Critical Casimir forces for $\mathcal{O}(n)$ systems with long-range interaction in the spherical limit

Abstract

We present exact results on the behavior of the thermodynamic Casimir force and the excess free energy in the framework of the $d$-dimensional spherical model with a power law long-range interaction decaying at large distances $r$ as $r^{-d-\sigma }$, where $\sigma <d<2\sigma$ and $0<\sigma \le 2$. For a film geometry and under periodic boundary conditions we consider the behavior of these quantities near the bulk critical temperature $T_c$, as well as for $T>T_c$ and $T<T_c$. The universal finite-size scaling function governing the behavior of the force in the critical region is derived and its asymptotics are investigated. While in the critical and under critical region the force is of the order of $L^{-d}$, for $T>T_c$ it decays as $L^{-d-\sigma }$, where $L$ is the thickness of the film. We consider both the case of a finite system that has no phase transition of its own, when $d-1<\sigma$, as well as the case with $d-1>\sigma$, when one observes a dimensional crossover from $d$ to a $d-1$ dimensional critical behavior. The behavior of the force along the phase coexistence line for a magnetic field H=0 and $T<T_c$ is also derived. We have proven analytically that the excess free energy is always negative and monotonically increasing function of $T$ and $H$. For the Casimir force we have demonstrated that for any $\sigma \ge 1$ it is everywhere negative, i.e. an attraction between the surfaces bounding the system is to be observed. At $T=T_c$ the force is an increasing function of $T$ for $\sigma >1$ and a decreasing one for $\sigma <1$. For any $d$ and $\sigma$ the minimum of the force at $T=T_c$ is always achieved at some $H\ne 0$.

arXiv comment13 pages, revtex, 8 figures

References in corpus (21)

• New Developments in the Casimir Effect
• Precision Measurement of the Casimir Force from 0.1 to 0.9 microns
• Measurement of the Casimir force between parallel metallic surfaces
• The `Friction' of Vacuum, and other Fluctuation-Induced Forces
• Improved Precision Measurement of the Casimir Force Using Gold Surfaces
• The Casimir Effect: Physical Manifestations of Zero Point Energy
• Fluctuation - induced forces in critical fluids
• Casimir Forces between Spherical Particles in a Critical Fluid and Conformal Invariance
• The critical Casimir force and its fluctuations in lattice spin models: exact and Monte Carlo results
• Universality of the thermodynamic Casimir effect
• Exact Three Dimensional Casimir Force Amplitude, $C$-function and Binder's Cumulant Ratio: Spherical Model Results
• Critical Casimir effect and wetting by helium mixtures
• Effects of weak surface fields on the density profiles and adsorption of a confined fluid near bulk criticality
• Influence of Capillary Condensation on the Near-Critical Solvation Force
• Solvation force for long ranged wall-fluid potentials
• Subleading long-range interactions and violations of finite size scaling
• Two-point correlation function in systems with van der Waals type interaction
• Critical behavior of systems with long-range interaction in restricted geometry
• Exact results for some Madelung type constants in the finite-size scaling theory
• Renormalization group treatment of the scaling properties of finite systems with subleading long-range interaction
• Finite-size scaling in systems with long-range interaction

Cited by in corpus (12)

• Critical Casimir Effect: Exact Results
• Critical Casimir forces and adsorption profiles in the presence of a chemically structured substrate
• The imperfect Bose gas in d dimensions: critical behavior and Casimir forces
• Casimir force in O(n) lattice models with a diffuse interface
• Casimir force in the O(n -> infinity) model with free boundary conditions
• Ferromagnetism in one dimension: Critical Temperature
• Thermodynamic Casimir Effect in Films: the Exchange Cluster Algorithm
• Line contribution to the critical Casimir force between a homogeneous and a chemically stepped surface
• Critical and near critical phase behaviour and interplay between the thermodynamic Casimir and van der Waals forces in confined non-polar fluid medium with competing surface and substrate potentials
• Thermodynamics from a scaling Hamiltonian
• Criticality in multicomponent spherical models : results and cautions
• Finite-Size Scaling and Long-Range Interactions