Diagrammatic perturbation approach to finite-size and surface critical behavior for Dirichlet boundary conditions

Statistical field theory with constraints: Application to critical Casimir forces in the canonical ensemble

The effect of imposing a constraint on a fluctuating scalar order parameter field in a system of finite volume is studied within statistical field theory. The canonical ensemble, corresponding to a fixed total integrated order parameter (e.g., the total number of particles), is obtained as a special case of the theory. A perturbative expansion is developed which allows one to systematically determine the constraint-induced finite-volume corrections to the free energy and to correlation functions. In particular, we focus on the Landau-Ginzburg model in a film geometry (i.e., in a rectangular parallelepiped with a small aspect ratio) with periodic, Dirichlet, or Neumann boundary conditions in the transverse direction and periodic boundary conditions in the remaining, lateral directions. Within the expansion in terms of ε=4-d, where d is the spatial dimension of the bulk, the finite-size contribution to the free energy of the confined system and the associated critical Casimir force are calculated to leading order in ε and are compared to the corresponding expressions for an unconstrained (grand canonical) system. The constraint restricts the fluctuations within the system and it accordingly modifies the residual finite-size free energy. The resulting critical Casimir force is shown to depend on whether it is defined by assuming a fixed transverse area or a fixed total volume. In the former case, the constraint is typically found to significantly enhance the attractive character of the force as compared to the grand canonical case. In contrast to the grand canonical Casimir force, which, for supercritical temperatures, vanishes in the limit of thick films, in the canonical case with fixed transverse area the critical Casimir force attains for thick films a negative value for all boundary conditions studied here. Typically, the dependence of the critical Casimir force both on the temperaturelike and on the fieldlike scaling variables is different in the two ensembles.

Pronounced minimum of the thermodynamic Casimir forces of O(n) symmetric film systems: analytic theory

Thermodynamic Casimir forces of film systems in the O(n) universality classes with Dirichlet boundary conditions are studied below bulk criticality. Substantial progress is achieved in resolving the long-standing problem of describing analytically the pronounced minimum of the scaling function observed experimentally in ^{4}He films (n=2) by Garcia and Chan [Phys. Rev. Lett. 83, 1187 (1999)] and in Monte Carlo simulations for the three-dimensional Ising model (n=1) by O. Vasilyev et al. [Europhys. Lett. 80, 60009 (2007)]. Our finite-size renormalization-group approach describes the film systems as the limit of finite-slab systems with vanishing aspect ratio. This yields excellent agreement with the depth and the position of the minimum for n=1 and semiquantitative agreement with the minimum for n=2. Our theory also predicts a pronounced minimum for the n=3 Heisenberg universality class.

Critical free energy and Casimir forces in rectangular geometries

We study the critical behavior of the free energy and the thermodynamic Casimir force in a L(∥)(d-1) × L block geometry in 2<d<4 dimensions with aspect ratio ρ=L/L(∥) on the basis of the O(n) symmetric ϕ4 lattice model with periodic boundary conditions and with isotropic short-range interactions. Exact results are derived in the large-n limit describing the geometric crossover from film (ρ=0) over cubic (ρ=1) to cylindrical (ρ=∞) geometries. For n=1, three perturbation approaches in the minimal renormalization scheme at fixed d are presented that cover both the central finite-size regime near T(c) for 1/4≲ρ≲3 and the region well above and below T(c). At bulk T(c), we predict the critical Casimir force in the vertical (L) direction to be negative (attractive) for a slab (ρ<1), positive (repulsive) for a rod (ρ>1), and zero for a cube (ρ=1). Our results for finite-size scaling functions agree well with Monte Carlo data for the three-dimensional Ising model by Hasenbusch for ρ=1 and by Vasilyev et al. for ρ=1/6 above, at, and below T(c).

Diversity of critical behavior within a universality class

We study spatial anisotropy effects on the bulk and finite-size critical behavior of the O(n) symmetric anisotropic phi;{4} lattice model with periodic boundary conditions in a d -dimensional hypercubic geometry above, at, and below Tc. The absence of two-scale factor universality is discussed for the bulk order-parameter correlation function, the bulk scattering intensity, and for several universal bulk amplitude relations. The anisotropy parameters are observable by scattering experiments at Tc. For the confined system, renormalization-group theory within the minimal subtraction scheme at fixed dimension d for 2<d<4 is employed. In contrast to the epsilon=4-d expansion, the fixed- d finite-size approach keeps the exponential form of the order-parameter distribution function unexpanded. For the case of cubic symmetry and for n=1 , our perturbation approach yields excellent agreement with the Monte Carlo (MC) data for the finite-size amplitude of the free energy of the three-dimensional Ising model at Tc by Mon [Phys. Rev. Lett. 54, 2671 (1985)]. The epsilon expansion result is in less good agreement. Below Tc, a minimum of the scaling function of the excess free energy is found. We predict a measurable dependence of this minimum on the anisotropy parameters. The relative anisotropy effect on the free energy is predicted to be significantly larger than that on the Binder cumulant. Our theory agrees quantitatively with the nonmonotonic dependence of the Binder cumulant on the ferromagnetic next-nearest-neighbor (NNN) coupling of the two-dimensional Ising model found by MC simulations of Selke and Shchur [J. Phys. A 38, L739 (2005)]. Our theory also predicts a nonmonotonic dependence for small values of the antiferromagnetic NNN coupling and the existence of a Lifshitz point at a larger value of this coupling. The nonuniversal anisotropy effects in the finite-size scaling regime are predicted to satisfy a kind of restricted universality. The tails of the large- L behavior at T++Tc violate both finite-size scaling and universality even for isotropic systems as they depend on the bare four-point coupling of the phi4 theory, on the cutoff procedure, and on subleading long-range interactions.

Nonuniversal finite-size scaling in anisotropic systems

We study the bulk and finite-size critical behavior of the O(n) symmetric phi4 theory with spatially anisotropic interactions of noncubic symmetry in d<4 dimensions. In such systems of a given (d,n) universality class, two-scale factor universality is absent in bulk correlation functions, and finite-size scaling functions including the Privman-Fisher scaling form of the free energy, the Binder cumulant ratio, and the Casimir amplitude are shown to be nonuniversal. In particular it is shown that, for anisotropic confined systems, isotropy cannot be restored by an anisotropic scale transformation.

Scaling and nonscaling finite-size effects in the Gaussian and the mean spherical model with free boundary conditions

We calculate finite-size effects of the Gaussian model in a Lx(d-1) box geometry with free boundary conditions in one direction and periodic boundary conditions in d-1 directions for 2<d<4. We also consider film geometry (L--> infinity ). Finite-size scaling is found to be valid for d<3 and d>3 but logarithmic deviations from finite-size scaling are found for the free energy and energy density at the Gaussian upper borderline dimension d*=3. The logarithms are related to the vanishing critical exponent 1-alpha-nu=(d-3)/2 of the Gaussian surface energy density. The latter has a cusplike singularity in d>3 dimensions. We show that these properties are the origin of nonscaling finite-size effects in the mean spherical model with free boundary conditions in d > or =3 dimensions. At bulk T(c), in d=3 dimensions we find an unexpected nonlogarithmic violation of finite-size scaling for the susceptibility chi approximately L3 of the mean spherical model in film geometry, whereas only a logarithmic deviation chi approximately L2 ln L exists for box geometry. The result for film geometry is explained by the existence of the lower borderline dimension d(l)=3, as implied by the Mermin-Wagner theorem, that coincides with the Gaussian upper borderline dimension d*=3. For 3<d<4 we find a power-law violation of scaling chi approximately L(d-1) at bulk T(c) for box geometry and a nonscaling temperature dependence chi(surface) approximately xi(d) of the surface susceptibility above T(c). For 2<d<3 dimensions we show the validity of universal finite-size scaling for the susceptibility of the mean spherical model with free boundary conditions for both box and film geometry and calculate the corresponding universal scaling functions for T > or =T(c).