Fluctuation-induced forces in strongly anisotropic critical systems

Strongly anisotropic critical systems have two (or more) correlation lengths $\xi_\alpha$ and $\xi_\beta$ that diverge as nontrivial powers $\xi_\alpha\sim \xi_\beta^\theta\to \infty$ upon approaching criticality. We investigate the effective (Casimir-like) forces that are induced between two confining parallel boundary planes at a distance $L$ by fluctuations in such systems at bulk criticality. Two fundamentally distinct orientations of boundary planes must be distinguished: parallel, for which the planes are parallel to all of the available $1\le m < d$ $\alpha$-directions, and perpendicular, for which they are perpendicular to an $\alpha$-direction, but parallel to all other $\alpha$- and $\beta$-directions. Using a RG approach, we show that universal Casimir amplitudes $\Delta^{BC}_{\|,\perp}$, depending on both the large-scale boundary condition (BC) at both plates and the type of surface plane orientation, can be introduced to characterize the asymptotic $L$-dependence of the critical fluctuation-induced force. This varies as $\mathcal{F}\sim -(\partial/\partial L)$$\,\Delta^{BC}_{\|,\perp}\,L^{-\zeta_{\|,\perp}}$, where the proportionality constant is nonuniversal. To corroborate these findings, $O(n)$ $\phi^4$ models with $m$-axial Lifshitz points are investigated below their upper critical dimension $d=4+m/2$. Explicit one- and two-loop results for $\Delta^{BC}_{\|,\perp}$ are presented for both orientations and periodic or Dirichlet-like boundary conditions, along with large-$n$ results.