O(N)-Universality Classes and the Mermin-Wagner Theorem

TL;DR

This paper explores how O(N)-symmetric models' universality classes change with dimension and field components, highlighting the effects of the Mermin-Wagner theorem and discovering new multi-critical potentials and classes.

Contribution

It provides a continuous analysis of O(N) universality classes across dimensions, revealing the emergence and disappearance of multi-critical potentials and infinite classes at N=0.

Findings

Multi-critical effective potentials exist for 2<d<3 and N≥1.

These potentials vanish at d=2 for N>1, aligning with the Mermin-Wagner theorem.

An infinite family of O(N=0) classes exists in two dimensions.

Abstract

We study how universality classes of O(N)-symmetric models depend continuously on the dimension d and the number of field components N. We observe, from a renormalization group perspective, how the implications of the Mermin-Wagner-Hohenberg theorem set in as we gradually deform theory space towards d=2. For fractal dimension in the range 2<d<3 we observe, for any N bigger than or equal to 1, a finite family of multi-critical effective potentials of increasing order. Apart for the N=1 case, these disappear in d=2 consistently with the Mermin-Wagner-Hohenberg theorem. Finally, we study O(N=0)-universality classes and find an infinite family of these in two dimensions.