Fluctuation-induced forces in strongly anisotropic critical systems

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 $\theta \mathrm{\ne }1$. Thus two fundamental orientations, perpendicular ($\perp$) and parallel ($\mathrm{\parallel }$), 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}$C\propto-(\partial/\partial L)\Delta{\perp,|},L^{-\zeta_{\perp,|}}$asthefilmthickness$L$ becomes large, where the proportionality constants involve nonuniversal amplitudes. The decay exponents ${\zeta }_{\perp ,\mathrm{\parallel }}$ and the Casimir amplitudes ${\mathrm{\Delta }}_{\perp ,\mathrm{\parallel }}$ 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 $d^{\ast }(m)=4+m\mathrm{/}2$ under various boundary conditions (BC). The exponents ${\zeta }_{\perp ,\mathrm{\parallel }}$ are determined, and explicit results to one- or two-loop order are presented for several Casimir amplitudes ${\mathrm{\Delta }}_{\perp ,\mathrm{\parallel }}^{\mathrm{B}\mathrm{C}}$. The large-$n$ limits of the Casimir amplitudes ${\mathrm{\Delta }}_{\mathrm{\parallel }}^{\mathrm{B}\mathrm{C}}\mathrm{/}n$ for periodic and Dirichlet BC are shown to be proportional to their critical-point analogues at dimension $d-m\mathrm{/}2$. The limiting values ${\mathrm{\Delta }}_{\mathrm{\parallel },\perp ,\mathrm{\infty }}^{\mathrm{P}\mathrm{B}\mathrm{C}}={\lim }_{n\to \mathrm{\infty }}{\mathrm{\Delta }}_{\mathrm{\parallel },\perp }^{\mathrm{P}\mathrm{B}\mathrm{C}}\mathrm{/}n$ are determined exactly for the uniaxial cases $(d,m)=(3,1)$ under periodic BC. Unlike ${\mathrm{\Delta }}_{\mathrm{\parallel },\mathrm{\infty }}^{\mathrm{P}\mathrm{B}\mathrm{C}}$, ${\mathrm{\Delta }}_{\perp ,\mathrm{\infty }}^{\mathrm{P}\mathrm{B}\mathrm{C}}$ is positive, so that the corresponding Casimir force is repulsive.