Crossover from low-temperature to high-temperature fluctuations. I. Thermodynamic Casimir forces of isotropic systems

Abstract: We study the crossover from low- to high-temperature fluctuations including critical fluctuations in confined isotropic 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. {\bf 110}, 107207 (2013)]. Our theory is formulated within the $\varphi^4$ lattice model in a $d$-dimensional block geometry with periodic boundary conditions. We derive the finite-size scaling functions $F^{\text ex}$ and $X$ of the excess free energy density and of the thermodynamic Casimir force, respectively, for $1\leq n \leq \infty$, $2<d\<4$. Applications are given for $ L_\parallel^{d-1} \times L$ slab geometries with a finite aspect ratio $\rho=L/L_\parallel$ as well as for the film limit $\rho \to 0$ at fixed $L$. For $n=1$ and $\rho=0$ the low-temperature limits of $F^{\text ex}$ and $X$ vanish whereas they are finite for $n\geq 2$ and $\rho = 0$ due to the effect of the Goldstone modes. For $n=1$ and $\rho\>0$ we find a finite low-temperature limit of $F^{\text ex}$ which deviates from that of the the Ising model. We attribute this deviation to the nonuniversal difference between the $\varphi^4$ model with continuous variables $\varphi$ and the Ising model with discrete spin variables $s=\pm1$. For $n\geq 2$ and $\rho>0$, a logarithmic divergence of $F^{\text ex}$ in the low-temperature limit is predicted, in excellent agreement with Monte Carlo (MC) data for the $d=3$ $XY$ model. For $2\leq n \leq \infty$ and $0\leq \rho<\rho_0=0.8567$ the Goldstone modes generate a negative (attractive) low-temperature Casimir force that vanishes for $\rho = \rho_0$ and becomes positive (repulsive) for $\rho > \rho_0$. Our predictions are compared with MC data for Ising, $XY$, and Heisenberg models in slab geometries with $0.01\leq\rho\leq1$. Good overall agreement is found.