Finite-size Nagle-Kardar model: Casimir force

Abstract

We derive exact results for the critical Casimir force (CCF) within the Nagle-Kardar model with periodic boundary conditions (PBC's). The model represents one-dimensional Ising chain with long-range equivalent-neighbor ferromagnetic interactions of strength $J_{l}/N>0$ superimposed on the nearest-neighbor interactions of strength $J_{s}$ which could be either ferromagnetic ($J_{s}>0$) or antiferromagnetic ($J_{s}<0$). In the infinite system limit the model exhibits in the plane $(K_s=βJ_s,K_l=βJ_l)$ a critical line $2 K_l=\exp{\left(-2 K_s\right)}, K_s>-\ln3/4$, which ends at a tricritical point $(K_l=-\sqrt{3}/2, K_s=-\ln3/4)$. The critical Casimir amplitudes are: $Δ_{\rm Cas}^{\rm (cr)}=1/4$ at the critical line, and $Δ_{\rm Cas}^{\rm (tr)}=1/3$ at the tricritical point. Quite unexpectedly, with the imposed PBC's the CCF exhibits very unusual behavior as a function of temperature and magnetic field. It is \textit{repulsive} near the critical line and tricritical point, decaying rapidly with separation from those two singular regimes fast away from them and becoming \textit{attractive}, displaying in which the maximum amplitude of the attraction exceeds the maximum amplitude of repulsion. This represents a violation of the widely-accepted ``boundary condition rule,'' which holds that the CCF is attractive for equivalent BC's and repulsive for conflicting BC's \textit{independently} of the actual bulk universality class of the phase transition under investigation.

Figures (3)

• Figure 1: The phase diagram in terms of the temperature, shown as a function of $J_s$ and $J_l$. The red line $T/J_l=2W_p(x)$ represents a line of critical points, while the green point marks the tricritical point C with coordinates $\{y_{\rm TP}=2/\sqrt{3}, x_{\rm TP}=-\ln(3)/(2\sqrt{3})\}$. A zero field first-order transition temperature (the blue line) meets the second order transition line at point $C$ and ends at $x=-0.5$.
• Figure 2: The behavior of the CCF pertinent to NK model as a function of $h$ for different fixed values of $K_s$ and $K_l$ with $N=100$. The red line corresponds to $K_s=0, K_l=0.5$, while the blue line corresponds to the tricritical point which emerges in the phase diagram at coordinate $K_s=-\ln 3/4,K_l=\sqrt{3}/2$ obtained from the conditions: the multipliers in front of $m^2$ and $m^4$ equal zero, see Eq.\ref{['mag1']}. The results are in a full agreement with the derived exact results - see Eq. (\ref{['eq:Cas-at-the-critical-line']}) and Eq. (\ref{['eq:Casimir-force-tricritical-point']}).
• Figure 3: The 3D visualization of the behavior of the CCF as a function of $h$ for different fixed values of $K_s$ and $K_l$ with $N=100$. Here $K_l\in (0,1.5]$ while $K_s=\ln\left[1/(2K_l)\right]/2$ (i.e. on the line of the critical points). The results are in a full agreement with the derived exact results - see Eq. (\ref{['eq:Cas-at-the-critical-line']}) and Eq. (\ref{['eq:Casimir-force-tricritical-point']}). As we see --- despite the boundary conditions being periodic, in the framework of the NK model the CCF can be both repulsive and attractive, depending on the values of $K_l, K_s$ and $h$. Obviously, the force is symmetric with respect to $h=0$ as a function of $h$.