Long-range vector models at large N

Abstract

We calculate various CFT data for the O(N) vector model with the long-range interaction, working at the next-to-leading order in the 1/N expansion. Our results provide additional evidence for the existence of conformal symmetry at the long-range fixed point, as well as the continuity of the CFT data at the long-range to short-range crossover point s* of the exponent parameter s. We also develop the N > 1 generalization of the recently proposed IR duality between the long-range and the deformed short-range models, providing further evidence for its non-perturbative validity in the entire region d/2 < s < s*.

Long-range fermions and critical dualities

“…For s slightly below s ⋆ , the IR fixed point λ ⋆ can be studied perturbatively. O(N ) generalization of this duality has recently been discussed in [16].…”

“…When s > d/2, the quartic coupling constant is relevant, and these models reach a non-trivial critical regime in the IR (when s < d/2, the IR regime is described by the mean-field theory). The O(N ) vector model version of these models has been explored in [14][15][16].…”

“…On the other hand, in the long-range domain d/2 < s < s ⋆ , 1 < d < 4, one obtains a rich critical IR behavior, controlled by the parameter s. While long-range models do not possess a stress-energy tensor, substantial evidence has been accumulated that this interacting critical regime is in fact described by a CFT [19][20][21][22] (see also [16] for the investigation of conformal symmetry in the large-N long-range O(N ) vector model, and [14,15,[23][24][25][26][27] for previous calculations of critical exponents). Moreover, the same CFT was argued in [19,20] to be the IR end-point of an RG flow triggered by coupling the shortrange vector field φ to the generalized free field χ of dimension (d + s)/2 via the bi-linear term λ d d x φχ.…”

“…The corresponding CFT can be studied using the 1/N expansion. In fact, the results of [14][15][16] for various anomalous dimensions and OPE coefficients, derived for general d…”

“…In [31] the background field method was proposed, that allows to easily calculate conformal triangles by using the propagator merging relations. 16 In such an approach, one of the fields is formally fixed to a non-dynamical background value. For instance, one can fix the Hubbard-Stratonovich field to a constant σ ≡ σ, while attaching the full fermionic propagators to the σ ψψ conformal triangle.…”