Dynamic critical behavior of model A in films: Zero-mode boundary conditions and expansion near four dimensions

The critical dynamics of relaxational stochastic models with nonconserved $n$-component order parameter $\bm{\phi}$ and no coupling to other slow variables ("model A") is investigated in film geometries for the cases of periodic and free boundary conditions. The Hamiltonian $\mathcal{H}$ governing the stationary equilibrium distribution is taken to be O(n) symmetric and to involve, in the case of free boundary conditions, the boundary terms $\int_{\mathfrak{B}_j}\mathring{c}_j \phi^2/2$ associated with the two confining surface planes $\mathfrak{B}_j$, $j=1,2$, at $z=0$ and $z=L$, where the enhancement variables $\mathring{c}_j$ are presumed to be subcritical or critical. A field-theoretic RG study of the dynamic critical behavior at $d=4-\epsilon$ bulk dimensions is presented, with special attention paid to the cases where the classical theories involve zero modes at $T_{c,\infty}$. This applies when either both $\mathring{c}_j$ take the critical value $\mathring{c}_{\text{sp}}$ associated with the special surface transition, or else periodic boundary conditions are imposed. Owing to the zero modes, the $\epsilon$ expansion becomes ill-defined at $T_{c,\infty}$. Analogously to the static case, the field theory can be reorganized to obtain a well-defined small-$\epsilon$ expansion involving half-integer powers of $\epsilon$, modulated by powers of $\ln\epsilon$. Explicit results for the scaling functions of $T$-dependent finite-size susceptibilities at temperatures $T\ge T_{c,\infty}$ and of layer and surface susceptibilities at the bulk critical point are given to orders $\epsilon$ and $\epsilon^{3/2}$, respectively. For the case of periodic boundary conditions, the consistency of the expansions to $O(\epsilon^{3/2})$ with exact large-$n$ results is shown.