Thermodynamic Casimir effects involving interacting field theories with zero modes

Daniel Grüneberg and H. W. Diehl
Phys. Rev. B 77, 115409 – Published 6 March 2008

Abstract

Systems with an O(n) symmetrical Hamiltonian are considered in a d-dimensional slab geometry of macroscopic lateral extension and finite thickness L that undergo a continuous bulk phase transition in the limit L. The effective forces induced by thermal fluctuations at and above the bulk critical temperature Tc, (thermodynamic Casimir effect) are investigated below the upper critical dimension d*=4 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 Tc, make conventional renormalization-group-improved perturbation theory in 4ϵ 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 TTc, as functions of LLξ, where ξ is the bulk correlation length. Scaling functions of the L-dependent residual free energy per area are obtained, whose L0 limits are in conformity with previous results for the Casimir amplitudes ΔC to O(ϵ32) and display a more reasonable small-L behavior inasmuch as they approach the critical value ΔC monotonically as L0. Extrapolations to d=3 for the Ising case n=1 with periodic boundary conditions are in fair agreement with Monte Carlo results. In the case of special-special boundary conditions, extrapolations to d=3 are hampered by the fact that the one-loop result for the inverse finite-size susceptibility becomes negative for some values of L when ϵ0.83.

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  • Received 22 October 2007

DOI:https://doi.org/10.1103/PhysRevB.77.115409

©2008 American Physical Society

Authors & Affiliations

Daniel Grüneberg and H. W. Diehl

  • Fachbereich Physik, Universität Duisburg-Essen, D-47048 Duisburg, Germany

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Issue

Vol. 77, Iss. 11 — 15 March 2008

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