Critical Casimir effect in films for generic non-symmetry-breaking boundary conditions

Abstract

Systems described by an O(n) symmetrical ${\varphi }^4$ Hamiltonian are considered in a $d$-dimensional film geometry at their bulk critical points. A detailed renormalization-group (RG) study of the critical Casimir forces induced between the film's boundary planes by thermal fluctuations is presented for the case where the O(n) symmetry remains unbroken by the surfaces. The boundary planes are assumed to cause short-ranged disturbances of the interactions that can be modelled by standard surface contributions $\propto {\mathbf{\varphi }}^2$ corresponding to subcritical or critical enhancement of the surface interactions. This translates into mesoscopic boundary conditions of the generic symmetry-preserving Robin type ${\partial }_n\mathbf{\varphi }={\mathring{c}}_j\mathbf{\varphi }$. RG-improved perturbation theory and Abel-Plana techniques are used to compute the $L$-dependent part $f_{\mathrm{res}}$ of the reduced excess free energy per film area $A\to \infty$ to two-loop order. When $d<4$, it takes the scaling form $f_{\mathrm{res}}\approx D(c_1{L}^{\Phi /\nu },c_2{L}^{\Phi /\nu })/L^{d-1}$ as $L\to \infty$, where $c_i$ are scaling fields associated with the surface-enhancement variables ${\mathring{c}}_i$, while $\Phi$ is a standard surface crossover exponent. The scaling function $D({\mathsf{c}}_1,{\mathsf{c}}_2)$ and its analogue $\mathcal{D}({\mathsf{c}}_1,{\mathsf{c}}_2)$ for the Casimir force are determined via expansion in $\epsilon =4-d$ and extrapolated to $d=3$ dimensions. In the special case ${\mathsf{c}}_1={\mathsf{c}}_2=0$, the expansion becomes fractional. Consistency with the known fractional expansions of D(0,0) and $\mathcal{D}(0,0)$ to order ${\epsilon }^{3/2}$ is achieved by appropriate reorganisation of RG-improved perturbation theory. For appropriate choices of $c_1$ and $c_2$, the Casimir forces can have either sign. Furthermore, crossovers from attraction to repulsion and vice versa may occur as $L$ increases.

Latex source file, 40 pages, 9 figures

References in corpus (15)

• The Casimir effect: from quantum to critical fluctuations
• Universal scaling functions of critical Casimir forces obtained by Monte Carlo simulations
• Monte Carlo simulation results for critical Casimir forces
• Tunability of Critical Casimir Interactions by Boundary Conditions
• Riemann Zeroes and Phase Transitions via the Spectral Operator on Fractal Strings
• Comment on the sign of the Casimir force
• Crossover from Attractive to Repulsive Casimir Forces and Vice Versa
• Thermodynamic Casimir effect for films in the 3D Ising universality class: Symmetry breaking boundary conditions
• Thermodynamic Casimir effects involving interacting field theories with zero modes
• Fluctuation-induced forces in periodic slabs: Breakdown of epsilon expansion at the bulk critical point and revised field theory
• Diversity of critical behavior within a universality class
• A Monte Carlo study of surface critical phenomena: The special point
• Dynamic critical behavior of model A in films: Zero-mode boundary conditions and expansion near four dimensions
• Critical Casimir amplitudes for $n$-component ${\varphi }^4$ models with O(n)-symmetry breaking quadratic boundary terms
• Casimir Physics: Geometry, Shape and Material

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