Fluctuation-induced forces in strongly anisotropic critical systems

TL;DR

This paper investigates fluctuation-induced forces in strongly anisotropic critical systems, revealing how boundary conditions and orientation influence universal Casimir amplitudes and decay exponents through RG analysis and large-n limits.

Contribution

It provides the first detailed RG analysis of Casimir forces in anisotropic systems with Lifshitz points, including explicit calculations of decay exponents and amplitudes for various boundary conditions.

Findings

Casimir forces decay as power laws with orientation-dependent exponents.

Explicit one- and two-loop results for Casimir amplitudes are obtained.

Large-n limits of Casimir amplitudes are derived and exact for uniaxial cases.

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 ($\|$), 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 $\mathcal{F}_C\propto-(\partial/\partial L)\Delta_{\perp,\|}\,L^{-\zeta_{\perp,\|}}$ as the film thickness $L$ becomes large, where the proportionality constants involve nonuniversal amplitudes. The decay exponents $\zeta_{\perp,\|}$ and the Casimir amplitudes $\Delta_{\perp,\|}$ 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 $\zeta_{\perp,\|}$ are determined, and explicit results to one- or two-loop order are presented for several Casimir amplitudes $\Delta^{\mathrm{BC}}_{\perp,\|}$. The large-$n$ limits of the Casimir amplitudes $\Delta_{\|}^{\mathrm{BC}}/n$ for periodic and Dirichlet BC are shown to be proportional to their critical-point analogues at dimension $d-m/2$. The limiting values $\Delta^{\mathrm{PBC}}_{\|,\perp,\infty}=\lim_{n\to\infty}\Delta_{\|,\perp}^{\mathrm{PBC}}/n$ are determined exactly for the uniaxial cases $(d,m)=(3,1)$ under periodic BC. Unlike $\Delta^{\mathrm{PBC}}_{\|,\infty}$, $\Delta^{\mathrm{PBC}}_{\perp,\infty}$ is positive, so that the corresponding Casimir force is repulsive.