Casimir force in the rotor model with twisted boundary conditions

TL;DR

This paper studies the Casimir force in a three-dimensional lattice XY model with twisted boundary conditions, showing how the force varies with temperature, boundary angle, and system thickness, including a phase transition at a specific twist angle.

Contribution

It introduces a mean field analysis of the Casimir force in the XY model with twisted boundaries, revealing controllable attractive or repulsive forces and a boundary-induced phase transition.

Findings

Casimir force depends continuously on boundary angle and temperature.

Force can be tuned to be attractive or repulsive.

A boundary-induced phase transition occurs at a specific twist angle.

Abstract

We investigate the three dimensional lattice XY model with nearest neighbor interaction. The vector order parameter of this system lies on the vertices of a cubic lattice, which is embedded in a system with a film geometry. The orientations of the vectors are fixed at the two opposite sides of the film. The angle between the vectors at the two boundaries is $\alpha$ where $0 \le \alpha \le \pi$. We make use of the mean field approximation to study the mean length and orientation of the vector order parameter throughout the film---and the Casimir force it generates---as a function of the temperature $T$, the angle $\alpha$, and the thickness $L$ of the system. Among the results of that calculation are a Casimir force that depends in a continuous way on both the parameter $\alpha$ and the temperature and that can be attractive or repulsive. In particular, by varying $\alpha$ and/or $T$ one controls \underline{both} the sign \underline{and} the magnitude of the Casimir force in a reversible way. Furthermore, for the case $\alpha=\pi$, we discover an additional phase transition occurring only in the finite system associated with the variation of the orientations of the vectors.