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 Ean is proportional to cosθ where θ is the angle by which the spins are rotated from their frozen metastable directions. Incorporating Ean into a "hydrodynamic" free energy the uniform spin-wave spectrum of the system is calculated as a function of the remanent magnetization M→r and an applied field H→ at arbitrary angles. The resulting three modes are mixed longitudinal and transverse unless H→±M→r. In the latter case, the longitudinal frequency is proportional to cos (θr2) where θr is the angle between 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 CuMn. We also discuss briefly the consequences for statics and dynamics of adding to Ean a term quadratic in cosθ, a generalization necessary in order to describe hysteresis.