Vortex fluctuations in the critical Casimir effect of superfluid and superconducting films

Multiparameter universality and intrinsic diversity of critical phenomena in weakly anisotropic systems

Recently a unified hypothesis of multiparameter universality for the critical behavior of bulk and confined anisotropic systems has been formulated [V. Dohm, Phys. Rev. E 97, 062128 (2018)2470-004510.1103/PhysRevE.97.062128]. We prove the validity of this hypothesis in d≥2 dimensions on the basis of the principle of two-scale-factor universality for isotropic systems at vanishing external field. We introduce an angular-dependent correlation vector and a generalized shear transformation that transforms weakly anisotropic systems to isotropic systems. As examples we consider the O(n)-symmetric φ^{4} model, Gaussian model, and n-vector model. By means of the inverse of the shear transformation we determine the general structure of the bulk order-parameter correlation function, of the singular bulk part of the critical free energy, and of critical bulk amplitude relations of anisotropic systems at and away from T_{c}. It is shown that weakly anisotropic systems exhibit a high degree of intrinsic diversity due to d(d+1)/2-1 independent parameters that cannot be determined by thermodynamic measurements. Exact results are derived for the d=2 Ising universality class and for the spherical and Gaussian universality classes in d≥2 dimensions. For the d=3 Ising universality class we identify the universal scaling function of the isotropic bulk correlation function from the nonuniversal result of the functional renormalization group. A proof is presented for the validity of multiparameter universality of the exact critical free energy and critical Casimir amplitude in a finite rectangular geometry of weakly anisotropic systems with periodic boundary conditions in the Ising universality class. This confirms the validity of recent predictions of self-similar structures of finite-size effects in the (d=2,n=1) universality class at T=T_{c} derived from conformal field theory [V. Dohm and S. Wessel, Phys. Rev. Lett. 126, 060601 (2021)PRLTAO0031-900710.1103/PhysRevLett.126.060601]. This also substantiates the previous notion of an effective shear transformation for anisotropic two-dimensional Ising models. Our theory paves the way for a quantitative theory of nonuniversal critical Casimir forces in anisotropic superconductors for which experiments have been proposed by G. A. Williams [Phys. Rev. Lett. 92, 197003 (2004)PRLTAO0031-900710.1103/PhysRevLett.92.197003].

Exact Critical Casimir Amplitude of Anisotropic Systems from Conformal Field Theory and Self-Similarity of Finite-Size Scaling Functions in d≥2 Dimensions

The exact critical Casimir amplitude is derived for anisotropic systems within the d=2 Ising universality class by combining conformal field theory with anisotropic φ^{4} theory. Explicit results are presented for the general anisotropic scalar φ^{4} model and for the fully anisotropic triangular-lattice Ising model in finite rectangular and infinite strip geometries with periodic boundary conditions. These results demonstrate the validity of multiparameter universality for confined anisotropic systems and the nonuniversality of the critical Casimir amplitude. We find an unexpected complex form of self-similarity of the anisotropy effects near the instability where weak anisotropy breaks down. This can be traced back to the property of modular invariance of isotropic conformal field theory for d=2. More generally, for d>2 we predict the existence of self-similar structures of the finite-size scaling functions of O(n)-symmetric systems with planar anisotropies and periodic boundary conditions both in the critical region for n≥1 as well as in the Goldstone-dominated low-temperature region for n≥2.

Crossover from low-temperature to high-temperature fluctuations: Universal and nonuniversal Casimir forces of isotropic and anisotropic systems

We study the crossover from low-temperature to high-temperature fluctuations including Goldstone-dominated and critical fluctuations in confined isotropic and weakly anisotropic O(n)-symmetric systems on the basis of a finite-size renormalization-group approach at fixed dimension d introduced previously [V. Dohm, Phys. Rev. Lett. 110, 107207 (2013)PRLTAO0031-900710.1103/PhysRevLett.110.107207]. Our theory is formulated within the φ^{4} lattice model in a d-dimensional block geometry with periodic boundary conditions. We calculate the finite-size scaling functions F^{ex} and X of the excess free-energy density and the thermodynamic Casimir force, respectively, for 1≤n≤∞, 2<d<4. Exact results are derived for n→∞. Applications are given for L_{∥}^{d-1}×L slab geometry with an aspect ratio ρ=L/L_{∥}>0 and for film geometry (ρ=0). Good overall agreement is found with Monte Carlo (MC) data for isotropic spin models with n=1,2,3. For ρ=0, the low-temperature limits of F^{ex} and X vanish for n=1, whereas they are finite for n≥2. For ρ>0 and n=1, we find a finite low-temperature limit of F^{ex}, which deviates from that of the Ising model. We attribute this deviation to the nonuniversal difference between the φ^{4} model with continuous variables and the Ising model with discrete variables. For n≥2 and ρ>0, a logarithmic divergence of F^{ex} in the low-temperature limit is predicted, in excellent agreement with MC data. For 2≤n≤∞ and ρ<ρ_{0}=0.8567 the Goldstone modes generate a negative low-temperature Casimir force that vanishes for ρ=ρ_{0} and becomes positive for ρ>ρ_{0}. For anisotropic systems a unified hypothesis of multiparameter universality is introduced for both bulk and confined systems. The dependence of their scaling functions on d(d+1)/2-1 microscopic anisotropy parameters implies a substantial reduction of the predictive power of the theory for anisotropic systems as compared to isotropic systems. An exact representation is derived for the nonuniversal large-distance behavior of the bulk correlation function of anisotropic systems and quantitative predictions are made. The validity of multiparameter universality is proven analytically for the d=2,n=1 universality class. A nonuniversal anisotropy-dependent minimum of the Casimir force scaling function X is found. Both the sign and magnitude of X and the shift of the film critical temperature are affected by the lattice anisotropy.

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.

Crossover from Goldstone to critical fluctuations: Casimir forces in confined O(n)-symmetric systems

We study the crossover between thermodynamic Casimir forces arising from long-range fluctuations due to Goldstone modes and those arising from critical fluctuations. Both types of forces exist in the low-temperature phase of O(n)-symmetric systems for n>1 in a d-dimensional L(||)(d-1) × L slab geometry with a finite aspect ratio ρ = L/L(||). Our finite-size renormalization-group treatment for periodic boundary conditions describes the entire crossover from the Goldstone regime with a nonvanishing constant tail of the finite-size scaling function far below T(c) up to the region far above T(c) including the critical regime with a minimum of the scaling function slightly below T(c). Our analytic result for ρ << 1 agrees well with Monte Carlo data for the three-dimensional XY model. A quantitative prediction is given for the crossover of systems in the 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).

Boundary conditions and the critical Casimir force on an Ising model film: exact results in one and two dimensions

Finite-size effects in certain critical systems can be understood as universal Casimir forces. Here, we compare the Casimir force for free, fixed, periodic, and antiperiodic boundary conditions in the exactly calculable case of the ferromagnetic Ising model in one and two dimensions. We employ a procedure which allows us to calculate the Casimir force with the aforementioned boundary conditions analytically in a transparent manner. Among other results, we find an attractive Casimir force for the case of periodic boundary conditions and a repulsive Casimir force in the antiperiodic case.

Optical-ellipsometric study of the nematic-to-smectic transition in 8CB films adsorbed on silicon

The nematic-to-smectic-A (NA) transition in 8CB (4-octyl-4'-cyanobiphenyl) is especially interesting because experimentally, it has been observed to be second order, but theoretically, it has been predicted that it must have a latent heat. The effect on the NA transition due to confinement in an adsorbed film has hitherto not been investigated. Previous study of adsorbed 8CB films on silicon for coverages less than 100 nm showed the existence of a broad coexistence region, identified by the formation of thick and thin islands on the surface that extends between the bulk NA and the isotropic-to-nematic transition temperatures. In this paper, optical and ellipsometric measurements of 8CB films as a function of temperature are used to identify the location of the NA transition in the film in relation to the coexistence region. The NA transition temperature in the film is found to occur at 32.2+/-0.4 degrees C independent of film thickness for films between 62 to 270 nm thick, based on the decrease in the film anisotropy. This decrease in the anisotropy is found to be surprisingly abrupt. For thicknesses below 62 nm, the NA transition line is joined to the thin-thick coexistence region found previously.

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.

Thinning of superfluid films below the critical point

Experiments on 4He films reveal an attractive Casimir-like force at the bulk lambda point and in the superfluid regime. Previous work has explained the magnitude of this force at the lambda transition and deep in the superfluid region but not the substantial attractive force immediately below the lambda point. Utilizing a simple mean-field calculation renormalized by critical fluctuations we obtain an effective Casimir force that is qualitatively consistent with the scaling function vartheta obtained by collapse of experimental data.

Critical Casimir force in 4He films: confirmation of finite-size scaling

We present new capacitance measurements of critical Casimir force-induced thinning of 4He films near the superfluid transition, focused on the region below Tlambda where the effect is the greatest. 4He films of 238, 285, and 340 A thickness are adsorbed on atomically smooth, N-doped silicon substrates. The Casimir force scaling function theta, deduced from the thinning of these three films, collapses onto a single universal curve, attaining a minimum theta=-1.30+/-0.03 at x=td1/nu=-9.7+/-0.8 A1/nu. The collapse confirms the finite-size scaling origin of the dip in the film thickness. Separately, we also confirm the presence down to 2.13 K of the Goldstone or surface fluctuation force, which makes the superfluid film approximately 2 A thinner than the normal film.

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.

Casimir forces, surface fluctuations, and thinning of superfluid film

Recent experiments on the wetting of 4He have shown that the film becomes thinner at the lambda transition and in the superfluid phase. The difference in thickness above and below the transition has been attributed to a Casimir interaction which is a consequence of a broken continuous symmetry in the bulk superfluid. However, the observed thinning of the film is larger than can be accounted by this Casimir force. We show that surface fluctuations give rise to an additional force, similar in form but larger in magnitude, which may explain the observations.