Large-napproach to thermodynamic Casimir effects in slabs with free surfaces
| Autor: | Alfred Hucht, H. W. Diehl, Felix M. Schmidt, Sergei B. Rutkevich, Daniel Grüneberg, Martin Hasenbusch |
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| Rok vydání: | 2014 |
| Předmět: |
High Energy Physics - Theory
Physics Statistical Mechanics (cond-mat.stat-mech) Critical phenomena FOS: Physical sciences Thermal fluctuations Physik (inkl. Astronomie) Models Theoretical Renormalization group Schrödinger equation Casimir effect symbols.namesake High Energy Physics - Theory (hep-th) Quantum mechanics symbols Thermodynamics Boundary value problem Wetting Scaling Condensed Matter - Statistical Mechanics |
| Zdroj: | Physical Review E. 89 |
| ISSN: | 1550-2376 1539-3755 |
| DOI: | 10.1103/physreve.89.062123 |
| Popis: | The classical $n$-vector $\phi^4$ model with $O(n)$ symmetrical Hamiltonian ${\cal H}$ is considered in a $\infty^2\times L$ slab geometry bounded by a pair of parallel free surface planes at separation $L$. The temperature-dependent scaling functions of the excess free energy and the thermodynamic Casimir force are computed in the large-$n$ limit for temperatures $T$ at, above, and below the bulk critical temperature $T_{\rm c}$. Their $n=\infty$ limits can be expressed exactly in terms of the eigensystem of a self-consistent one-dimensional Schr\"odinger equation. This equation is solved by numerical means for two distinct discretized versions of the model: in the first ("model A"), only the coordinate $z$ across the slab is discretized and the integrations over momenta conjugate to the lateral coordinates are regularized dimensionally; in the second ("model B"), a simple cubic lattice with periodic boundary conditions along the lateral directions is used. Renormalization-group ideas are invoked to show that, in addition to corrections to scaling $\propto L^{-1}$, anomalous ones $\propto L^{-1}\ln L$ should occur. They can be considerably decreased by taking an appropriate $g\to\infty$ ($T_{\rm c}\to\infty$) limit of the $\phi^4$ interaction constant $g$. Depending on the model A or B, they can be absorbed completely or to a large extent in an effective thickness $L_{\rm eff}=L+\delta L$. Excellent data collapses and consistent high-precision results for both models are obtained. The approach to the low-temperature Goldstone values of the scaling functions is shown to involve logarithmic anomalies. The scaling functions exhibit all qualitative features seen in experiments on the thinning of wetting layers of ${}^4$He and Monte Carlo simulations of $XY$ models, including a pronounced minimum of the Casimir force below $T_{\rm c}$. Comment: 24 pages, 6 figures |
| Databáze: | OpenAIRE |
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