Fluctuation Induced Force in the Nagle–Kardar Model in Ensemble with Conserved Magnetization

For finite systems the thermodynamic potentials are not equivalent. Thus, in any of them the fluctuations induce different forces: Casimir force (CF) in the grand canonical ensemble (GCE) and Helmholtz force (HF) in the canonical (CE) one with fixed magnetization. Here, we consider the HF within the Nagle–Kardar model with periodic boundary conditions. The model represents a one-dimensional Ising chain with both nearest nearest-neighbor short-ranged and long-ranged interactions present in it. The nearest-neighbor interactions of strength Js could be either ferromagnetic (Js>0), or antiferromagnetic (Js<0). The long-ranged equivalent-neighbor ferromagnetic interactions are of strength Jl/N>0. In the thermodynamic limit the model exhibits in the (Ks=βJs, Kl=βJl) plane a second-order phase transition line 2Kl=exp−2Ks, with Ks>−ln(3)/4, which ends at a tricritical point (Kl=3/2, Ks=−ln(3)/4). The obtained results demonstrate that the temperature behavior of the HF differs essentially from the one of the CF. Furthermore, we show that the CE and GCE are not equivalent, even in the thermodynamic limit. Finally, in the {Ks,Kl,m} phase space we determine the regions of stable states of the system.