Spin-resonance frequencies in spin-glasses with random anisotropies

In the case of Heisenberg spin-glasses with weak Dzyaloshinskii-Moriya anisotropic interactions, the macroscopic anisotropy energy $E_{\mathrm{an}}$ is proportional to $cos\theta$ where $\theta$ is the angle by which the spins are rotated from their frozen metastable directions. Incorporating $E_{\mathrm{an}}$ into a "hydrodynamic" free energy the uniform spin-wave spectrum of the system is calculated as a function of the remanent magnetization ${\overrightarrow{\mathrm{M}}}_r$ and an applied field $\overrightarrow{\mathrm{H}}$ at arbitrary angles. The resulting three modes are mixed longitudinal and transverse unless $\overrightarrow{\mathrm{H}}\parallel \pm {\overrightarrow{\mathrm{M}}}_r$. In the latter case, the longitudinal frequency is proportional to cos ($\frac{{\theta }_r}{2}$) where ${\theta }_r$ is the angle between ${\overrightarrow{\mathrm{M}}}_r$ and the cooling field. The predicted field for resonance has a rich angular dependence and is consistent with recent electron-spin-resonance measurements on $\mathrm{Cu}\mathrm{Mn}$. We also discuss briefly the consequences for statics and dynamics of adding to $E_{\mathrm{an}}$ a term quadratic in $cos\theta$, a generalization necessary in order to describe hysteresis.