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Casimir and Helmholtz forces in one-dimensional Ising model with Dirichlet (free) boundary conditions

D. M. Dantchev, N. S. Tonchev, J. Rudnick

Abstract

Attention in the literature has increasingly turned to the issue of the dependence on ensemble and boundary conditions of fluctuation-induced forces. We have recently investigated this problem in the one-dimensional Ising model with periodic and antiperiodic boundary conditions (Annals of Physics {\bf 459}, 169533 (2023)). Significant variations of the behavior of Casimir and Helmholtz forces was observed, depending on both ensemble and boundary conditions. Here we extend our study by considering the problem in the important case of Dirichlet (also termed free, or missing neighbors) boundary conditions. The advantage of the mathematical formulation of the problem in terms of Chebyshev polynomials is demonstrated and, in this approach, expressions for the partition functions in the canonical and the grand canonical ensembles are presented. We prove analytically that the Casimir force is attractive for all values of temperature and external ordering field, while the Helmholtz force can be both attractive and repulsive.

Casimir and Helmholtz forces in one-dimensional Ising model with Dirichlet (free) boundary conditions

Abstract

Attention in the literature has increasingly turned to the issue of the dependence on ensemble and boundary conditions of fluctuation-induced forces. We have recently investigated this problem in the one-dimensional Ising model with periodic and antiperiodic boundary conditions (Annals of Physics {\bf 459}, 169533 (2023)). Significant variations of the behavior of Casimir and Helmholtz forces was observed, depending on both ensemble and boundary conditions. Here we extend our study by considering the problem in the important case of Dirichlet (also termed free, or missing neighbors) boundary conditions. The advantage of the mathematical formulation of the problem in terms of Chebyshev polynomials is demonstrated and, in this approach, expressions for the partition functions in the canonical and the grand canonical ensembles are presented. We prove analytically that the Casimir force is attractive for all values of temperature and external ordering field, while the Helmholtz force can be both attractive and repulsive.
Paper Structure (12 sections, 80 equations, 3 figures)

This paper contains 12 sections, 80 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Gibbs free energy and Casimir force within the grand canonical ensemble with DBC's
  3. On the critical behavior of the infinite Ising chain and the leading finite-size corrections
  4. On the behavior of the Casimir force
  5. Helmholtz free energy and Helmholtz force within the canonical ensemble with Dirichlet BC's
  6. On the derivation of $Z^{(D)}_C(N,K,M)$ via transfer matrix method
  7. Concluding remarks and discussion
  8. On the relation of $Z^{(D)}_{\rm GC}(N,K,h)$ to $Z^{\rm (per)}_{\rm GC}(N,K,h)$ and $Z^{\rm (anti)}_{\rm GC}(N,K,h)$
  9. Proof of the inequality $0<A(K,h)<\cosh(h)$
  10. Rationalized presentation of the canonical partition function in terms of Gauss hypergeometric functions
  11. On the relation of $Z^{(D)}_{\rm C}(N,K,M)$ to $Z^{\rm (per)}_{\rm C}(N,K,M)$ and $Z^{\rm (anti)}_{\rm C}(N,K,M)$
  12. Proof of the equality of Eqs. \ref{['Zd']} and \ref{['eq:ZD-in-CE-integral']}

Figures (3)

  • Figure 1: On the left panel: The behavior of the scaling function $X_{\rm Cas}^{(D)}(x_t,x_h)$ of the Casimir force as a function of the scaling variables $x_t$ and $x_h$. We observe that the function is negative for all values of $x_t$ and $x_h$. On the right panel: The behavior of the function $N F_{\rm Cas}^{(D)}$ with $N=100$.
  • Figure 2: On the left panel: The behavior of the scaling function $X_{\rm H}^{(D)}(x_t,m)$ of the Helmholtz force as a function of the scaling variable $x_t$ and $m$ for $m=0.1, m=0.3, m=0.6$ and $m=0.9$. We observe that the function is negative for small values of $x_t$ for $m^2<3/4$ (see the $x_t\to 0$ asymptote in Eq. (\ref{['eq:asymptotes-of_helmholtz-scaling-function']})). For $m^2>3/4$ and large values of $x_t$ the force is always repulsive. On the right panel: The behavior of the function $X_{\rm H}^{(D)}(x_t,m)$. We observe that, depending on the values of $x_t$ and $m$ the force can be both attractive and repulsive. This behavior is remarkably different from that of the Casimir force which is always attractive for the same boundary conditions.
  • Figure 3: Left panel: The full $K$ behavior of the Helmholtz force illustrated fo $N=100$ and few values of $m$: $m=0.1, m=0.3, m=0.6$ and $m=0.9$. Right panel: The $K$ dependence of the Helmholtz force for different values of $N$ for $N=100, 200, 300, 400$ and $N=500$. As we see - larger the $N$, stronger is the repulsion for small values of $K$.