Crossover from low-temperature to high-temperature fluctuations. II. Nonuniversal thermodynamic Casimir forces of anisotropic systems

Abstract: The finite-size renormalization-group approach for isotropic O$(n)$-symmetric systems introduced previously [V. Dohm, Phys. Rev. Lett. {\bf 110}, 107207 (2013)] is extended to weakly anisotropic O$(n)$-symmetric systems. Our theory is formulated within the $\varphi^4$ model with lattice anisotropy in a $d$-dimensional block geometry with periodic boundary conditions. It describes the crossover from low- to high-temperature fluctuations including Goldstone-dominated and critical fluctuations for $1\leq n \leq \infty$ in $2<d\<4$ dimensions. An exact representation is derived for the large-distance behavior of the bulk correlation function of anisotropic systems in terms of the principal correlation lengths and an anisotropy matrix ${\bf \bar A}$. This includes the long-ranged correlations with an anisotropic algebraic decay at low temperatures due to the Goldstone modes for $n\>1$. We calculate the finite-size scaling functions of the excess free energy and thermodynamic Casimir force. Exact results are derived in the large-$n$ limit. 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 $\rho \to 0$. For weakly anisotropic systems two-scale-factor universality is replaced by multiparameter universality. This implies a substantial reduction of the predictive power of bulk and finite-size theory for anisotropic systems as compared to isotropic systems. The validity of multiparameter universality is confirmed analytically for a nontrivial example of the $d=2,n=1$ universality class. Anisotropy-dependent minima of the Casimir force scaling function are found below $T_c$. Both the sign and magnitude of the Casimir amplitude in the Goldstone and critical regimes are affected by the lattice anisotropy. Quantitative predictions are made that can be tested by Monte Carlo simulations.