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 $\varphi$ 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 $O\left(n\right)$ 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 $z=L$. Both enhancement variables ${\mathring{c}}_j$ are presumed to be subcritical or critical, so that no long-range surface order can occur above the bulk critical temperature $T_{c,\infty }$. A field-theoretic renormalization-group study of the dynamic critical behavior at $d=4-ϵ$ 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 $ϵ$ 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-$ϵ$ expansion involving half-integer powers of $ϵ$, modulated by powers of $\text{ln}ϵ$. This is achieved through the construction of an effective $\left(d-1\right)$-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 $T\ge T_{c,\infty }$ 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 $O\left({ϵ}^{3/2}\right)$ 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 $B^{kl}{\partial }_k\mathbf{\varphi }\cdot {\partial }_l\mathbf{\varphi }$ to those with a conventional ${\left(\nabla \mathbf{\varphi }\right)}^2$ term.

Figure 1

(Color online) Graphs contributing to minus the effective action (4.8). Lines with and without an arrow represent the response and correlation propagators $R_L$ and $C_L$, respectively. Red dotted lines indicate $k=0$ components; blue broken lines indicate $k\ne 0$ components. For further explanations, see main text.