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

TL;DR

This paper proves a hypothesis of multiparameter universality in weakly anisotropic critical systems, showing their intrinsic diversity and deriving exact results for various universality classes, with implications for finite-size effects and Casimir forces.

Contribution

It introduces an angular-dependent correlation vector and shear transformation to relate anisotropic systems to isotropic ones, extending universality concepts to anisotropic critical phenomena.

Findings

Validation of multiparameter universality hypothesis.

Exact results for 2D Ising, spherical, and Gaussian classes.

Universal scaling functions and Casimir amplitude proofs.

Abstract

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.