The critical dynamics at elastic phase transitions of second order is studied by the renormalization-group theory. The dynamical theory is based on a stochastic equation of motion for damped phonons. For systems with one-dimensional soft sectors usual dynamical scaling holds with the critical dynamical exponent $z=2\mathrm{}$. For two-dimensional sectors logarithmic corrections appear. Novel features are found for an isotropic-phonon model. The dynamical susceptibility and the characteristic frequency depend singularly on an irrelevant parameter, which, however, diverges at the fixed point. Consequently dynamical scaling breaks down; e.g., the characteristic frequency is no longer a homogeneous function of the wave number $k$ and the inverse correlation length ${\xi }^{-1\mathrm{}}$. Instead it is a homogeneous function of these variables and an irrelevant parameter. Consequently, the critical dynamics can be characterized by the relaxational exponent $z=2+c\eta$ at $T_c$, while in the hydrodynamic region the sound frequency is characterized by $z=2-\frac{1}{2}\eta$ and the damping by yet another exponent. This breakdown of scaling is also reflected by the fact that different fixing conditions, i.e., different choices of the frequency, lead to different values of $z$. All of these apparently different transformations lead to the same modified dynamical scaling relations.

• Received 1 February 1979

DOI:https://doi.org/10.1103/PhysRevB.20.1229