Critical dynamics of ferromagnets in6−εdimensions: General discussion and detailed calculation

We report in detail a study of the critical dynamics of a model ferromagnet in $6-\epsilon$ dimensions both above and below $T_c$, for small $\epsilon$. The precession of spins in the local magnetic field plays a major role in the dynamics. Its effect can be ignored at 6 or more dimensions, as many authors have previously observed. At a dimension slightly less than 6, the effect is nontrivial but tractable by perturbation theory. We carry out a renormalization-group analysis and determine the dynamic exponent $z=4-\frac{\epsilon }{2}=1+\frac{d}{2}$, accurate to $O\left(\epsilon \right)$. We also solve the equation of motion to obtain the spin response function. The frequency and damping of spin waves below $T_c$ are determined. Forms of equations of motion and the extent of universality in critical dynamics are discussed.