Finite-size Nagle-Kardar model: Casimir force

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 $(K_s=\beta J_s,K_l=\beta J_l)$ 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: $\Delta_{\rm Cas}^{\rm (cr)}=1/4$ at the critical line, and $\Delta_{\rm Cas}^{\rm (tr)}=1/3$ 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.