Diversity of critical behavior within a universality class

TL;DR

This paper investigates how spatial anisotropy influences critical behavior in the O(n) symmetric $^4$ lattice model, revealing nonuniversal effects, agreement with Monte Carlo data, and predicting measurable anisotropy-dependent features.

Contribution

It provides a detailed analysis of anisotropy effects on critical phenomena, combining renormalization-group theory with Monte Carlo data, and predicts measurable anisotropy-dependent features in finite-size scaling.

Findings

Excellent agreement with Monte Carlo data for the 3D Ising model at $T_c$

Predicted non-monotonic dependence of the Binder cumulant on NNN couplings

Identified a Lifschitz point at larger antiferromagnetic NNN coupling

Abstract

We study spatial anisotropy effects on the bulk and finite-size critical behavior of the O$(n)$ symmetric anisotropic $\phi^4$ lattice model with periodic boundary conditions in a $d$-dimensional hypercubic geometry above, at and below $T_c$. The absence of two-scale factor universality is discussed for the bulk order-parameter correlation function, the bulk scattering intensity, and for several universal bulk amplitude relations. For the confined system, renormalization-group theory within the minimal subtraction scheme at fixed dimension $d$ for $2<d<4$ is employed. For the case of cubic symmetry and for $n=1$ our perturbation approach yields excellent agreement with the Monte Carlo (MC) data for the finite-size amplitude of the free energy of the three-dimensional Ising model at $T_c$ by Mon [Phys. Rev. Lett. {\bf 54}, 2671 (1985)]. Below $T_c$ a minimum of the scaling function of the excess free energy is found. We predict a measurable dependence of this minimum on the anisotropy parameters. The relative anisotropy effect on the free energy is predicted to be significantly larger than that on the Binder cumulant. Our theory agrees quantitatively with the non-monotonic dependence of the Binder cumulant on the ferromagnetic next-nearest neighbor (NNN) coupling of the two-dimensional Ising model found by MC simulations of Selke and Shchur [J. Phys. {\bf A 38}, L739 (2005)]. Our theory also predicts a non-monotonic dependence for small values of the {\it antiferromagnetic} NNN coupling and the existence of a Lifschitz point at a larger value of this coupling. The nonuniversal anisotropy effects in the finite-size scaling regime are predicted to satisfy a kind of restricted universality. The tails of the large-$L$ behavior at $T \neq T_c$ violate both finite-size scaling and universality.