We study the crossover from low-temperature to high-temperature fluctuations including Goldstone-dominated and critical fluctuations in confined isotropic and weakly anisotropic $\mathrm{O}\left(n\right)$-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 ${\phi }^4$ lattice model in a $d$-dimensional block geometry with periodic boundary conditions. We calculate the finite-size scaling functions $F^{\text{ex}}$ and $X$ of the excess free-energy density and the thermodynamic Casimir force, respectively, for $1\le n\le \infty , 2<d<4$. Exact results are derived for $n\to \infty$. Applications are given for $L_{\parallel }^{d-1}\times L$ slab geometry with an aspect ratio $\rho =L/L_{\parallel }>0$ and for film geometry ($\rho =0$). Good overall agreement is found with Monte Carlo (MC) data for isotropic spin models with $n=1,2,3$. For $\rho =0$, the low-temperature limits of $F^{\text{ex}}$ and $X$ vanish for $n=1$, whereas they are finite for $n\ge 2$. For $\rho >0$ and $n=1$, we find a finite low-temperature limit of $F^{\text{ex}}$, which deviates from that of the Ising model. We attribute this deviation to the nonuniversal difference between the ${\phi }^4$ model with continuous variables and the Ising model with discrete variables. For $n\ge 2$ and $\rho >0$, a logarithmic divergence of $F^{\text{ex}}$ in the low-temperature limit is predicted, in excellent agreement with MC data. For $2\le n\le \infty$ and $\rho <{\rho }_0=0.8567$ the Goldstone modes generate a negative low-temperature Casimir force that vanishes for $\rho ={\rho }_0$ and becomes positive for $\rho >{\rho }_0$. 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.

• Received 17 August 2017
• Revised 28 February 2018

DOI:https://doi.org/10.1103/PhysRevE.97.062128