Critical Casimir Effect: Exact Results

Abstract

In any medium there are fluctuations due to temperature or due to the quantum nature of its constituents. If a material body is immersed into such a medium, its shape and the properties of its constituents modify the properties of the surrounding medium and its fluctuations. If in the same medium there is a second body then -- in addition to all direct interactions between them -- the modifications due to the first body influence the modifications due to the second body. This mutual influence results in a force between these bodies. If the excitations of the medium, which mediate the effective interaction between the bodies, are massless, this force is long-ranged and nowadays known as a Casimir force. If the fluctuating medium consists of the confined electromagnetic field in vacuum, one speaks of the quantum mechanical Casimir effect. In the case that the order parameter of material fields fluctuates - such as differences of number densities or concentrations - and that the corresponding fluctuations of the order parameter are long-ranged, one speaks of the critical Casimir effect. This holds, e.g., in the case of systems which undergo a second-order phase transition and which are thermodynamically located near the corresponding critical point, or for systems with a continuous symmetry exhibiting Goldstone mode excitations. Here we review the currently available exact results concerning the critical Casimir effect in systems encompassing the one-dimensional Ising, XY, and Heisenberg models, the two-dimensional Ising model, the Gaussian and the spherical models, as well as the mean field results for the Ising and the XY model. Special attention is paid to the influence of the boundary conditions on the behavior of the Casimir force.

218 pages, 68 figures

References in corpus (106)

• Statistical mechanics and dynamics of solvable models with long-range interactions
• Quantum criticality
• Quantum magnetism and criticality
• Quantum Criticality
• The Casimir effect: from quantum to critical fluctuations
• One-Dimensional van der Waals Quantum Materials -- State of the Art and Perspectives
• Casimir-Lifshitz force out of thermal equilibrium
• Universal scaling functions of critical Casimir forces obtained by Monte Carlo simulations
• Critical Casimir force in ${}^4$He films: confirmation of finite-size scaling
• Critical Casimir forces in colloidal suspensions on chemically patterned surfaces
• Monte Carlo simulation results for critical Casimir forces
• Tunability of Critical Casimir Interactions by Boundary Conditions
• Experiment and theory in the Casimir effect
• Boundary critical behavior of the three-dimensional Heisenberg universality class
• The thermodynamic Casimir force: A Monte Carlo study of the crossover between the ordinary and the normal surface universality class
• 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
• Fluctuation-Induced Casimir Forces in Granular Fluids
• Bose-Einstein Condensation in the Relativistic Ideal Bose Gas
• Critical Casimir Effect in superfluid wetting films
• Thinning of superfluid films below the critical point
• Critical Casimir effect for colloids close to chemically patterned substrates
• Salt-induced changes of colloidal interactions in critical mixtures
• Finite-size scaling above the upper critical dimension
• Classical and fluctuation-induced electromagnetic interactions in micronscale systems: designer bonding, antibonding, and Casimir forces
• Thermodynamic Casimir effects involving interacting field theories with zero modes
• Aspect-ratio dependence of thermodynamic Casimir forces
• Control of the Casimir Force Using Semiconductor Test Bodies
• Thermodynamic Casimir Forces between a Sphere and a Plate: Monte Carlo Simulation of a Spin Model
• Interplay of critical Casimir and dispersion forces
• Effective forces between colloids at interfaces induced by capillary wave-like fluctuations
• Off-critical Casimir effect in Ising Slabs with Symmetric Boundary Conditions in d=3
• Critical Casimir Forces and Colloidal Phase Transitions in a Near-Critical Solvent : A Simple Model Reveals a Rich Phase Diagram
• Trapping colloids near chemical stripes via critical Casimir forces
• Quantum Critical Paraelectrics and the Casimir Effect in Time
• Critical Casimir forces steered by patterned substrates
• The Casimir effect in the nanoworld
• Critical Casimir forces and adsorption profiles in the presence of a chemically structured substrate
• Fluctuation-induced forces in periodic slabs: Breakdown of epsilon expansion at the bulk critical point and revised field theory
• Finite-size scaling above the upper critical dimension in Ising models with long-range interactions
• Diversity of critical behavior within a universality class
• Extraordinary-log surface phase transition in the three-dimensional $XY$ model
• Relaxation phenomena at criticality
• Local theory for ions in binary liquid mixtures
• Finite size behaviors of critical Ising model on a rectangle with free boundaries
• Fluctuations of the Casimir-like force between two membrane inclusions
• Fisher's scaling relation above the upper critical dimension
• Design of a Casimir-driven parametric amplifier
• Non-reciprocal energy transfer through the Casimir effect
• Critical adsorption and critical Casimir forces for geometrically structured confinements
• Finite size scaling of the 5D Ising model with free boundary conditions
• Energy transfer between colloids via critical interactions
• Tunable critical Casimir forces counteract Casimir-Lifshitz attraction
• Relation between the thermodynamic Casimir effect in Bose-gas slabs and critical Casimir forces
• Transient Casimir forces from quenches in thermal and active matter
• Solvation force for long ranged wall-fluid potentials
• Field-theoretical approach to the Casimir-like interaction in a one-dimensional Bose gas
• Exact thermodynamic Casimir forces for an interacting three-dimensional model system in film geometry with free surfaces
• Fluctuation-induced forces in strongly anisotropic critical systems
• Critical Casimir Interactions Between Spherical Particles in the Presence of the Bulk Ordering Fields
• Geometric explanation of anomalous finite-size scaling in high dimensions
• Bose-Einstein Condensation and Casimir Effect of Trapped Ideal Bose Gas in between two Slabs
• Dynamic critical behavior of model A in films: Zero-mode boundary conditions and expansion near four dimensions
• Casimir force in O(n) lattice models with a diffuse interface
• Quantum spherical model with competing interactions
• Critical Casimir force in slab geometry with finite aspect ratio: analytic calculation above and below $T_c$
• Scaling of solvation force in 2D Ising strip
• Revealing polymerization kinetics with colloidal dipatch particles
• Crossover from Goldstone to critical fluctuations: Casimir forces in confined O$(\mathbf{n})$ symmetric systems
• Casimir and van der Waals forces: Advances and problems
• Three-body critical Casimir forces
• Direct simulation of critical Casimir forces
• Thermodynamic Casimir Effect in Films: the Exchange Cluster Algorithm
• Finite-size effects on the behavior of the susceptibility in van der Waals films under $+,+$ boundary conditions
• Exact expressions for the partition function of the one-dimensional Ising model in the fixed-$M$ ensemble
• Finite-size effects in the spherical model of finite thickness
• Critical Casimir forces in the presence of random surface fields
• Exact Critical Casimir Amplitude of Anisotropic Systems from Conformal Field Theory and Self-Similarity of Finite-Size Scaling Functions in $d\ge 2$ Dimensions
• Exact results for the behavior of the thermodynamic Casimir force in a model with a strong adsorption
• Critical Casimir force scaling functions of the two-dimensional Ising model at finite aspect ratios
• Dynamics and steady states of a tracer particle in a confined critical fluid
• 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
• Lateral critical Casimir force in 2D Ising strip with inhomogeneous walls
• Comment on "Casimir force in the $O(n\to \infty )$ model with free boundary conditions"
• Finite size corrections to scaling of the formation probabilities and the Casimir effect in the conformal field theories
• Finite-size effects in presence of gravity: The behavior of the susceptibility in ${}^{3}He$ and ${}^{4}He$ films near the liquid-vapor critical point
• Something Can Come of Nothing: Surface Approaches to Quantum Fluctuations and the Casimir Force
• 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
• Multiparameter universality and conformal field theory for anisotropic confined systems: test by Monte Carlo simulations
• On The Critical Casimir Interaction Between Anisotropic Inclusions On A Membrane
• Macroscopic approach to the Casimir friction force
• Fluctuations of the critical Casimir force
• Exact solution for the order parameter profiles and the Casimir force in ${}^4$He superfluid films in an effective field theory
• Critical Casimir forces between planar and crenellated surfaces
• Controlling liquid-liquid phase behavior with an active fluid
• Casimir forces for the ideal Bose gas in anisotropic optical lattices: the effect of alternating sign upon varying dimensionality
• Phonon Casimir effect in polyatomic systems
• Reply to Comment on "Casimir force in the $O(n\to \infty )$ model with free boundary conditions"
• Strong Casimir-like Forces in Flocking Active Matter
• Spin models in three dimensions: Adaptive lattice spacing
• Magnetization profiles at the upper critical dimension as solutions of the integer Yamabe problem
• Variance of the Casimir force in an ideal Bose gas
• Surface critical behavior of the three-dimensional O(3) model
• Effective binding potential from Casimir interactions: the case of the Bose gas
• Fluctuation-Induced Forces in Disordered Landau-Ginzburg Model
• Reversing the Critical Casimir force by shape deformation

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• Casimir and Helmholtz forces in one-dimensional Ising model with Dirichlet (free) boundary conditions
• Thermodynamically Consistent Phase-Field Theory Including Nearest-Neighbor Pair Correlations Explains Failure of Mean-Field Reasoning
• The ordinary surface universality class of the three-dimensional O($N$) model
• Casimir Forces in CFT with Defects and Boundaries
• Critical surface adsorption of confined binary liquids with locally conserved mass and composition
• Helicity modulus and chiral symmetry breaking for boundary conditions with finite twist
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• Bound of Casimir Effect by Holography
• Quantum and Critical Casimir Effects: Bridging Fluctuation Physics and Nanotechnology
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• Boundary operator product expansion coefficients of the three-dimensional Ising universality class
• Effective binding potential from Casimir interactions: the case of the Bose gas
• Casimir energy on the sphere and 6D CFT trace anomaly
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