Exact Three Dimensional Casimir Force Amplitude, $C$-function and Binder's Cumulant Ratio: Spherical Model Results

TL;DR

This paper provides exact calculations of the Casimir amplitude and Binder's cumulant ratio for a 3D spherical model with film geometry, exploring their relations to the finite-temperature C-function and demonstrating the negativity of the Casimir force near criticality.

Contribution

It offers exact analytical results for universal quantities in the 3D spherical model and discusses their relation to the finite-temperature C-function and Casimir force behavior.

Findings

Casimir amplitude $ o$ approximately -0.153051

Binder's cumulant ratio $ o$ explicit analytical form

Casimir force is negative near and below $T_c$

Abstract

The three dimensional mean spherical model on a hypercubic lattice with a film geometry $L\times \infty ^2$ under periodic boundary conditions is considered in the presence of an external magnetic field $H$. The universal Casimir amplitude $\Delta $ and the Binder's cumulant ratio $B$ are calculated exactly and found to be $\Delta =-2\zeta (3)/(5\pi)\approx -0.153051$ and $B=2\pi /(\sqrt{5}\ln ^3[(1+\sqrt{5})/2]).$ A discussion on the relations between the finite temperature $C$-function, usually defined for quantum systems, and the excess free energy (due to the finite-size contributions to the free energy of the system) scaling function is presented. It is demonstrated that the $C$-function of the model equals 4/5 at the bulk critical temperature $T_c$. It is analytically shown that the excess free energy is a monotonically increasing function of the temperature $T$ and of the magnetic field $|H|$ in the vicinity of $T_c.$ This property is supposed to hold for any classical $d$-dimensional $O(n),n>2,$ model with a film geometry under periodic boundary conditions when $d\leq 3$. An analytical evidence is also presented to confirm that the Casimir force in the system is negative both below and in the vicinity of the bulk critical temperature $T_c.$