Fluctuation-induced forces in strongly anisotropic critical systems

Abstract

Strongly anisotropic critical systems are considered in a $d$-dimensional film geometry. Such systems involve two (or more) distinct correlation lengths $ξ_β$ and $ξ_α$ that scale as nontrivial powers of each other, i.e.\ $ξ_α\simξ_β^θ$ with anisotropy index $θ\ne 1$. Thus two fundamental orientations, perpendicular ($\perp$) and parallel ($\|$), for which the surface normal is oriented along an $α$- and $β$-direction, respectively, must be distinguished. The confinement of critical fluctuations caused by the film's boundary planes is shown to induce effective forces $\mathcal{F}_C$ that decay as $\mathcal{F}_C\propto-(\partial/\partial L)Δ_{\perp,\|}\,L^{-ζ_{\perp,\|}}$ as the film thickness $L$ becomes large, where the proportionality constants involve nonuniversal amplitudes. The decay exponents $ζ_{\perp,\|}$ and the Casimir amplitudes $Δ_{\perp,\|}$ are universal but depend on the type of orientation. To corroborate these findings, $n$-vector models with an $m$-axial bulk Lifshitz point are investigated by means of RG methods below the upper critical dimension $d^*(m)=4+m/2$ under various boundary conditions (BC). The exponents $ζ_{\perp,\|}$ are determined, and explicit results to one- or two-loop order are presented for several Casimir amplitudes $Δ^{\mathrm{BC}}_{\perp,\|}$. The large-$n$ limits of the Casimir amplitudes $Δ_{\|}^{\mathrm{BC}}/n$ for periodic and Dirichlet BC are shown to be proportional to their critical-point analogues at dimension $d-m/2$. The limiting values $Δ^{\mathrm{PBC}}_{\|,\perp,\infty}=\lim_{n\to\infty}Δ_{\|,\perp}^{\mathrm{PBC}}/n$ are determined exactly for the uniaxial cases $(d,m)=(3,1)$ under periodic BC. Unlike $Δ^{\mathrm{PBC}}_{\|,\infty}$, $Δ^{\mathrm{PBC}}_{\perp,\infty}$ is positive, so that the corresponding Casimir force is repulsive.