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

Abstract

The critical dynamics of relaxational stochastic models with nonconserved $n$-component order parameter $\mathbf{\varphi }$ 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 ${\int }_{{\mathfrak{B}}_j}{\mathring{c}}_j{\varphi }^2/2$ 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.

Latex file with 8 eps files included; text added in conclusions and abstract, typos corrected

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Cited by in corpus (1)

• Critical Casimir amplitudes for $n$-component ${\varphi }^4$ models with O(n)-symmetry breaking quadratic boundary terms