Finite-size Nagle-Kardar model: Casimir force

TL;DR

This paper derives exact critical Casimir force results for a one-dimensional long-range Ising model with periodic boundary conditions, revealing unexpected repulsive behavior near criticality and a violation of the boundary condition rule.

Contribution

It provides the first exact analysis of the critical Casimir force in the Nagle-Kardar model with periodic boundary conditions, showing unusual force behavior.

Findings

CCF is repulsive near the critical line and tricritical point

CCF becomes attractive away from critical regimes

Violates the boundary condition rule for Casimir forces

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 $(K_s=\beta J_s,K_l=\beta J_l)$ 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.