Crossover from Attractive to Repulsive Casimir Forces and Vice Versa

TL;DR

This paper investigates how critical Casimir forces between boundary planes in a film can switch from attractive to repulsive depending on boundary conditions and film thickness, using renormalization group analysis.

Contribution

It provides explicit two-loop RG calculations of the Casimir force behavior at the bulk critical point for generic boundary conditions in $d<4$ dimensions.

Findings

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

Crossovers from attraction to repulsion occur as film thickness varies.

Explicit two-loop results for the force scaling function at $d=4-\e$ dimensions.

Abstract

Systems described by an O(n) symmetrical $\phi^4$ Hamiltonian are considered in a $d$-dimensional film geometry at their bulk critical points. The critical Casimir forces between the film's boundary planes $\mathfrak{B}_j, j=1,2$, are investigated as functions of film thickness $L$ for generic symmetry-preserving boundary conditions $\partial_n\bm{\phi}=\mathring{c}_j\bm{\phi}$. The $L$-dependent part of the reduced excess free energy per cross-sectional area takes the scaling form $f_{\text{res}}\approx D(c_1L^{\Phi/\nu},c_2L^{\Phi/\nu})/L^{d-1}$ when $d<4$, where $c_i$ are scaling fields associated with the variables $\mathring{c}_i$, and $\Phi$ is a surface crossover exponent. Explicit two-loop renormalization group results for the function $D(\mathsf{c}_1,\mathsf{c}_2)$ at $d=4-\epsilon$ dimensions are presented. These show that (i) the Casimir force can have either sign, depending on $\mathsf{c}_1$ and $\mathsf{c}_2$, and (ii) for appropriate choices of the enhancements $\mathring{c}_j$, crossovers from attraction to repulsion and vice versa occur as $L$ increases.