Exact three-dimensional Casimir force amplitude,<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>C</mml:mi></mml:math>function, and Binder’s cumulant ratio: Spherical model results

Abstract

The three-dimensional mean spherical model on a hypercubic lattice with a film geometry $L\ifmmode\times\else\texttimes\fi{}{\ensuremath{\infty}}^{2}$ under periodic boundary conditions is considered in the presence of an external magnetic field $H.$ The universal Casimir amplitude $\ensuremath{\Delta}$ and the Binder's cumulant ratio $B$ are calculated exactly and found to be $\ensuremath{\Delta}=\ensuremath{-}2\ensuremath{\zeta}(3)/(5\ensuremath{\pi})\ensuremath{\approx}\ensuremath{-}0.153051$ and $B=2\ensuremath{\pi}/{\sqrt{5}{\mathrm{ln}}^{3}[(1+\sqrt{5})/2]}.$ 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 $O(n),n&gt;2,$ model with a film geometry under periodic boundary conditions when $d&lt;~3.$ 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 ${T}_{c}.$