Ordering dynamics of Heisenberg spins with torque: Crossover, spin waves, and defects

We study the effect of a torque induced by the local molecular field on the phase ordering dynamics of the Heisenberg model when the total magnetization is conserved. The torque drives the zero-temperature ordering dynamics to a new fixed point, characterized by exponents $z=2\mathrm{}$ and $\lambda \approx 5.$ This “torque-driven” fixed point is approached at times such that $t\gg g^2,$ where g is the strength of the torque. All physical quantities, like the domain size $L\left(t\right)\mathrm{}$ and the equal and unequal time correlation functions, obey a crossover scaling form over the entire range of g. An attempt to understand this crossover behavior from the approximate Gaussian closure scheme fails completely, implying that the dynamics at late times cannot be understood from the dynamics of defects alone. We provide convincing arguments that the spin configurations can be decomposed in terms of defects and spin waves which interact with each other even at late times. In the absence of the torque term, the spin waves decay faster, but even so we find that the Gaussian closure scheme is inconsistent. In the latter case the inconsistency may be remedied by including corrections to a simple Gaussian distribution. For completeness we include a discussion of the ordering dynamics at $T_c,$ where the torque is shown to be relevant, with exponents $z=4\mathrm{}-\varepsilon /2\mathrm{}$ and $\lambda =d$ (where $\varepsilon =6\mathrm{}-d).\mathrm{}$ We show to all orders in perturbation theory that $\lambda =d$ as a consequence of the conservation law.