Critical Casimir amplitudes for $n$-component ${\varphi }^4$ models with O(n)-symmetry breaking quadratic boundary terms

Abstract

Euclidean $n$-component ${\varphi }^4$ theories whose Hamiltonians are O(n) symmetric except for quadratic symmetry breaking boundary terms are studied in films of thickness $L$. The boundary terms imply the Robin boundary conditions ${\partial }_n{\varphi }_{\alpha }={\mathring{c}}_{\alpha }^{(j)}{\varphi }_{\alpha }$ at the boundary planes ${\mathfrak{B}}_{j=1,2}$ at $z=0$ and $z=L$. Particular attention is paid to the cases in which $m_j$ of the $n$ variables ${\mathring{c}}_{\alpha }^{(j)}$ take the special value ${\mathring{c}}_{m_j\text{-sp}}$ corresponding to critical enhancement while the remaining ones are subcritically enhanced. Under these conditions, the semi-infinite system bounded by ${\mathfrak{B}}_j$ has a multicritical point, called $m_j$-special, at which an $O(m_j)$ symmetric critical surface phase coexists with the O(n) symmetric bulk phase, provided $d$ is sufficiently large. The $L$-dependent part of the reduced free energy per area behaves as ${\Delta }_C/L^{d-1}$ as $L\to \infty$ at the bulk critical point. The Casimir amplitudes ${\Delta }_C$ are determined for small $\epsilon =4-d$ in the general case where $m_{c,c}$ components ${\varphi }_{\alpha }$ are critically enhanced at both boundary planes, $m_{c,D}+m_{D,c}$ components are enhanced at one plane but satisfy asymptotic Dirichlet boundary conditions at the respective other, and the remaining $m_{D,D}$ components satisfy asymptotic Dirichlet boundary conditions at both ${\mathfrak{B}}_j$. Whenever $m_{c,c}>0$, these expansions involve integer and fractional powers ${\epsilon }^{k/2}$ with $k\ge 3$ (mod logarithms). Results to $O({\epsilon }^{3/2})$ for general values of $m_{c,c}$, $m_{c,D}+m_{D,c}$, and $m_{D,D}$ are used to estimate the ${\Delta }_C$ of 3D Heisenberg systems with surface spin anisotropies when $(m_{c,c},m_{c,D}+m_{D,c})=(1,0)$, $(0,1)$, and $(1,1)$.

arXiv commentLatex source file with 5 eps files; version with minor amendments and corrected typos

References in corpus (18)

• New Developments in the Casimir Effect
• The Theory of Boundary Critical Phenomena
• Casimir effect for scalar fields under Robin boundary conditions on plates
• Casimir forces in binary liquid mixtures
• Monte Carlo simulation results for critical Casimir forces
• The Thermodynamic Casimir Effect in ${}^4$He Films near $T_{\lambda }$: Monte Carlo Results
• Ordinary, extraordinary, and normal surface transitions: extraordinary-normal equivalence and simple explanation of $|T-T_c{|}^{2-\alpha }$ singularities
• Surface and bulk transitions in three-dimensional O(n) models
• The critical Casimir force and its fluctuations in lattice spin models: exact and Monte Carlo results
• Crossover from Attractive to Repulsive Casimir Forces and Vice Versa
• Thermodynamic Casimir effects involving interacting field theories with zero modes
• Excess free energy and Casimir forces in systems with long-range interactions of van-der-Waals type: General considerations and exact spherical-model results
• 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
• Surface scaling behavior of isotropic Heisenberg systems: Critical exponents, structure factor, and profiles
• Dynamic critical behavior of model A in films: Zero-mode boundary conditions and expansion near four dimensions
• Dynamic surface critical behavior of isotropic Heisenberg ferromagnets: boundary conditions, renormalized field theory, and computer simulation results
• Thermodynamic Casimir Effect in the large-n limit

Cited by in corpus (13)

• Aspect-ratio dependence of thermodynamic Casimir forces
• Critical Casimir forces and adsorption profiles in the presence of a chemically structured substrate
• Large-$n$ approach to thermodynamic Casimir effects in slabs with free surfaces
• Fluctuation-induced forces in strongly anisotropic critical systems
• Critical Casimir effect in films for generic non-symmetry-breaking boundary conditions
• Critical free energy and Casimir forces in rectangular geometries
• Thermodynamic Casimir Effect in Films: the Exchange Cluster Algorithm
• Crossover from low-temperature to high-temperature fluctuations. II. Nonuniversal thermodynamic Casimir forces of anisotropic systems
• Crossover from low-temperature to high-temperature fluctuations. I. Thermodynamic Casimir forces of isotropic systems
• Anisotropic scaling of the two-dimensional Ising model I: the torus
• Dynamics of the critical Casimir force for a conserved order parameter after a critical quench
• Inverse scattering-theory approach to the exact large-$n$ solutions of $O(n)$ ${\varphi }^4$ models on films and semi-infinite systems bounded by free surfaces
• The three-dimensional $O(n)$ ${\varphi }^4$ model on a strip with free boundary conditions: exact results for a nontrivial dimensional crossover in the limit $n\to \infty$