Critical Probability Distributions of the Order Parameter from the Functional Renormalization Group

Approach to the lower critical dimension of the φ^{4} theory in the derivative expansion of the functional renormalization group

We revisit the approach to the lower critical dimension d_{lc} in the Ising-like φ^{4} theory within the functional renormalization group by studying the lowest approximation levels in the derivative expansion of the effective average action. Our goal is to assess how the latter, which provides a generic approximation scheme valid across dimensions and found to be accurate in d≥2, is able to capture the long-distance physics associated with the expected proliferation of localized excitations near d_{lc}. We show that the convergence of the fixed-point effective potential is nonuniform in the field when d→d_{lc} with the emergence of a boundary layer around the minimum of the potential. This allows us to make analytical predictions for the value of the lower critical dimension d_{lc} and for the behavior of the critical temperature as d→d_{lc}, which are both found in fair agreement with the known results. This confirms the versatility of the theoretical approach.

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 97, 062128 (2018)2470-004510.1103/PhysRevE.97.062128]. We prove the validity of this hypothesis in d≥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≥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)PRLTAO0031-900710.1103/PhysRevLett.126.060601]. 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)PRLTAO0031-900710.1103/PhysRevLett.92.197003].

Anomalous Dynamical Scaling Determines Universal Critical Singularities

Anomalous diffusion phenomena occur on length scales spanning from intracellular to astrophysical ranges. A specific form of decay at a large argument of the probability density function of rescaled displacement (scaling function) is derived and shown to imply universal singularities in the normalized cumulant generator. Exact calculations for continuous time random walks provide paradigmatic examples connected with singularities of second order phase transitions. In the biased case scaling is restricted to displacements in the drift direction and singularities have no equilibrium analogue.