Abstract
Systems with an symmetrical Hamiltonian are considered in a -dimensional slab geometry of macroscopic lateral extension and finite thickness that undergo a continuous bulk phase transition in the limit . The effective forces induced by thermal fluctuations at and above the bulk critical temperature (thermodynamic Casimir effect) are investigated below the upper critical dimension by means of field-theoretic renormalization-group methods for the case of periodic and special-special boundary conditions, where the latter correspond to the critical enhancement of the surface interactions on both boundary planes. As shown previously [Europhys. Lett. 75, 241 (2006)], the zero modes that are present in Landau theory at make conventional renormalization-group-improved perturbation theory in dimensions ill-defined. The revised expansion introduced there is utilized to compute the scaling functions of the excess free energy and the Casimir force for temperatures as functions of , where is the bulk correlation length. Scaling functions of the -dependent residual free energy per area are obtained, whose limits are in conformity with previous results for the Casimir amplitudes to and display a more reasonable small- behavior inasmuch as they approach the critical value monotonically as . Extrapolations to for the Ising case with periodic boundary conditions are in fair agreement with Monte Carlo results. In the case of special-special boundary conditions, extrapolations to are hampered by the fact that the one-loop result for the inverse finite-size susceptibility becomes negative for some values of when .
3 More- Received 22 October 2007
DOI:https://doi.org/10.1103/PhysRevB.77.115409
©2008 American Physical Society