Dynamic critical behavior of model A in films: Zero-mode boundary conditions and expansion near four dimensions

Abstract

The critical dynamics of relaxational stochastic models with nonconserved $n$-component order parameter $\mathbf{\varphi }$ and no coupling to other slow variables ("model A") is investigated in film geometries for the cases of periodic and free boundary conditions. The Hamiltonian $\mathcal{H}$ governing the stationary equilibrium distribution is taken to be O(n) symmetric and to involve, in the case of free boundary conditions, the boundary terms ${\int }_{{\mathfrak{B}}_j}{\mathring{c}}_j{\varphi }^2/2$ associated with the two confining surface planes ${\mathfrak{B}}_j$, $j=1,2$, at $z=0$ and $z=L$, where the enhancement variables ${\mathring{c}}_j$ are presumed to be subcritical or critical. A field-theoretic RG study of the dynamic critical behavior at $d=4-\epsilon$ bulk dimensions is presented, with special attention paid to the cases where the classical theories involve zero modes at $T_{c,\infty }$. This applies when either both ${\mathring{c}}_j$ take the critical value ${\mathring{c}}_{\text{sp}}$ associated with the special surface transition, or else periodic boundary conditions are imposed. Owing to the zero modes, the $\epsilon$ expansion becomes ill-defined at $T_{c,\infty }$. Analogously to the static case, the field theory can be reorganized to obtain a well-defined small-$\epsilon$ expansion involving half-integer powers of $\epsilon$, modulated by powers of $\ln \epsilon$. Explicit results for the scaling functions of $T$-dependent finite-size susceptibilities at temperatures $T\ge T_{c,\infty }$ and of layer and surface susceptibilities at the bulk critical point are given to orders $\epsilon$ and ${\epsilon }^{3/2}$, respectively. For the case of periodic boundary conditions, the consistency of the expansions to $O({\epsilon }^{3/2})$ with exact large-$n$ results is shown.

Latex file with 8 eps files included; text added in conclusions and abstract, typos corrected

References in corpus (33)

• The Theory of Boundary Critical Phenomena
• Casimir effect for scalar fields under Robin boundary conditions on plates
• Monte Carlo simulation results for critical Casimir forces
• The transition temperature of the dilute interacting Bose gas for $N$ internal degrees of freedom
• The Thermodynamic Casimir Effect in ${}^4$He Films near $T_{\lambda }$: Monte Carlo Results
• Theory of finite temperature crossovers near quantum critical points close to, or above, their upper-critical dimension
• Foam Diagram Summation at Finite Temperature
• Crossover from Attractive to Repulsive Casimir Forces and Vice Versa
• Critical Casimir Effect in superfluid wetting films
• Thinning of superfluid films below the critical point
• Critical dynamics in thin films
• 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
• Relaxation phenomena at criticality
• Universal Short-Time Behavior in Critical Dynamics near Surfaces
• Universality of the thermodynamic Casimir effect
• Surface Critical Behavior of Binary Alloys and Antiferromagnets: Dependence of the Universality Class on Surface Orientation
• Theory of a spherical quantum rotors model: low--temperature regime and finite-size scaling
• Subleading long-range interactions and violations of finite size scaling
• Dynamical Relaxation and Universal Short-Time Behavior in Finite Systems: The Renormalization Group Approach
• Casimir force in O(n) lattice models with a diffuse interface
• Surface critical behavior of bcc binary alloys
• Non-universal size dependence of the free energy of confined systems near criticality
• Finite-size effects in the spherical model of finite thickness
• Dynamic surface critical behavior of systems with conserved bulk order parameter: Detailed RG analysis of the semi-infinite extensions of model B with and without nonconservative surface terms
• On the Finite Size Scaling in Disordered Systems
• Dynamic surface critical behavior of isotropic Heisenberg ferromagnets: boundary conditions, renormalized field theory, and computer simulation results
• Scaling behavior for finite O(n) systems with long-range interaction
• Renormalization group treatment of the scaling properties of finite systems with subleading long-range interaction
• Dynamic surface scaling behavior of isotropic Heisenberg ferromagnets
• Dynamic critical phenomena in disordered systems with finite geometry

Cited by in corpus (19)

• Finite-size scaling at quantum transitions
• Coherent and dissipative dynamics at quantum phase transitions
• Critical Casimir Effect: Exact Results
• Critical Casimir forces and adsorption profiles in the presence of a chemically structured substrate
• Critical Casimir torques and forces acting on needles in two spatial dimensions
• Fluctuation-induced forces in strongly anisotropic critical systems
• Ensemble dependence of Critical Casimir Forces in Films with Dirichlet Boundary Conditions
• Critical Casimir effect in films for generic non-symmetry-breaking boundary conditions
• Finite-size effects in film geometry with nonperiodic boundary conditions: Gaussian model and renormalization-group theory at fixed dimension
• Fluctuation-induced forces in confined ideal and imperfect Bose gases
• Critical free energy and Casimir forces in rectangular geometries
• Critical Casimir amplitudes for $n$-component ${\varphi }^4$ models with O(n)-symmetry breaking quadratic boundary terms
• Crossover from low-temperature to high-temperature fluctuations. I. Thermodynamic Casimir forces of isotropic systems
• Crossover from low-temperature to high-temperature fluctuations. II. Nonuniversal thermodynamic Casimir forces of anisotropic systems
• Surface-induced non-equilibrium dynamics and critical Casimir forces for model B in film geometry
• Lifshitz-point correlation length exponents from the large-n expansion
• Dynamics of the critical Casimir force for a conserved order parameter after a critical quench
• Anisotropy and universality: Critical Binder cumulant of the two-dimensional Ising model
• Universal anisotropic finite-size critical behavior of the two-dimensional Ising model on a strip and of d-dimensional models on films