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 (PBCs). The model represents a 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 2K_{l}=exp(-2K_{s}),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_{Cas}^{(cr)}=1/4 at the critical line, and Δ_{Cas}^{(tr)}=1/3 at the tricritical point. Quite unexpectedly, with the imposed PBCs, the CCF exhibits very unusual behavior as a function of temperature and magnetic field. It is repulsive near the critical line and tricritical point, decaying rapidly with separation from those two singular regimes, and it becomes attractive when 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 BCs and repulsive for conflicting BCs independently of the actual bulk universality class of the phase transition under investigation.