Fluctuation-induced forces in periodic slabs: Breakdown of expansion at the bulk critical point and revised field theory

¹ Fachbereich Physik, Universität Duisburg-Essen - Campus Essen D-45117 Essen, Germany
² Institute for Condensed Matter Physics - 79011 Lviv, Ukraine

Received: 2 May 2006
Accepted: 23 May 2006

Abstract

Systems described by n-component models in a slab geometry of finite thickness L are considered at and above their bulk critical temperature . The renormalization-group improved perturbation theory commonly employed to investigate the fluctuation-induced forces (“thermodynamic Casimir effect”) in bulk dimensions is re-examined. It is found to be ill-defined beyond two-loop order because of infrared singularities when the boundary conditions are such that the free propagator in slab geometry involves a zero-energy mode at bulk criticality. This applies to periodic boundary conditions and the special-special ones corresponding to the critical enhancement of the surface interactions on both confining plates. The field theory is reorganized such that a small-ϵ expansion results which remains well behaved down to . The leading contributions to the critical Casimir amplitudes and beyond two-loop order are , where is the value of the renormalized coupling at the infrared-stable fixed point. Besides integer powers of ϵ, the small-ϵ expansions of these amplitudes involve fractional powers , with , and powers of . Explicit results to order are presented for and , which are used to estimate their values at .