Spin-wave interaction in two-dimensional ferromagnets with dipolar forces

We discuss the spin-wave interaction in two-dimensional (2D) Heisenberg ferromagnet (FM) with dipolar forces at $T_C\gg T\ge0$ using 1/S expansion. A comprehensive analysis is carried out of the first 1/S corrections to the spin-wave spectrum. In particular, similar to 3D FM discussed in our previous paper A.V. Syromyatnikov, PRB {\bf 74}, 014435 (2006), we obtain that the spin-wave interaction leads to the {\it gap} in the spectrum $\epsilon_{\bf k}$ renormalizing greatly the bare gapless spectrum at small momenta $k$. Expressions for the spin-wave damping $\Gamma_{\bf k}$ are derived self-consistently and it is concluded that magnons are well-defined quasi-particles in both quantum and classical 2D FMs at small $T$. We observe thermal enhancement of both $\Gamma_{\bf k}$ and $\Gamma_{\bf k}/\epsilon_{\bf k}$ at small momenta. In particular, a peak appears in $\Gamma_{\bf k}$ and $\Gamma_{\bf k}/\epsilon_{\bf k}$ at small $k$ and at any given direction of $\bf k$. If $S\sim1$ the height of the peak in $\Gamma_{\bf k}/\epsilon_{\bf k}$ is not larger than a value proportional to $T/D\ll1$, where $D$ is the spin-wave stiffness. In the case of large spins $S\gg1$ the peak in $\Gamma_{\bf k}/\epsilon_{\bf k}$ cannot be greater than that of the classical 2D FM found at $k=0$ which height is small only {\it numerically}: $\Gamma_{\bf 0}/\epsilon_{\bf 0}\approx0.16$ for the simple square lattice. Frustrating next-nearest-neighbor exchange coupling increases $\Gamma_{\bf 0}/\epsilon_{\bf 0}$ in classical 2D FM only slightly. We find expressions for spin Green's functions and the magnetization. The latter differs from the well-known result by S.V. Maleev, Sov. Phys. JETP {\bf 43}, 1240 (1976). The effect of the exchange anisotropy is also discussed briefly.