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 both the sign 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.

Figure 1

Renderings of the moments ($d=3$, $N=30$, $\alpha =\pi$) for temperatures above (top) and below bottom) the temperature where a kink occurs in the Casimir force.

Figure 2

The Casimir force in the lattice model ($d=3$, $N=30$) as a function of reduced temperature $t=(T-T_{c,\mathrm{bulk}})/T_{c,\mathrm{bulk}}$ for various values of the twist angle $\alpha$.

Figure 3

A plot of the dimensionless scaling function for the Casimir force $X_{\mathrm{Cas}}$ vs $x_t$ (also dimensionless) for several values of $\alpha$. The dotted curve is the Casimir scaling function in the Ising-like case of a critical fluid under $(+,-)$ boundary conditions. For $x_t$ above a certain value, this curve coincides with the one for the model studied here when twisted by an angle $\alpha \approx \pi$.

Figure 4

The scaling functions for the Casimir force in the Ginzburg-Landau model (solid curves), overlaid with those from the lattice model with $N=50$ (data points), for several values of $\alpha$ (from top to bottom: $\alpha =0.98\pi$, $\alpha =2\pi /3$, $\alpha =\pi /2$, $\alpha =\pi /3$, $\alpha =0$).

Figure 5

Casimir force curves in the Ginzburg-Landau model for several values of $\alpha$, overlayed with their respective asymptotic expressions (dotted) given by Eq. (3.23) as proven in Appendix .

Figure 6

Top: The positions $x_{t,0}^{\left(\alpha \right)}$ of the zeros of the Casimir force in the $(x_t,\alpha )$ plane. Bottom: The Casimir amplitude ${\Delta }_{\mathrm{Cas}}^{\left(\alpha \right)}$ as a function of the twist angle $\alpha$. This curve has been initially reported in [7] based on representation (3.29) derived there. The Casimir amplitude changes sign at $\alpha =\pi /3$.

Figure 7

The dimensionless quantities $X_0$ (solid), $X_{\varphi }/10$ (long dashing; scale reduced for ease of plotting) and $X_{\varphi }/X_0$ (short dashing) as functions of $x_t$ when $\alpha \approx \pi$. The vertical dotted line indicates $x_t=x_{t,\mathrm{kink}}$.

Figure 8

The loci of points in the $(p,\tau )$-plane for which the roots $x_{\pm }$ possess the properties discussed in the main text. When any of the roots approaches the thick blue line, $X_0\to \infty$. One observes that this is only possible for $\tau <0$, i.e., when $\tau X_0^2=x_t\to -\infty$.

Figure 9

Plots of the amplitude profile and the angle of the order parameter for $\alpha =\pi /3$ and some choices of $x_t$. We observe that, when the temperature increases, the value of the amplitude in the middle of the system decreases. The twist of the local variables through the system spans over the total system almost uniformly for low temperatures, while for higher ones it concentrates more and more in the middle of the system where the amplitude is at its smallest values.