# Critical Casimir force scaling function of the mean spherical model in $2<d<3$ dimensions for nonperiodic boundary conditions

## Abstract

Little is known about finite-size effects of critical systems below the bulk transition temperature $T_c$ for realistic boundary conditions, such as of Dirichlet or Neumann type. However, the exactly solvable mean spherical model can play a significant role in elucidating finite-size effects in the entire scaling regime from above to below $T_c$. We present the first investigation of finite-size effects in this model with nonperiodic boundary conditions for temperatures below $T_c$. To avoid nonscaling effects present in $d=3$ dimensions 1, we continue the model to $2<d<3$, where scaling is known to hold even for nonperiodic boundary conditions 2. Explicit results are obtained in film geometry for the universal scaling functions of the excess free energy and the Casimir force for Dirichlet and Neumann boundary conditions. For appropriate bulk correlation lengths $\xi_+$ and $\xi_-$ above and below $T_c$, respectively, we find an unexpected size dependence $\propto\xi_\pm^-1/\nuL^-2$ of the Casimir force for large $L/\xi_\pm$. The behavior above $T_c$ originates from the combined action of the spherical constraint and the presence of a surface free energy. It differs from the predictions of standard finite-size theories 3. The behavior below $T_c$ arises from a nonzero energy of the lowest mode. The results for both above and below $T_c$ differ qualitatively from the predictions of current finite-size theories for the critical Casimir force above and below the lambda point of $^4$He 4. Possible applications of our results include the weakly interacting Bose gas in the precritical regions above and below $T_c$.

Item Type: Book Section 2013 Kastening, Boris and Dohm, Volker Critical Casimir force scaling function of the mean spherical model in \$2 Material Science > Advanced Thin Film Technology11 Department of Materials and Earth Sciences > Material Science11 Department of Materials and Earth Sciences 09 Jan 2013 09:31 http://st23.statphys23.org/webservices/abstract/preview_pop.... Support by DLR is gratefully acknowledged. EP3 XMLHTML CitationReference ManagerT2T_XMLSimple MetadataDublin CoreASCII CitationAtomBibTeXEndNoteRDF+XMLMODSMultiline CSVJSON

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