Domain growth in the Heisenberg ferromagnet:: Effective vector theory of the S=1/2 model

We derive an effective vector theory of the spin S=1/2 Heisenberg ferromagnet in an external magnetic field using the Majorana representation for the spin operators and decoupling the interaction term via a Hubbard-Stratonovich transformation. This theory contains both cubic and quartic bosoniclike field terms. We analyze the problem in the Hartree approximation, similarly to the analysis by Boyanovsky [Phys. Rev. E 48, 767 (1993); Phys. Rev. D 48, 800 (1993)] for the scalar case. The time dependence of the radius of the stable phase domain (bubble) in this bosonic theory is studied in the cases of different dimensionalities and weak magnetic field H. The role of the cubic terms in the process of domain growth is analyzed. It is shown that the field components perpendicular to H acquire a larger amplitude than the component parallel to H, at early times. The domain radius grows as root t for times smaller than the spinodal time or in the case of a very weakly coupled theory. A simple scaling analysis shows that the time dependence changes to R similar to t at long times.