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

TL;DR

This paper investigates the dynamic critical behavior of model A in film geometries near four dimensions, addressing zero-mode boundary conditions and developing an epsilon expansion that accounts for these modes.

Contribution

It introduces a reorganized field-theoretic RG approach to handle zero modes in dynamic critical phenomena, providing explicit scaling functions and extending the epsilon expansion to fractional powers.

Findings

Derived scaling functions for susceptibilities at criticality.

Established the consistency of epsilon^{3/2} expansion with large-n results.

Addressed ill-defined expansions due to zero modes and proposed a solution.

Abstract

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.