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Spin Waves in AntiferromagnetsDispersion RelationConsider 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 HolsteinPrimakoff 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^{z} in (3.53) by these operators, one gets
In a first approximation, one supposes that the number of excited spin waves n is small with respect to 2S, 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 groundstate 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_{к}. One has 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 TemperaturesIf 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. EnergyFor 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 groundstate energy _zj_ns_ (Neel state). This correction is due to quantum fluctuations in analogy with the zeropoint phonon energy. At low temperatures, one calculates the magnon energy by the use of a lowtemperature 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 temperaturesTo 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 zeropoint 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 FerrimagnetsIn 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 intrasublattice interactions (nextnearest neighbors). To simplify the presentation, one supposes in the following ]_{2}^{Л} = /_{2}^{fl} = J2 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 meanfield spirit, the operators S^{z}m and Sf by their averaged values < S^{z}m > and < 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, Z1 = 8 and Z_{2} = 6 the numbers of nearest and nextnearest neighbors. One recognizes here that pi and рг connect a site to its nearest neighbors (NN) and nextnearest neighbors (NNN), respectively. For a nontrivial solution of (3.102)—(3.103), one imposes the secular equation from which, one obtains We examine some particular cases:
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_{y} for k_{z} = 0. 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 bodycentered cubic lattice versus k_{x} = k_{y} with k_{z} = 0, a = —1/3, e = 0. Spin Waves in HelimagnetsHelimagnets are a family of materials in which the spins are not collinear in the lowГ 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 nextnearest 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 if1 < cosd < 1, i.e. J/ (/_{2}) < 4 or I/2I//1 > 1/4 = s_{c}. There are two degenerate configurations corresponding to clockwise and counterclockwise 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 threedimensional bodycentered 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 nextnearest 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 "easyplane 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 Rjj 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 bodycentered 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_{x} = k_{y} = 0. ConclusionThe 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 higherorder 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 magnonmagnon 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 temperaturedependence of physical properties. One notes also that the completely antiparallel spin configuration (Neel state) is not the real ground state of quantum antiferromagnets. Zeropoint 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 noncollinear 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:
ProblemsProblem 1. Prove (3.51)(3.52). Problem 2. Chain of Heisenberg spins:
Problem 3. Heisenberg spin systems in two dimensions: Consider the Heisenberg spin model on a twodimensional lattice with a ferromagnetic interaction J between nearest neighbors.
Problem 4. Prove Eqs. (3.131)(3.133). Problem 5. Consider the Ising spin model on a "UnionJack” 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_{2}, Уз). Indicate the phases where the centered spins are undefined (partial disorder). Problem 6. Determine the groundstate 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_{c} = 1/3. 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 longrange magnetic ordering at finite temperature in two dimensions? (cf. Problem 3). Problem 9. Show that the operators a^{+} and a defined in the HolsteinPrimakoff 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 nextnearest neighbors defined in e, Eq. (3.107), is larger than a critical constant. 
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