Macroscopic spin dynamics of spin-glasses with remanence and anisotropy

Real spin-glasses possess both anisotropy and remanence. To interpret spin wave, electron spin resonance (ESR), and related experiments on such systems, one must develop an appropriate macroscopic spin dynamics, thus extending the hydrodynamic theory of Halperin and Saslow. It is found convenient to employ a local spin coordinate system represented by $\hat{n}$ and $\theta$, so that the macroscopic variables are ($\hat{n}, \theta$) and the magnetization $\overrightarrow{\mathrm{m}}$. The equations of motion, for small disturbances about the remanence ${\overrightarrow{\mathrm{m}}}_0$, are analyzed. Longitudinal spin waves and transverse spin waves are obtained; for pure uniaxial anisotropy, the transverse spin waves possess an anisotropy-induced gap. For a static field $\overrightarrow{\mathrm{H}}$ along ${\overrightarrow{\mathrm{m}}}_0$ (so $\overrightarrow{\mathrm{m}}={\overrightarrow{\mathrm{m}}}_0+{\chi }_{\parallel }\overrightarrow{\mathrm{H}}$), the ESR line satisfies $\omega =\gamma H+{\omega }_a$, where ${\omega }_a\propto m^{-1\mathrm{}}$. The phenomenological free energy is made consistent with these results. The effective transverse susceptibility ${\tilde{\chi }}_{\perp }$, for small transverse fields, is also calculated. It is noted that measurement of $m_0$, ${\chi }_{\parallel }$, and the zero-field values of ${\tilde{\chi }}_{\perp }$ and the ESR frequency completely determine the parameters of the theory, so that a measurement of the field dependence of ${\tilde{\chi }}_{\perp }$ can serve as a check on the theory. Spin waves at finite wave vectors, and the effects of remanence on the low-temperature specific heat are also discussed. It is noted that, on the basis of the ESR data, one may conclude that there is an exchange coupling between ${\overrightarrow{\mathrm{m}}}_0$ and ${\chi }_{\parallel }\overrightarrow{\mathrm{H}}$, for small transverse fields.