Conformal Invariance and Vector Operators in the O(N) Model

Abstract

It is widely expected that, for a large class of models, scale invariance implies conformal invariance. A sufficient condition for this to happen is that there exists no integrated vector operator, invariant under all internal symmetries of the model, with scaling dimension −1. In this article, we compute the scaling dimensions of vector operators with lowest dimensions in the O(N ) model. We use three different approximation schemes: ǫ expansion, large N limit and third order of the Derivative Expansion of Non-Perturbative Renormalization Group equations. We find that the scaling dimensions of all considered integrated vector operators are always much larger than −1. This strongly supports the existence of conformal invariance in this model. For the Ising model, an argument based on correlation functions inequalities was derived, which yields a lower bound for the scaling dimension of the vector perturbations. We generalize this proof to the case of the O(N ) model with N ∈ {2, 3, 4}.

Precision calculation of critical exponents in the O(N) universality classes with the nonperturbative renormalization group

“…Whenever needed, scaling relations are used in order to express results in terms of η and ν. (47) of this work. We find, for the controversial value of ν, a result that turns out to be compatible with the most precise MC simulations and CB and incompatible with experiments from [23].…”

Scale without conformal invariance in membrane theory

“…2 Redundant operators, whose insertion is equivalent to an infinitesimal change of variables, are allowed: even if their dimension is exactly equal to −1, they do not destroy conformal invariance but, rather, modify the transformation of fields under the elements of the conformal group [29]. This is consistent with the fact that the scaling dimension of redundant operators can actually be chosen at will, by suitable design of the specific renormalization group transformation [31].…”

“…Analyses of the relation between scale and conformal invariance in more general classes of theories are thus crucial for several physical applications (see Refs. [2,6,[25][26][27][28][29][30] for some of the results and methods).…”

“…A basis from which we can formulate similar arguments is provided by the results of Refs. [25,28,29] which, instead of analyzing the energy-momentum tensor, used non-perturbative renormalization group techniques. Refs.…”

“…Refs. [28,29] showed that, for critical scalar and O(N ) models, scale implies conformal invariance if no vector eigenoperator with scaling dimension −1 exists 2 . This vector quantity plays a role analogue to the space integral of the virial current.…”