Casimir force in $O\left(n\right)$ systems with a diffuse interface

We study the behavior of the Casimir force in $O\left(n\right)$ 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. We argue that in such an anisotropic system the Casimir force, the free energy, and the helicity modulus will differ from those of the corresponding isotropic system, even at the bulk critical temperature, despite that these systems both belong to the same universality class. We suggest a relation between the scaling functions pertinent to the both systems. Explicit exact analytical results for the scaling functions, as a function of the temperature $T$, of the free energy density, Casimir force, and the helicity modulus are derived for the $n\to \infty$ limit of $O\left(n\right)$ models with antiperiodic boundary conditions applied along the finite dimension $L$ of the film. We observe that the Casimir amplitude ${\Delta }_{\text{Casimir}}(d\mid J_{\perp },J_{\parallel })$ of the anisotropic $d$-dimensional system is related to that of the isotropic system ${\Delta }_{\text{Casimir}}\left(d\right)$ via ${\Delta }_{\text{Casimir}}(d\mid J_{\perp },J_{\parallel })={(J_{\perp }∕J_{\parallel })}^{(d-1)∕2}{\Delta }_{\text{Casimir}}\left(d\right)$. For $d=3$ we derive the exact Casimir amplitude ${\Delta }_{\text{Casimir}}(3,\mid J_{\perp },J_{\parallel })=[{\mathrm{Cl}}_2(\pi ∕3)∕3-\zeta \left(3\right)∕\left(6\pi \right)](J_{\perp }∕J_{\parallel })$, as well as the exact scaling functions of the Casimir force and of the helicity modulus ${\rm Y}(T,L)$. We obtain that ${\beta }_c{\rm Y}(T_c,L)=(2∕{\pi }^2)[{\mathrm{Cl}}_2(\pi ∕3)∕3+7\zeta \left(3\right)∕\left(30\pi \right)](J_{\perp }∕J_{\parallel })L^{-1}$, where $T_c$ is the critical temperature of the bulk system. We find that the contributions in the excess free energy due to the existence of a diffuse interface result in a repulsive Casimir force in the whole temperature region.

Figure 1

The scaling function $X_{\text{Casimir}}\left(x_t\right)$ of the Casimir force $F_{\text{Casimir}}^{\left(a\right)}(\beta ,N_{\perp }\mid d=3,\mathbf{b})$ for $d=3$. Note that $X_{\text{Casimir}}\left(x_t\right)>0$ for all $x_t$. The asymptotic behavior of $X_{\text{Casimir}}\left(x_t\right)$ for $x_t⪡-1$ is given according to Eq. (3.51).

Figure 2

The scaling function $X_{{\rm Y}}\left(x_t\right)$ of the helicity modulus ${\rm Y}(T,L)$ for $d=3$. One observes that it is a monotonically decreasing function of $x_t$. The asymptote of $X_{{\rm Y}}\left(x_t\right)$ for $x_t⪡-1$ is given in Eq. (4.13).

Figure 3

The scaling functions $X_{\text{Casimir}}\left(x_t\right)$ of the Casimir forces $F_{\text{Casimir}}^{\left(a\right)}(\beta ,N_{\perp }\mid d=3,\mathbf{b})$ and $F_{\text{Casimir}}^{\left(p\right)}(\beta ,N_{\perp }\mid d=3,\mathbf{b})$ for $d=3$. The difference is due to the contributions stemming from the helicity modulus. We see that this contribution is rather strong and dominates the behavior of the force under antiperiodic boundary conditions converting it from attractive (under periodic boundary condition) into a repulsive one (under antiperiodic boundary conditions).