Casimir interaction of rod-like particles in a two-dimensional critical system

TL;DR

This paper analyzes the fluctuation-induced Casimir forces between rod-like particles in a two-dimensional critical Ising system using conformal mapping techniques, providing analytical and numerical results across different geometries.

Contribution

It introduces a conformal mapping approach to compute Casimir interactions for needles of arbitrary length and orientation in a 2D critical system, extending previous models.

Findings

Analytical expressions for forces and torques between needles of various lengths.

Numerical results showing crossover from small to large separation regimes.

Validation of results against known limits and small-particle expansion.

Abstract

We consider the fluctuation-induced interaction of two thin, rod-like particles or "needles" immersed in a two-dimensional critical fluid of Ising symmetry right at the critical point. Conformally mapping the plane containing the needles onto a simpler geometry in which the stress tensor is known, we analyze the force and torque between needles of arbitrary length, separation, and orientation. For infinite and semi-infinite needles we utilize the mapping of the plane bounded by the needles onto the half plane, and for two needles of finite length the mapping onto an annulus. For semi-infinite and infinite needles the force is expressed in terms of elementary functions, and we also obtain analytical results for the force and torque between needles of finite length with separation much greater than their length. Evaluating formulas in our approach numerically for several needle geometries and surface universality classes, we study the full crossover from small to large values of the separation to length ratio. In these two limits the numerical results agree with results for infinitely long needles and with predictions of the small-particle operator expansion, respectively.