Abstract
The critical dynamics of relaxational stochastic models with nonconserved -component order parameter and no coupling to other slow variables (“model ”) is investigated in film geometries for the cases of periodic and free boundary conditions. The Hamiltonian governing the stationary equilibrium distribution is taken to be symmetric and to involve, in the case of free boundary conditions, the boundary terms associated with the two confining surface planes , , at and . Both enhancement variables are presumed to be subcritical or critical, so that no long-range surface order can occur above the bulk critical temperature . A field-theoretic renormalization-group study of the dynamic critical behavior at bulk dimensions is presented, with special attention paid to the cases where the classical theories involve zero modes at . This applies when either both take the critical value associated with the special surface transition or else periodic boundary conditions are imposed. Owing to the zero modes, the expansion becomes ill-defined at . Analogously to the static case, the field theory can be reorganized to obtain a well-defined small- expansion involving half-integer powers of , modulated by powers of . This is achieved through the construction of an effective -dimensional action for the zero-mode component of the order parameter by integrating out its orthogonal component via renormalization-group improved perturbation theory. Explicit results for the scaling functions of temperature-dependent finite-size susceptibilities at temperatures and of layer and surface susceptibilities at the bulk critical point are given to orders and , respectively. They show that dependent shifts of the multicritical special point occur along the temperature and enhancement axes. For the case of periodic boundary conditions, the consistency of the expansions to with exact large- results is shown. We also discuss briefly the effects of weak anisotropy, relating theories whose Hamiltonian involves a generalized square gradient term to those with a conventional term.
- Received 28 October 2008
DOI:https://doi.org/10.1103/PhysRevB.79.104301
©2009 American Physical Society