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 ${\xi }_{\beta }$ and ${\xi }_{\alpha }$ that scale as nontrivial powers of each other, i.e.\ ${\xi }_{\alpha }\sim {\xi }_{\beta }^{\theta }$ with anisotropy index $\theta \ne 1$. Thus two fundamental orientations, perpendicular ($\perp$) and parallel ($\Vert$), for which the surface normal is oriented along an $\alpha$- and $\beta$-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 ${\mathcal{F}}_C\propto -(\partial /\partial L){\Delta }_{\perp ,\Vert }{L}^{-{\zeta }_{\perp ,\Vert }}$ as the film thickness $L$ becomes large, where the proportionality constants involve nonuniversal amplitudes. The decay exponents ${\zeta }_{\perp ,\Vert }$ and the Casimir amplitudes ${\Delta }_{\perp ,\Vert }$ 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 $d^{\ast }(m)=4+m/2$ under various boundary conditions (BC). The exponents ${\zeta }_{\perp ,\Vert }$ are determined, and explicit results to one- or two-loop order are presented for several Casimir amplitudes ${\Delta }_{\perp ,\Vert }^{\mathrm{BC}}$. The large-$n$ limits of the Casimir amplitudes ${\Delta }_{\Vert }^{\mathrm{BC}}/n$ for periodic and Dirichlet BC are shown to be proportional to their critical-point analogues at dimension $d-m/2$. The limiting values ${\Delta }_{\Vert ,\perp ,\infty }^{\mathrm{PBC}}={\lim }_{n\to \infty }{\Delta }_{\Vert ,\perp }^{\mathrm{PBC}}/n$ are determined exactly for the uniaxial cases $(d,m)=(3,1)$ under periodic BC. Unlike ${\Delta }_{\Vert ,\infty }^{\mathrm{PBC}}$, ${\Delta }_{\perp ,\infty }^{\mathrm{PBC}}$ is positive, so that the corresponding Casimir force is repulsive.

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