Exact Three Dimensional Casimir Force Amplitude and Binder’s Cumulant Ratio: Spherical Model Results

The three dimensional mean spherical model on a hypercubic lattice with a film geometry L × ∞2 under periodic boundary conditions is studied in the presence of an external magnetic field H. The universal Casimir amplitude ∆ and the Binder’s cumulant ratio B are calculated exactly and found to be ∆ = −2ζ(3)/(5π) ≈ −0.153051 and B = 2π/( √ 5 ln[(1 + √ 5)/2]). It is analytically shown that the excess free energy (due to the finite-size contributions to the free energy of the system) is a monotonically increasing function of the temperature T and of the magnetic field |H| in the vicinity of the bulk critical temperature Tc. 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 ≤ 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 Tc. A discussion on the relations between the finite temperature Cfunction, usually defined for quantum systems, and the excess free energy scaling function is presented. PACS number(s): 05.20.-y, 05.50.+q, 75.10.Hk