Spin Wave Theory for the Classical Easy Plane Ferromagnet in Zero Field: Exact Results at Low Temperature

Abstract

We present a spin wave calculation of the dynamical response of the classical easy plane ferromagnet

$$H = - {\text{J}}\sum\limits_i {{{\mathop {\text{S}}\limits^ \to }_i}\cdot{{\mathop {\text{S}}\limits^ \to }_{i + 1}} + D\sum\limits_i {{{\left( {s_i^z} \right)}^2}} ,} $$

(1)

J,D > 0 for KT ≪ $\sqrt {8{\text{DJ}}} {S^2}$. One can calculate the dynamical response function

$$[R_q^a\left( z \right) = \int\limits_o^\infty {{e^{izt}}} \left\langle {s_q^a\left( t \right)s_{ - q}^a} \right\rangle dt$$

(2)

asymptotically exactly, for all wavevectors, as T → 0. The method used is an extension of that used to treat the fully isotropic system [1]. The response function can be written as [2]

$$R_q^a\left( z \right) = i\left\langle {s_q^as_{ - q}^a} \right\rangle {\left[ {z - \omega _{q,a}^2/(z + \Upsilon _q^a(z)} \right]^{ - 1}}$$

(3)

.