O(N) models with boundary interactions and their long range generalizations

Abstract

We study the critical properties of scalar field theories in d+1 dimensions with O(N ) invariant interactions localized on a d-dimensional boundary. By a combination of large N and epsilon expansions, we provide evidence for the existence of non-trivial O(N ) BCFTs in 1 < d < 4. Due to having free fields in the bulk, these models possess bulk higher- spin currents which are conserved up to terms localized on the boundary. We suggest that this should lead to a set of protected spinning operators on the boundary, and give evidence that their anomalous dimensions vanish. We also discuss the closely related long-range O(N ) models in d dimensions, and in particular study a weakly coupled description of the d = 1 long range O(N ) model near the upper critical value of the long range parameter, which is given in terms of a non-local non-linear sigma model. By combining the known perturbative descriptions, we provide some estimates of critical exponents in d = 1.