The free propagator of strongly anisotropic systems with free surfaces

TL;DR

This work develops an explicit Gaussian propagator for strongly anisotropic systems at a Lifshitz point in a film geometry with perpendicular free surfaces and free boundary conditions. By solving a fourth-order boundary-value problem with Everitt’s Green’s-function framework, the authors derive a closed-form propagator $G_{\perp}^{\rm FF}(p,q;z,z')$ and verify it in the uniaxial limit, enabling a consistent calculation of the one-loop residual free energy and Casimir amplitude. They reproduce the known one-loop result for the perpendicular orientation and provide a concrete route to higher-order perturbative corrections via the propagator, laying groundwork for an $\varepsilon$-expansion treatment of anisotropic Lifshitz-critical Casimir effects. The results offer a robust tool for predicting fluctuation-induced forces in anisotropic confined systems and may inform both theoretical developments and experimental analyses near Lifshitz points.

Abstract

A brief overview of fluctuation-induced forces in statistical systems with film geometry at the critical point and the calculation of Casimir amplitudes, which characterize these forces quantitatively, is presented. Particular attention is paid to the special features of strongly anisotropic $m$-axis systems at the Lifshitz point, specifically, in the case of a "$perpendicular$" orientation of surfaces with free boundary conditions. Beyond the simplest one-loop approximation, calculations of Casimir amplitudes are impossible without knowledge of the Gaussian propagator, which corresponds to the lines of Feynman diagrams in the perturbation theory. We present an explicit expression for such a propagator in the case of an anisotropic system confined by parallel surfaces $perpendicular$ to one of the anisotropy axes. Using this propagator, we reproduce the one-loop result derived earlier in an essentially different way. The knowledge of the propagator provides the possibility of higher-order calculations in perturbation theory.