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

Volker Dohm

Phys. Rev. E 108, 044149 – Published 30 October 2023

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 97, 062128 (2018)]. We prove the validity of this hypothesis in $d \geq 2$ dimensions on the basis of the principle of two-scale-factor universality for isotropic systems at vanishing external field. 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 $φ^{4}$ model, Gaussian model, and $n$ -vector model. By means of the inverse of the shear transformation we determine the general 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 at and away from $T_{c}$ . It is shown that weakly anisotropic systems exhibit a high degree of intrinsic diversity due to $d (d + 1) / 2 - 1$ independent parameters that cannot be determined by thermodynamic measurements. Exact results are derived for the $d = 2$ Ising universality class and for the spherical and Gaussian universality classes in $d \geq 2$ dimensions. 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 free energy and critical Casimir amplitude in a finite 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 in the $(d = 2, n = 1)$ universality class at $T = T_{c}$ derived from conformal field theory [V. Dohm and S. Wessel, Phys. Rev. Lett. 126, 060601 (2021)]. 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 for which experiments have been proposed by G. A. Williams [Phys. Rev. Lett. 92, 197003 (2004)].