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

Abstract

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 ${\Delta }_{\mathrm{Casimir}}(d|J_{\perp },J_{\parallel })$ of the anisotropic $d$-dimensional system is related to that one of the isotropic system ${\Delta }_{\mathrm{Casimir}}(d)$ via ${\Delta }_{\mathrm{Casimir}}(d|J_{\perp },J_{\parallel })=(J_{\perp }/J_{\parallel }{)}^{(d-1)/2}{\Delta }_{\mathrm{Casimir}}(d)$. For $d=3$ we find the exact Casimir amplitude ${\Delta }_{\mathrm{Casimir}}=[{\mathrm{Cl}}_2(\pi /3)/3-\zeta (3)/(6\pi )](J_{\perp }/J_{\parallel })$, as well as the exact scaling functions of the Casimir force and of the helicity modulus ${\rm Y}(T,L)$. We obtain that ${\beta }_c{\rm Y}(T_c,L)=(2/{\pi }^2)[{\mathrm{Cl}}_2(\pi /3)/3+7\zeta (3)/(30\pi )](J_{\perp }/J_{\parallel })L^{-1}$, 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.

15 pages, 3 figures

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