Casimir force in O(n) lattice models with a diffuse interface

On the example of the spherical model we study, as a function of the temperature $T$, the behavior of the Casimir force in O(n) systems with a diffuse interface and slab geometry $\infty^{d-1}\times L$, where $2<d<4$ is the dimensionality of the system. We consider a system with nearest-neighbor anisotropic interaction constants $J_\parallel$ parallel to the film and $J_\perp$ across it. The model represents the $n\to\infty$ limit of O(n) models with antiperiodic boundary conditions applied across the finite dimension $L$ of the film. We observe that the Casimir amplitude $\Delta_{\rm Casimir}(d|J_\perp,J_\parallel)$ of the anisotropic $d$-dimensional system is related to that one of the isotropic system $\Delta_{\rm Casimir}(d)$ via $\Delta_{\rm Casimir}(d|J_\perp,J_\parallel)=(J_\perp/J_\parallel)^{(d-1)/2} \Delta_{\rm Casimir}(d)$. For $d=3$ we find the exact Casimir amplitude $\Delta_{\rm Casimir}= [ {\rm Cl}_2 (\pi/3)/3-\zeta (3)/(6 \pi)](J_\perp/J_\parallel)$, as well as the exact scaling functions of the Casimir force and of the helicity modulus $\Upsilon(T,L)$. We obtain that $\beta_c\Upsilon(T_c,L)=(2/\pi^{2}) [{\rm Cl}_2(\pi/3)/3+7\zeta(3)/(30\pi)] (J_\perp/J_\parallel)L^{-1}$, where $T_c$ is the critical temperature of the bulk system. We find that the effect of the helicity is thus strong that the Casimir force is repulsive in the whole temperature region.