Multiparameter universality and intrinsic diversity of critical phenomena in weakly anisotropic systems

Recently a unified hypothesis of multiparameter universality for the critical behavior of bulk and confined anisotropic systems has been formulated [V. Dohm, Phys. Rev. E {\bf 97}, 062128 (2018)]. We prove the validity of this hypothesis on the basis of the principle of two-scale-factor universality for isotropic systems. We introduce an angular-dependent correlation vector and a generalized shear transformation that transforms weakly anisotropic systems to isotropic systems. As examples we consider the $O(n)$-symmetric $\varphi^4$, Gaussian, and $n$-vector model. We determine the structure of the bulk order-parameter correlation function, of the singular bulk part of the critical free energy, and of critical bulk amplitude relations of anisotropic systems. It is shown that weakly anisotropic systems exhibit a high degree of intrinsic diversity due to $d(d+1)/2-1$ independent parameters. Exact results are derived for the $d=2$ Ising universality class and for the spherical and Gaussian universality classes. For the $d=3$ Ising universality class we identify the universal scaling function of the isotropic bulk correlation function from the nonuniversal result of the functional renormalization group. A proof is presented for the validity of multiparameter universality of the exact critical Casimir amplitude in a rectangular geometry of weakly anisotropic systems with periodic boundary conditions in the Ising universality class. This confirms the validity of recent predictions of self-similar structures of finite-size effects at $T=T_c$ derived from conformal field theory. This also substantiates the previous notion of an effective shear transformation for anisotropic two-dimensional Ising models. Our theory paves the way for a quantitative theory of nonuniversal critical Casimir forces in anisotropic superconductors.