Casimir force in the rotor model with twisted boundary conditions

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.