Quantum critical dynamics with mode coupling. I. Ferromagnets

We examine the low-temperature critical dynamics of two ferromagnetic models of the disordered state having "mode-mode coupling." A method is developed to incorporate mode-mode coupling contributions in an $O\left(\epsilon \right)$ momentum-space renormalization-group analysis of a quantum system. For each model, both quantum and classical critical regions may exist. The characteristic dimensionality changes from $d=6\mathrm{}$ classically to $d=10\mathrm{}$ in the quantum limit. The dynamic critical index associated with the quantum fixed point is $z=4-\frac{1}{3}\mathrm{}(10-d)\mathrm{}$. The scaling behavior of the spin transport coefficient $\Gamma$ is examined in the $T=0\mathrm{}$ limit. We discuss the properties of the quantum-to-classical crossover, including the "metastable" nature of the quantum fixed points.