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

References in corpus (7)

• Monte Carlo simulation results for critical Casimir forces
• Comment on the sign of the Casimir force
• Crossover from Attractive to Repulsive Casimir Forces and Vice Versa
• Critical Casimir Effect in superfluid wetting films
• Thermodynamic Casimir effects involving interacting field theories with zero modes
• Interplay of critical Casimir and dispersion forces
• Finite-size effects in the spherical model of finite thickness

Cited by in corpus (2)

• Dynamic critical behavior of model A in films: Zero-mode boundary conditions and expansion near four dimensions
• Critical Casimir force in slab geometry with finite aspect ratio: analytic calculation above and below $T_c$