Exact three-dimensional Casimir force amplitude, C 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×∞2 under periodic boundary conditions is considered in the presence of an external magnetic field H. The universal Casimir amplitude Δ and the Binder's cumulant ratio B are calculated exactly and found to be Δ=-2ζ(3)/(5π)~-0.153051 and B=2π/\{5ln3[(1+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 Tc. 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 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.
- Publication:
-
Physical Review E
- Pub Date:
- August 1998
- DOI:
- 10.1103/PhysRevE.58.1455
- arXiv:
- arXiv:cond-mat/9803155
- Bibcode:
- 1998PhRvE..58.1455D
- Keywords:
-
- 05.50.+q;
- 05.20.-y;
- 75.10.Hk;
- Lattice theory and statistics;
- Classical statistical mechanics;
- Classical spin models;
- Condensed Matter - Statistical Mechanics;
- High Energy Physics - Theory
- E-Print:
- 12 pages revtex, one eps figure, submitted to Phys. Rev E A set of references added with the text needed to incorporate them. Small changes in the title and in the abstract