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

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The critical dynamics of relaxational stochastic models with nonconserved $n$-component order parameter $\ensuremath{\phi}$ 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 ${\ensuremath{\int}}_{{\mathfrak{B}}_{j}}{\stackrel{\r{}}{c}}_{j}{\ensuremath{\phi}}^{2}/2$ associated with the two confining surface planes ${\mathfrak{B}}_{j}$, $j=1,2$, at $z=0$ and $z=L$. Both enhancement variables ${\stackrel{\r{}}{c}}_{j}$ are presumed to be subcritical or critical, so that no long-range surface order can occur above the bulk critical temperature ${T}_{c,\ensuremath{\infty}}$. A field-theoretic renormalization-group study of the dynamic critical behavior at $d=4\ensuremath{-}ϵ$ bulk dimensions is presented, with special attention paid to the cases where the classical theories involve zero modes at ${T}_{c,\ensuremath{\infty}}$. This applies when either both ${\stackrel{\r{}}{c}}_{j}$ take the critical value ${\stackrel{\r{}}{c}}_{\text{sp}}$ associated with the special surface transition or else periodic boundary conditions are imposed. Owing to the zero modes, the $ϵ$ expansion becomes ill-defined at ${T}_{c,\ensuremath{\infty}}$. Analogously to the static case, the field theory can be reorganized to obtain a well-defined small-$ϵ$ expansion involving half-integer powers of $ϵ$, modulated by powers of $\text{ln}\text{ }ϵ$. This is achieved through the construction of an effective $(d\ensuremath{-}1)$-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 $T\ensuremath{\ge}{T}_{c,\ensuremath{\infty}}$ and of layer and surface susceptibilities at the bulk critical point are given to orders $ϵ$ and ${ϵ}^{3/2}$, respectively. They show that $L$ 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 $O({ϵ}^{3/2})$ with exact large-$n$ results is shown. We also discuss briefly the effects of weak anisotropy, relating theories whose Hamiltonian involves a generalized square gradient term ${B}^{kl}{\ensuremath{\partial}}_{k}\mathbf{\ensuremath{\phi}}\ensuremath{\cdot}{\ensuremath{\partial}}_{l}\mathbf{\ensuremath{\phi}}$ to those with a conventional ${(\ensuremath{\nabla}\mathbf{\ensuremath{\phi}})}^{2}$ term.