Critical Casimir effect in films for generic non-symmetry-breaking boundary conditions

TL;DR

This paper presents a detailed renormalization-group analysis of the critical Casimir effect in films with symmetry-preserving boundary conditions, calculating force behaviors and crossovers near the bulk critical point.

Contribution

It provides the first comprehensive RG-based calculation of the critical Casimir force for generic Robin boundary conditions in O(n) symmetric systems, including scaling functions and crossover phenomena.

Findings

Casimir force can be attractive or repulsive depending on boundary parameters.

Scaling functions are determined via epsilon expansion and extrapolated to three dimensions.

Crossovers between attraction and repulsion occur as film thickness varies.

Abstract

Systems described by an O(n) symmetrical $\phi^4$ Hamiltonian are considered in a $d$-dimensional film geometry at their bulk critical points. A detailed renormalization-group (RG) study of the critical Casimir forces induced between the film's boundary planes by thermal fluctuations is presented for the case where the O(n) symmetry remains unbroken by the surfaces. The boundary planes are assumed to cause short-ranged disturbances of the interactions that can be modelled by standard surface contributions $\propto \bm{\phi}^2$ corresponding to subcritical or critical enhancement of the surface interactions. This translates into mesoscopic boundary conditions of the generic symmetry-preserving Robin type $\partial_n\bm{\phi}=\mathring{c}_j\bm{\phi}$. RG-improved perturbation theory and Abel-Plana techniques are used to compute the $L$-dependent part $f_{\mathrm{res}}$ of the reduced excess free energy per film area $A\to\infty $ to two-loop order. When $d<4$, it takes the scaling form $f_{\mathrm{res}}\approx D(c_1L^{\Phi/\nu},c_2L^{\Phi/\nu})/L^{d-1}$ as $L\to\infty$, where $c_i$ are scaling fields associated with the surface-enhancement variables $\mathring{c}_i$, while $\Phi$ is a standard surface crossover exponent. The scaling function $D(\mathsf{c}_1,\mathsf{c}_2)$ and its analogue $\mathcal{D}(\mathsf{c}_1,\mathsf{c}_2)$ for the Casimir force are determined via expansion in $\epsilon=4-d$ and extrapolated to $d=3$ dimensions. In the special case $\mathsf{c}_1=\mathsf{c}_2=0$, the expansion becomes fractional. Consistency with the known fractional expansions of D(0,0) and $\mathcal{D}(0,0)$ to order $\epsilon^{3/2}$ is achieved by appropriate reorganisation of RG-improved perturbation theory. For appropriate choices of $c_1$ and $c_2$, the Casimir forces can have either sign. Furthermore, crossovers from attraction to repulsion and vice versa may occur as $L$ increases.