# Spin Waves in Antiferromagnets

## Dispersion Relation

Consider the following Heisenberg Hamiltonian:

where interactions are limited to pairs of nearest neighbors (/, *m) *with an antiferromagnetic exchange integral / > 0. *H* is the amplitude of a magnetic field applied in the *z* direction. / and *m *indicate the sites belonging, respectively, to t and *i* sublattices. Note that we do not use the factor 2 in front of *j* in the Hamiltonian (3.53).

We use the Holstein-Primakoff method in the same manner as in the case of ferromagnets shown in (3.26)-(3.28) but with a distinction of up and down sublattices. For the up sublattice, one has

where

For the down sublattice, one defines where

The operators *a,* a^{+}, *b* and *b ^{+}* obey the commutation relations (Problem 9 in Section 3.6). Replacing operators

*S*and

^{±}*S*in (3.53) by these operators, one gets

^{z}

In a first approximation, one supposes that the number of excited spin waves *n* is small with respect to 2*S,* namely cr_{;}^{+}a/ <£ 2S and

*b+b _{m}*

where *Z* is the coordination number and the following relation has been used

where R is the vector connecting the spin at / to a nearest neighbor belonging to the other sublattice, and *N/2* the total number of spins in a sublattice.

The first term of (3.65) is the classical ground-state energy where neighboring spins are perfectly antiparallel (Neel ground state). One introduces now the following Fourier transformations

As in the case of ferromagnets, the Fourier components я_{к}, я^, *b _{k }*and obey the boson commutation relations. Using the above Fourier transforms, Eq. (3.65) becomes

where

and

One sees that *V.* of Eq. (3.71) does not have a diagonal form of the "harmonic oscillator" namely ajc/k and *b£b _{k},* because of the existence of the term

*a*+

_{k}b_{k}*a£b£.*One can diagonalize

*V.*using the following transformation

where *в _{к}* is a variable to be determined. The inverse transformation gives

One can verify that the new operators also obey the commutation relations (see Problem 10 of Section 3.6). Replacing (3.78)-(3.81) in (3.71), one has

*V.* is diagonal if the coefficient before the term *a _{k}p_{k}* + is zero. This requirement allows us to determine the variable

*в*One has

_{к}.Expressing sinh(2%) and cosh(20*) as functions of tanh(26>_{((}) then using Eq. (3.83), one obtains

One recognizes that for a given wave vector **k, **there are two spin wave modes corresponding to

This is the magnon dispersion relation of antiferromagnets. Without an applied field, these modes are degenerate. Note that for small *k, *using *y£* ~ (1 + *ak ^{2}* H----) one obtains

As said before, this result for antiferromagnets is different from a *k ^{2}* obtained for ferromagnets. One expects therefore that thermodynamic properties are different for the two cases in particular at low temperatures where small

*к*modes dominate. This will be indeed seen below.

## Properties at Low Temperatures

If one knows the dispersion relation e_{k} one can in principle use formulas of statistical mechanics to study properties of a system as a function of the temperature (see Appendix A). One writes the partition function *Z* as follows [see Eq. (A.9) with a change of the notation to avoid Z, the coordination number used above]:

where one has used

The free energy is written as

One uses the above expression of *F* to calculate various thermodynamic properties as seen below.

### Energy

For *h =* 0, one has

where one has used 1 = *j =* number of microscopic states in the first Brillouin zone which is equal the number of spins in each sublattice. At *T = 0, n _{k} = n'_{k} = 0* one obtains

The second term is a correction to the classical ground-state energy _zj_ns_ (Neel state). This correction is due to quantum fluctuations in analogy with the zero-point phonon energy.

At low temperatures, one calculates the magnon energy by the use of a low-temperature expansion. One gets

where о is a coefficient proportional to *Z] S.* One notes that the power Г^{4} is different from that of the ferromagnetic case [Eq. (3.51)]. This difference stems from e_{k} ос *к* for small *к* in antiferromagnets.

### Magnetization at low temperatures

To calculate the sublattice magnetization, one writes for the f sublattice

where one has used successively the Fourier transformation and relations (3.78)-(3.81). One expresses now cosh^{2}^ and sinh^{2}^ in terms of yk using (3.83), then one uses

to obtain *M.* At low temperatures, using ^ with e_{k} a *к* for small *k,* one gets

where *AS* = sinh^{2} *в _{к}* is independent of

*T*, and

*A*a coefficient.

One sees that at Г =0, the magnetization is *S* - *AS* which is smaller than the spin magnitude *S. AS* is called the zero-point spin contraction. *AS* depends on the lattice: *AS* ~ 0.197 for an antiferromagnetic square lattice, *AS* ~ 0.078 for a cubic antiferromagnet of NaCl type.

Note that the sublattice magnetization of an antiferromagnet decreases as Г^{2} while the ferromagnetic magnetization decreases as Г^{3/2} [Eq. (3.48)].

# Spin Waves in Ferrimagnets

In this section, one calculates the magnon dispersion relation in the case of a ferrimagnet. In principle, one can use the Holstein- Primakoff method as described above for antiferromagnets. However, the purpose here is to obtain the dispersion relation in a simplest manner. So, one will use the method of equation of motion as described hereafter.

One considers here a simple model of ferrimagnet which is composed of two sublattices of *A* and *В* Heisenberg spins occupying, respectively, the corner sites / and the center sites *m* of a body- centered cubic lattice. The *A* sublattice contains f spins of amplitude *S _{A}* and the

*В*sublattice contains | spins of amplitude

*Sg.*The Hamiltonian (2.72) is rewritten as

where /1 denotes the interaction between *A* and *В* spins (nearest neighbors), *j£* and _/_{2}^{B} denote the intra-sublattice interactions (next-nearest neighbors). To simplify the presentation, one supposes in the following ]_{2}^{Л} = /_{2}^{fl} *= J**2**-*

The Heisenberg equation of motion for the operator S,^{_}(t) reads

The equation of motion for 5+ can be obtained from this equation by exchanging / -о- *m, I'* 0 *m'* and *S ^{+}* -o-

*S~.*It is noted that in this equation, one has replaced, in the mean-field spirit, the operators

*S*and

^{z}m*Sf*by their averaged values

*< S*and

^{z}m >*< Sf >.*Using the following Fourier transformations for 5, and 5+

where *BZ* indicates the first Brillouin zone, one obtains for the body- centered cubic lattice the following coupled equations

where one has used

*a* being the lattice constant, *Z*1 = 8 and Z_{2} = 6 the numbers of nearest and next-nearest neighbors. One recognizes here that *pi *and *рг* connect a site to its nearest neighbors (NN) and next-nearest neighbors (NNN), respectively.

For a non-trivial solution of (3.102)—(3.103), one imposes the secular equation

from which, one obtains We examine some particular cases:

- (1) For
*к*= 0, one has F_ = 0 and*E*= -16a, independent of e, namely_{+}*]*_{2}- - (2) For
*k*=_{x}a*k*=_{y}a = k_{z}a*я,*one has*E+*= 8(1 — a) - 12e(l + a) and = —8(1 + a) + 12e(l — a). - (3) If
*s*= 0, then for*k*=_{x}a = k_{y}a = k_{z}a*я,*one has*E*8(1 — a) and = —8(1 + a)._{+}=

These results show a gap in the magnon spectrum at к = 0 with a width proportional to *a.* Figure 3.5 shows the magnon spectrum versus *k _{x}* =

*k*for

_{y}*k*= 0.

_{z}The energy, the heat capacity and the magnetization at low temperatures can be calculated using the method of the preceding section for antiferromagnet.

Figure 3.5 Magnon spectrum of a ferrimagnet of body-centered cubic lattice versus *k _{x} = k_{y}* with

*k*0,

_{z}=*a = —*1/3,

*e =*0.

# Spin Waves in Helimagnets

Helimagnets are a family of materials in which the spins are not collinear in the low-Г ordered phase as in other systems considered above. Due to a competition between various kinds of interaction, the neighboring spins make an angle which is neither zero nor *л.* Helimagnets are thus frustrated systems which present many unexpected properties [84, 85].

We consider here the simplest example of helimagnet which is a chain with a ferromagnetic interaction *J[>* 0) between nearest neighbors and an antiferromagnetic interaction /_{2}(< 0) between next-nearest neighbors. When *e* = I/2I//1 is larger than a critical value *s _{c},* the spin configuration of the ground state becomes non collinear. One shows that the helical configuration displayed in Fig. 3.6 is obtained by minimizing the following interaction energy

where one has supposed that the angle between nearest neighbors is *в.*

The solutions are ^{•}

Ferromagnetic and antiferromagnetic configurations:

Replacing the above solutions into Eq. (3.114), we see that the antiferromagnetic solution *(в =* 7r) corresponds to the maximum of *E.* It is to be discarded.

• Helical configuration:

Figure 3.6 Helical configuration when г = I/2I//1 *> **s _{c} *

*=*1 /4 (У1 > 0. /2 < 0).

The last solution is possible if-1 < *cosd* < 1, i.e. *J/* (|/_{2}|) < 4 or I/2I//1 > 1/4 = *s _{c}.* There are two degenerate configurations corresponding to clockwise and counter-clockwise turning angles. The ferromagnetic solution has an energy lower than that of the helical solution for I/2I//1 < 1/4. It is therefore more stable in this range of parameters. For I/21/У1 > 1/4, the helical configuration is more stable.

For the magnon spectrum, let us consider a three-dimensional body-centered cubic lattice with Heisenberg spins interacting with each other via (i) a ferromagnetic interaction /1 > 0 between nearest neighbors, (ii) an antiferromagnetic interaction *J _{2} <* 0 between next-nearest neighbors only along the

*у*axis. The Hamiltonian is given by

Figure 3.7 Local coordinates defined for two spins S, and S_{;}. The axis *ц* is common for the two spins.

where *D >* 0 is a very small anisotropy of the type "easy-plane anisotropy” which stabilizes the spins in the *xz* plane. The ground state can be calculated in the same manner as in the case of a chain given above. We find cos *в = ^* so that the helical configuration in the *у* direction is stable when *s = J _{2}/Ji > s_{c} = 1.* Note that the spins in the same

*xz*plane are parallel with each other.

Let *r/j,* £,) be the unit vectors making a direct trihedron at the site *i,* namely *гц* is parallel to the *у* axis as shown in Fig. 3.7. One supposes in addition that the quantization axis of the spin S, coincides with the local axis *Q.*

One uses now the following transformation in the local coordinates associated with S, and *Sj*

One writes

Their scalar product becomes

To be general, the angle *Q* should depend on positions of S, and Sy. One defines cos *Q =* cos(Q • R,y) where Q is the vector of modulus *Q, *perpendicular to the plane of the angle *Q,* namely plane (f, £), and R*jj* the vector connecting the positions of S, and Sy. One shall keep in the following *J* (R,y) as interaction between S, and Sy which will be replaced by *J* i and *J _{2}* depending on R,y at the end of the calculation. Equation (3.117) is rewritten as

With this Hamiltonian, one can choose an appropriate method to calculate the magnon dispersion relation. One can use the Holstein- Primakoff method by replacing the operators S^{1}*^{1} and *S ^{z}* by (3.26)- (3.29), or Green's function method (Chapter 4) or simply by the method of equation of motion (Section 3.3). Using the Holstein- Primakoff method, one obtains the magnon dispersion relation

where

The Hamiltonian (3.125) can be diagonalized by introducing the new operators a_{k} and just as in the antiferromagnetic case studied above

where a_{k} and obey the boson commutation relations [see similar transformation in Eqs. (3.74)-(3.77)]. Hamiltonian (3.125) is diagonal if one takes

One then has

where the energy of the magnon of mode к is

In the case of the body-centered cubic lattice, one has

where = 1 has been used, *a* and *c* being the lattice constants (one uses *c* for the helical axis). Figure 3.8 shows the magnon spectrum for /г/У 1 corresponding to *Q = n/3.* One observes that the magnon frequency is zero not only at *к* = 0 but also at *k _{z} = Q.*

Figure 3.8 Magnon spectrum *E* = *hw*_{k}, Eq. (3.133), versus *k _{z}c* in a helimagnet defined by the Hamiltonian (3.117), with

*Q =*,t/3,

*k*0.

_{x}= k_{y}=# Conclusion

The theory of magnons presented in this chapter allows us to calculate the spin wave dispersion relation which is used to study thermodynamic properties of magnetic systems at low temperatures in a precise manner. When the temperature increases, it is necessary to take into account higher-order terms in the Hamiltonian which represent interactions between magnons. The calculation then becomes more complicated and needs other approximations to decouple chains of operators to renormalize the harmonic, or free- magnon, spectrum. One can also use Green's function method which allows us to include implicitly magnon-magnon interactions up to the transition temperature (see Chapter 4). The method involves, however, some decoupling schemes which make the results near the transition less precise.

Let us summarize some main points of this chapter. First, on the dispersion relation, results of antiferromagnets are quite different from those of ferromagnets. For instance, at small k, one has for antiferromagnets e_{k} a *k,* while for ferromagnets e_{k} a *k ^{2}. *This difference yields different temperature-dependence of physical properties. One notes also that the completely antiparallel spin configuration (Neel state) is not the real ground state of quantum antiferromagnets. Zero-point quantum fluctuations reduce the spin amplitude at Г = 0 in antiferromagnets. For ferrimagnets, only a simple example has been used to illustrate the magnon spectrum. A gap due to the difference of spin magnitudes is frequently found. However, it should be emphasized that real ferrimagnets often have much more complicated lattice structures.

We have also presented some aspects of helimagnets where the magnon spectrum has been shown. It is important to note that systems with non-collinear spin configurations have been and still are subject of intensive investigations since 30 years [84, 85].

Finally, to close this chapter, let us outline some aspects which are important to know:

- (1)
*Magnetic anisotropy:*The Heisenberg model is isotropic.lt does not tell us in which direction the spins should align themselves. It is a habit to suppose that the spins are on the*z*axis for calculation. But all other directions are equivalent. In real materials, there often exists a preferential direction which is called "easy-magnetization axis.” This magnetic "anisotropy” stems from complicated microscopic origins such as spin-orbit interaction, dipole-dipole interaction, etc. [149]. - (2)
*Long-range interactions:*In this chapter, one has supposed for simplicity that interactions are limited to nearest neighbors. The calculations can be of course extended to interactions up to second, third,... nearest neighbors. However, for infinite-range interaction, specific methods should be used. - (3)
*Low-dimensional systems:*In the case of one dimension, a system of spins with short-range interactions are disordered for non-zero*T*whatever the spin model is. In the case of two dimensions, systems of discrete spins such as Ising and Potts models have a phase transition at a finite temperature [25]. Systems of vector spins such as Heisenberg and*XY*models are not ordered at Г ^ 0. This has been rigorously shown by a theorem of Mermin-Wagner [231]. One can see this in an approximative manner: In two dimensions one replaces 4*nk*in (3.46) by^{2}dk*2лkdk,*the integral then becomes divergent at the lower bound (k*-*■*0), causing undefined M except when*T*= 0. - (4) There exist other methods to study temperature-dependent properties of magnetic systems such as low- and high- temperature series expansions [95], renormalization-group analysis and numerical simulations. These are shown in Chapter 5.

# Problems

Problem 1. Prove (3.51)-(3.52).

Problem 2. Chain of Heisenberg spins:

- (a) Calculate the magnon spectrum e(k) for a chain of Heisenberg spins of lattice constant
*a*with ferromagnetic interactions*J*between nearest neighbors and*J*between next-nearest neighbors. Plot e(/c) versus_{2}*к*within the first Brillouin zone (BZ). - (b) The spectrum e(/c) obtained in the previous question is supposed to be valid when
*J*becomes antiferromagnetic as long as e(/c) > 0. Show that the ferromagnetic order becomes unstable when_{2}*J*is larger than a critical value. Determine this value and compare to_{2}*s*given below Eq. (3.116)._{c}

Problem 3. Heisenberg spin systems in two dimensions:

Consider the Heisenberg spin model on a two-dimensional lattice with a ferromagnetic interaction *J* between nearest neighbors.

- (a) Calculate the magnon spectrum e(k) as a function of k. Check that e (k) a
*k*as^{2}*к**—>*0. - (b) Write down the formal expression connecting the magnetization
*M*to the temperature*T.*Show that*M*is undefined as soon as*T*becomes non-zero. Comments.

Problem 4. Prove Eqs. (3.131)-(3.133).

Problem 5. Consider the Ising spin model on a "Union-Jack” lattice, namely the square lattice in which one square out of every two has a centered site. Define sublattice 1 containing the centered sites, and sublattice 2 containing the remaining sites (namely the cornered sites). Let *Ji* be the interaction between a centered spin and its nearest neighbors, *J _{2}* and /з the interactions between two nearest spins on the

*у*and

*x*axes of the sublattice 2, respectively. Determine the phase diagram of the ground state in the space (

*Ji*,

*J*Уз). Indicate the phases where the centered spins are undefined (partial disorder).

_{2},Problem 6. Determine the ground-state spin configuration of a triangular antiferromagnet with XY spins interacting with each other via an exchange *J i* between nearest neighbors. Show that the spin configuration is the 120° structure shown in Fig. 18.6 (left).

Problem 7. Determine the ground state spin configuration of the 2D Villain's model with XY spins defined in Fig. 18.6 (right). Write the energy of the elementary plaquette. By minimizing this energy, determine the ground state as a function of the antiferromagnetic interaction *] _{A}p* = ->]J f where

*>i*is a positive coefficient. Determine the angle between two neighboring spins as a function of

*>].*Show that the critical value of

*>j*beyond which the spin configuration is not collinear is

*tj*= 1/3.

_{c}Problem 8. Uniaxial anisotropy

(a) Show that if one includes in the Heisenberg Hamiltonian for ferromagnets the following anisotropy term — D^_{f}(S?)^{2 }where *D* is a positive constant and the sum is performed over all spins, one obtains the following magnon spectrum

where *d =* [see the definitions of other notations in Eq. (3.40)].

(b) Is it possible to have a long-range magnetic ordering at finite temperature in two dimensions? (cf. Problem 3).

Problem 9. Show that the operators *a ^{+}* and

*a*defined in the Holstein-Primakoff approximation, Eqs. (3.26)-(3.28), respect rigorously the commutation relations between the spin operators.

Problem 10. Show that the operators defined in Eqs. (3.74)-(3.77) obey the commutation relations.

Problem 11. Show that the magnon spectrum (3.113) becomes unstable when the interaction between next-nearest neighbors defined in e, Eq. (3.107), is larger than a critical constant.