Crossover from Attractive to Repulsive Casimir Forces and Vice Versa

Understand

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.