Advanced Solid State Physics

Sx,Sy<< Sz. It is now reasonable to neglect product terms ofSxSx,SySyandSxSyrenamingSz
toS. This results in the linearized system valid in the low energy limit

d

dt

S~xk =^2 J

~

S(2Syk−Sky− 1 +−Syk+1) (149)

d

dt

S~ky = −^2 J

~

S(2Skx−Skx− 1 +−Sxk+1) (150)

d

dt

S~kz = 0 (151)

. To solve this system we use plane wave solutions

Sxp = x 0 ei(pka−ωt) (152)

Spy = y 0 ei(pka−ωt) (153)

wherepis a natural number andais the lattice constant. Plugging in, the resulting equations are

−iωx 0 =

2 JS

~

(2−e−ika−eika)y 0 =

4 JS

~

(1−cos(ka))y 0 (154)

−iωy 0 = −

2 JS

~

(2−e−ika−eika)x 0 =−

4 JS

~

(1−cos(ka))x 0 (155)

. The solution to this set of equations (setting the determinant to zero) is

ω=

4 JS

~

(1−cos(ka)) (156)

which is the magnon dispersion relation in one dimension considering only nearest neighbor interaction.
The eigenfunctions for the S components take the form

Sxp = ξcos(pka−ωt) (157)

Spy = ξsin(pka−ωt) (158)

Spz = η (159)

which is a spin precession around the z axis. Expanding the cosine in the dispersion relation one gets
aE∝k^2 behaveiour for long wavelengths (see figure 78).

To take a look at the thermal properties of such a system we consider the overall number of magnons
which are excited at temperature T

Nexc=

∑

k

nk=

∫

1 .BZ

D(ω)< n(ω)> dω (160)

where< n(ω)>is the Bose-Einstein statistics

< n(ω)>=

1

e

~ω

kBT− 1

(161)