of order 1/N , that is why the same regime is obtained for the continuum of such states as for free two-magnon states.

However, I1 can only take the values 0, 1, 2, . . . , I2 − 2, since the solution for I1 = I2 − 1 would be k1 = k2, implying a vanishing wavefunction, as we have already seen. Therefore, if the solutions found so far are collected for all values of I2, we end up with N − 1 less states than we should. To find the missing ones, we must allow k1 and k2 to take complex values. We shall not go into the details of the calculation here, just convey the results: for each value of the total (resultant) wave number k there exists a complex conjugate pair k1, k2 = k1 such that the corresponding excitation energy is below the bottom of the continuum of two-magnon excitations. For S = 1/2

the excitation energy can be given analytically as
ω = 21 J [1 − cos ka] .	(15.2.96)

In Fig. 15.3 these excitation energies are indicated by the curve below the continuum. Because of the complex wave number, these excitations correspond to states in which large amplitudes belong to configurations where the two spin flips have taken place on identical or adjacent lattice sites. These excitations can thus be regarded as bound states of two magnons. Their energy is below the continuum, because it is energetically more favorable to flip a spin that already has a flipped spin as a neighbor.

15.3 Antiferromagnetic Magnons

The description of the ground state and low-lying excited states of ferromagnetic materials was facilitated by the possibility of building the global ground state from local ground states of pairs of neighboring spins. For antiferromagnetic coupling this is no longer the case. The local ground state of two antiferromagnetically coupled spins is the singlet configuration of the spins. In this state the expectation value of the spin operator vanishes for each spin. Such local ground states cannot serve as building blocks for a global ground state with nonzero sublattice magnetization. The determination of the ground state is in fact rather di cult. Discussion of this point will be deferred to the next subsection. Here, we shall start from the classical, so-called Néel state, in which the z component of each spin is S in sublattice A, and −S in sublattice B. In fact, this is not an eigenstate of the Heisenberg Hamiltonian, for neighboring spins cannot have maximum projections in opposite directions because

15.3 Antiferromagnetic Magnons

541

of the term

SixSjx + Siy Sjy = 21 Si+Sj− + Si−Sj+ .

(15.3.1)

For lack of a better starting point, we shall adopt this classical ground state, and try to generalize the results obtained for ferromagnets.

15.3.1 Diagonalization of the Hamiltonian

Excited states can be studied by means of the Holstein–Primako or the Dyson–Maleev transformation, familiar from the treatment of ferromagnets. However, di erent assignments must be used on the two sublattices, with dominantly upward and downward spins (“up” and “down” sublattices). At low temperatures, the z component of the spins in sublattice A is dominantly S, while for those in sublattice B it is −S, hence excitations above the Néel state are created by operator S− in sublattice A, and operator S+ in sublattice B. Consequently, spin operators can be represented by boson operators as

+

√

†

1/2

+

√

†

†

1/2

√

ai ,

√

2Sbj

1

SAi = 2S 1 − ai ai /2S

SBj =

1 − bj bj /2S

,

1/2

/2

z

†

1

−

ai†ai /2S

z

=

†−

bj ,

SA−i = 2Sai†

,

SB−j

2S 1

bj†bj /2S

SAi = S − ai ai ,

SBj = −S + bj bj ,

(15.3.2)

or

+

√

1 − ai†ai /2S ai ,

+

√

SAi

= √2S

SBj

= √2Sbj† ,

z

†

z

†−

SA−i

=

SB−j

=

1

bj†bj /2S bj ,

2Sai† ,

2S

SAi = S − ai ai ,

SBj = −S + bj bj .

(15.3.3) Whichever choice is adopted, the correct commutation relations will be recovered for the spin operators, as long as the ladder operators ai and bj are bosonic in character.

We shall restrict our discussion to nearest-neighbor interactions and bipartite lattices, where the nearest neighbors of an up spin in a two-sublattice antiferromagnet are down spins in the other sublattice. Expanding the Hamiltonian through bilinear terms – that is, neglecting four-operator terms yielding magnon-magnon interactions – we have

H0 = 2N zJ S2 − 2JS	iδ	ai†ai + bi†+δ bi+δ + ai bi+δ + ai†bi†+δ , (15.3.4)

where δ is the vector to the nearest neighbors. In terms of the Fourier transforms,

542 15

Elementary Excitations in Magnetic Systems

ak

=

/

N

i A e−ik·Ri ai ,

ak† =

/

N

i A eik·Ri ai†

,

2

2

(15.3.5)

bk

=

/ N

bk† =

/ N

j B e−ik·Rj bj ,

j B eik·Rj bj†

,

2

2

the Hamiltonian takes the form

H0 = 2N zJ S2 − 2JzS a†kak + b†−kb−k + γk ak b−k + a†k b†−k .

k

(15.3.6) As transition to the boson operators had to be performed di erently in the two sublattices, this Hamiltonian is not yet diagonal. We have to make one more transformation, the unitary Bogoliubov transformation.7 Introducing two new creation and annihilation operators that mix the operators of the

two sublattices,

αk = ukak − vkb−† k ,	βk = ukb−k − vkak† ,	(15.3.7)
αk† = ukak† − vkb−k ,	βk† = ukb−† k − vkak .

The coe cients are assumed to be real; in retrospect, this will prove justified. The new operators satisfy bosonic commutation relations if u2k − vk2 = 1. The inverse transformation formulas are then

a

= u

α + v β†

,

b

−k

= u

β

+ v α† ,

(15.3.8)

k

k k k k

k

k

k k

ak† = ukαk† + vk βk ,

b−† k = ukβk† + vk αk .

Substituting these into (15.3.6), we find that if condition

2ukvk + γk uk2 + vk2 = 0

(15.3.9)

is met, only the diagonal elements in the Hamiltonian survive. Then

H0 = 2N zJ S2 + 2N zJ S +

k

ωk αk†

αk + βk†

βk + 1 ,

(15.3.10)

where

ωk = 2|J|zS

uk2 + vk2 + 2γkukvk .

(15.3.11)

satisfying the auxiliary condition

k

− k

The solution of (15.3.9)

uk2 = 21 # 1 − γk2 + 1$,

vk2 = 21 # 1 − γk2 − 1$ .

(15.3.12)

1

1

Inserting this into the excitation energy formula,

7 N. N. Bogoliubov, 1958.

15.3 Antiferromagnetic Magnons

543

ωk = 2|J|zS&

1

1

γ2

−

1 k

γ2

'

= 2|J|zS 1 − γk2 .

(15.3.13)

γ2

"

k

k

−

−

In the long-wavelength limit, where γk is close to unity, the square root can be approximated as

"

=

≈ √

.

1 − γk2

(15.3.14)

1 + γk

1 − γk

2

1 − γk

The geometry-dependent factor 1 − γk was determined for cubic crystals in equations (15.2.12), (15.2.14) and (15.2.15), and in these cases

"1 − γk2	2
	≈ / z ka .	(15.3.15)

Thus, as has been found in the classical limit, the dispersion relation of antiferromagnetic magnons starts linearly:

	ωk ≈ 2JS	√			(15.3.16)
		2zka .

This is why antiferromagnets are strikingly di erent from ferromagnets, as far as thermodynamic behavior is concerned; in some aspects they are similar to a phonon gas.

In two-sublattice antiferromagnets two types of spin waves (or antiferromagnons) may therefore propagate, whose energies are equal in the absence of anisotropies and external magnetic fields. Spins in both sublattices participate in each type of excitation, however, spins in the “up” sublattice play the major role in one type, while those in the “down” sublattice in the other. If we had considered more complex antiferromagnets with more sublattices, we would have found not just two excitation branches but as many as there are sublattices. Nor will it be true any longer that each excitation branch starts at zero. Using the same terminology as for phonons, we can speak of acoustic and optical magnons.

15.3.2 The Antiferromagnetic Ground State

When expressed in terms of spin-wave creation and destruction operators, the Hamiltonian (15.3.10) of the antiferromagnetic system has, in addition to the magnon number operator, a term – “+1” in the brackets – that corresponds to the zero-point energy of the two types of magnons. In the ferromagnetic ground state, with all spins lined up parallel, there is no zero-point energy contribution. Its presence in antiferromagnets indicates that the Néel state is not the true ground state.

Taking the zero-point energy contribution into account, we have

E0 = 2N zJ S(S + 1) +	k	ωk = 2N zJS	S + z		(15.3.17)
				β

544 15 Elementary Excitations in Magnetic Systems

for the ground-state energy, where

z	= N	k	1 −		1 − γk2	.	(15.3.18)
β	1			"

For simple and body-centered cubic lattices β ≈ 0.58. The number of nearest neighbors (the coordination number) is six in the first case and eight in the second, and so β/z < 0.1. The correction provided by zero-point vibrations is therefore less than 10% of the Néel state energy.

Due to quantum fluctuations present in the true ground state, the expectation value of the spins is slightly smaller in magnitude than the ground-state S value. To determine the zero-point spin contraction, consider the next formula for the spins in sublattice A:

2
SAz i = S − ai†ai = S −		ak† ak .	(15.3.19)
	N
		k

Expressing the aks in terms of the magnon creation and annihilation operators, we have

SAz i = S − N2 u2kα†k αk + vk2 βkβk† + ukvk α†k β−† k + ukvk β−kαk .

k

(15.3.20) The ground-state spin reduction is the expectation value of this expression at T = 0. The only nonvanishing contribution comes from the term βkβk† = 1 − βk† βk, whence

Sz = S − SAz i = N		vk2 = N		&		1 γ2 − 1' .			(15.3.21)
2		1				1
						−
	k		k				k

Transforming the sum over wave vectors into an integral, and evaluating the latter numerically, 0.078 is obtained for simple cubic lattices, and 0.059 for body-centered cubic lattices. (The deviation from the mean-field theory is smaller for bcc lattices because of the larger number of nearest neighbors.) Hence, even in the worst case – that is in a cubic antiferromagnet built up of S = 1/2 spins – the spin is contracted by no more than 15% in the ground state. As we shall see later, this is not the case in lower-dimensional systems.

15.3.3 Antiferromagnetic Magnons at Finite Temperature

Because of the linear dispersion relation of antiferromagnetic magnons, the thermodynamic behavior of antiferromagnets is expected to di er from that of ferromagnets. Indeed, the temperature dependence of sublattice magnetization and specific heat are governed by di erent power laws in the two types of material.