PHYSICAL REVIEW E 97, 062128 (2018) Crossover from low-temperature to high-temperature fluctuations: Universal and nonuniversal Casimir forces of isotropic and anisotropic systems Volker Dohm Institute for Theoretical Physics, RWTH Aachen University, D-52056 Aachen, Germany (Received 17 August 2017; revised manuscript received 28 February 2018; published 18 June 2018) 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)]. 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. DOI: 10.1103/PhysRevE.97.062128 I. INTRODUCTION AND SUMMARY Macroscopic forces arise from microscopic fluctuations in confined systems if the fluctuations have long-range corre- lations. The most prominent example for such fluctuation- induced macroscopic forces is the Casimir force [1,2], which is generated by vacuum fluctuations of the electromagnetic field, i.e., the quantum field of massless photons, confined between two neutral metallic plates. Analogous phenomena exist in various condensed matter systems at finite temperatures [3,4], where classical thermal fluctuations rather than quantum fluctuations have long-range correlations, which then generate so-called thermodynamic Casimir forces. We consider two important examples which result from two fundamentally different sources: from long-range classical fluctuations due to massless “Goldstone modes” [5,6] and from long-range critical fluctuations at a critical temperature T c [7]. Both types of fluctuations exist in O(n)-symmetric systems with n  2 undergoing a second-order phase transition gov- erned by the thermodynamic fluctuations of an n-component order parameter. (For bulk theories on systems with Goldstone modes near T c , see Refs. [8–10].) Prominent examples of confined O(n)-symmetric systems where both types of Casimir forces have been observed or predicted to exist are isotropic superfluids [11,12] and anisotropic high-T c superconductors [13–15]. The critical behavior of both systems belongs to the same (n = 2) universality class. A Casimir force ∝1/L 3 in a 4 He film of thickness L was observed [11] both far below as well as close to the superfluid transition which was confirmed by Monte Carlo (MC) data for isotropic XY models [16–18]. Arguments have been presented [13] that a Casimir force, in the form of an electrical potential difference, appears also at the junction between an anisotropic high-T c superconducting film and the bulk superconductor due to the transfer of Cooper pairs from the film to the bulk. Quantitative estimates indicated this effect to be directly measurable. While analytic results have been derived that separately describe such Casimir forces either in the Goldstone regime [12,17,19] or above T c [20–25], there is a serious lack of knowledge concerning an analytic theory (beyond the mean- field approximation) describing the crossover between these two types of Casimir forces. This is related to the notorious difficulty of treating Goldstone modes in confined systems near T c . Substantial progress has been achieved recently [26] by a simultaneous description of the Goldstone and critical regimes within the isotropic ϕ 4 theory with periodic boundary conditions (BC), where an analytic finite-size scaling func- tion was presented that is valid for general n in the entire 2470-0045/2018/97(6)/062128(42) 062128-1 ©2018 American Physical Society