Finite-size scaling of the superfluid density of 4He confined between silicon wafers

Diminution of the ordering in plastic and liquid crystalline phases by confinement

In a compound exhibiting plastic (crystal B) and liquid crystalline (smectic A) phases, we report calorimetric and X-ray features that are drastically affected by confinement and orientation in Anopore membranes. Data on the untreated membranes in which the molecules aligned parallel to the pore axes, show a significant diminution in the correlation length corresponding to the positional order of the plastic phase suggesting that finite size effects can perhaps transform the plastic phase to a hexatic one. In the orthogonal case, having the molecules in the plane of the membrane, a new phase is induced, whose structural possibilities are discussed. Calorimetric measurements corroborate these results and also bring out the different thermal behaviors, not only between the bulk and the confined cases, but also between the situations wherein the orientations of the molecules are different. These findings are expected to open up a new path to understand the melting phenomenon, especially that which occurs in lower dimensions.

Casimir force in O(n) systems with a diffuse interface

We study the behavior of the Casimir force in O(n) systems with a diffuse interface and slab geometry infinity;{d-1}xL , where 2<d<4 is the dimensionality of the system. We consider a system with nearest-neighbor anisotropic interaction constants J_{ parallel} parallel to the film and J_{ perpendicular} across it. We argue that in such an anisotropic system the Casimir force, the free energy, and the helicity modulus will differ from those of the corresponding isotropic system, even at the bulk critical temperature, despite that these systems both belong to the same universality class. We suggest a relation between the scaling functions pertinent to the both systems. Explicit exact analytical results for the scaling functions, as a function of the temperature T , of the free energy density, Casimir force, and the helicity modulus are derived for the n-->infinity limit of O(n) models with antiperiodic boundary conditions applied along the finite dimension L of the film. We observe that the Casimir amplitude Delta_{Casimir}(dmid R:J_{ perpendicular},J_{ parallel}) of the anisotropic d -dimensional system is related to that of the isotropic system Delta_{Casimir}(d) via Delta_{Casimir}(dmid R:J_{ perpendicular},J_{ parallel})=(J_{ perpendicular}J_{ parallel});{(d-1)2}Delta_{Casimir}(d) . For d=3 we derive the exact Casimir amplitude Delta_{Casimir}(3,mid R:J_{ perpendicular},J_{ parallel})=[Cl_{2}(pi3)3-zeta(3)(6pi)](J_{ perpendicular}J_{ parallel}) , as well as the exact scaling functions of the Casimir force and of the helicity modulus Upsilon(T,L) . We obtain that beta_{c}Upsilon(T_{c},L)=(2pi;{2})[Cl_{2}(pi3)3+7zeta(3)(30pi)](J_{ perpendicular}J_{ parallel})L;{-1} , where T_{c} is the critical temperature of the bulk system. We find that the contributions in the excess free energy due to the existence of a diffuse interface result in a repulsive Casimir force in the whole temperature region.

Scaling function for the critical diffusion coefficient of a critical fluid in a finite geometry

The long-wavelength diffusion coefficient of a critical fluid confined between two parallel plates, separated by a distance L, is strongly affected by the finite size. Finite size scaling leads us to expect that the vanishing of the diffusion coefficient as xi(-1) for xi<<L, xi being the correlation length, would crossover to L-1 for xi>>L. We show that this is not strictly true. There is a logarithmic scaling violation. We construct a Kawasaki-like scaling function that connects the thermodynamic regime to the extreme critical (xi>>L) regime.

Universal finite-size scaling functions with exact nonuniversal metric factors

Using exact partition functions and finite-size corrections for the Ising model on finite square, plane triangular, and honeycomb lattices and extending a method [J. Phys. 19, L1215 (1986)] to subtract leading singular terms from the free energy, we obtain universal finite-size scaling functions for the specific heat, internal energy, and free energy of the Ising model on these lattices with exact nonuniversal metric factors.

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).

Superfluid fraction of (3)He-(4)He mixtures confined at 0.0483 microm between silicon wafers

We report measurements of the superfluid fraction rho(s)/rho of films of (3)He-(4)He mixtures confined between silicon wafers at 0.0483 microm separation. The data obtained using adiabatic fountain resonance (AFR) can be used to test for the first time expectations of correlation-length scaling in the case of planar mixtures. For the mixtures, the data for rho(s)/rho collapse well on a universal function. The dissipation associated with AFR can also be scaled, and indicates two-dimensional crossover. These results are in contrast to pure (4)He, where over a wider range of confinements, the data for rho(s)/rho are found not to scale.

Crossover scaling: A renormalization group approach

We derive a theory of crossover scaling based on a scaling variable g ξ g , where g is the anisotropy parameter inducing the crossover and ξ g is the correlation length in the presence of g . Our considerations are field theoretic and based on a renormalization group with a g dependent differential generator that interpolates between qualitatively different degrees of freedom. ξ g is a nonlinear scaling field for this renormalization group and interpolates between ( T – T c ( g )) – v 0 and ( T – T c ( g )) – v ∞ ( v 0 and v ∞ being the isotropic and anisotropic exponents respectively). By expanding about a ‘floating’ fixed point we can make corrections to scaling small throughout the crossover. In this formulation effective scaling exponents obey standard scaling laws, e. g. γ eff = v eff (2 – ɳ eff ). We discuss its advantages giving for various crossovers explicit supporting perturbative calculations of the susceptibility, which is found to conform to the general form derived from the g dependent renormalization group.