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

Abstract

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.

Multiparameter universality and directional nonuniversality of exact anisotropic critical correlation functions of the two-dimensional Ising universality class

“…Recently [9] the notion of multiparameter universality, originally introduced for critical amplitude relations [7], was formulated for the scaling structure of G(x, t) within the anisotropic ϕ 4 theory where G(x, t) depends on up to d(d + 1)/2 + 1 nonuniversal parameters in d dimensions, i.e., up to four or seven parameters in two or three dimensions, respectively. It was hypothesized that multiparameter universality of G(x, t) is valid not only for "soft-spin" ϕ 4 models but also for all weakly anisotropic systems within a given universality class including fixedlength spin models such as Ising (n = 1), XY (n = 2), and Heisenberg (n = 3) models.…”

“…Our results are expected to make an impact on scaling theories for G(x, t) of real anisotropic systems such as magnetic materials [14], superconductors [15], alloys [16], and solids with structural phase transitions [17], where angular-dependent correlation functions are measurable quantities. Our anisotropy matrix also enters the finitesize effects of confined anisotropic systems [9].…”

“…is the correlation length of the isotropic system obtained after a shear transformation that consists of a rotation and a rescaling in the e (α) directions [5][6][7]9].…”

Ensemble dependence of critical Casimir forces in films with Dirichlet boundary conditions

“…For a discussion of the influence of anisotropy as well as of long-ranged (van der Waals) interactions on the critical behavior we refer to Refs. [35][36][37][38][39][40][41].…”

“…The inability to capture this actual dimensional crossover is a well-known shortcoming of many analytical approaches, i.e., MFT and beyond [2,4,53,54] (see, however, Refs. [41,55]), whereas simulations can deal with this issue successfully.…”