Noncompact sigma-models: Large N expansion and thermodynamic limit

Abstract

Noncompact SO(1, N) sigma-models are studied in terms of their large N expansion in a lattice formulation in dimensions d ≥ 2. Explicit results for the spin and current two-point functions as well as for the Binder cumulant are presented to next to leading order on a finite lattice. The dynamically generated gap is negative and serves as a coupling-dependent infrared regulator which vanishes in the limit of infinite lattice size. The cancellation of infrared divergences in invariant correlation functions in this limit is nontrivial and is in d = 2 demonstrated by explicit computation for the above quantities. For the Binder cumulant the thermodynamic limit is finite and is given by 2/(N +1) in the order considered. Monte Carlo simulations suggest that the remainder is small or zero. The potential implications for "criticality" and "triviality" of the theories in the SO(1, N) invariant sector are discussed. * Membre du CNRS

Continuum limit of fishnet graphs and AdS sigma model

“…which is the canonical way of boosting the system along a big circle ⊂ S d+1 , see [52,53]. Therefore, in agreement with the discussions in [54][55][56], the compact and non-compact problems are the same, perturbatively, if not for the sign of the coupling, e 2 ↔ −e 2 . At this stage, we could import the result directly from the perturbative studies of the sphere sigma model [52,53] by doing the continuation to negative coupling.…”

“…We can follow the same lines for the hyperboloid. In fact, the results for the sphere carry over to the hyperboloid, since compact and non-compact models only differ perturbatively in the sign of the coupling constant, as expected on geometrical grounds [54][55][56]. We recall how this comes about below.…”

Perturbative and nonperturbative correspondences between compact and noncompact sigma-models

“…Heuristically the origin of the 'dual' generating functional W d − [H] can be understood by a dualization of the 'spatial' spin components n x , x ∈ Λ, and a formal contour deformation in field space [9]. The formal nature of the latter is responsible for (3.8), although the large N expansion coefficients of W − [H] are correctly reproduced.…”

“…This "large N correspondence" is computationally useful because the computations in the compact model, which do not require gauge fixing, are much simpler. The 'detour' over finite volume cannot be avoided as the correspondence is difficult to interpret directly in the infinite volume limit, see [9]. In addition is important to appreciate that although the gap equations are related by flipping the sign of the large N coupling λ = (N +1)/β, the leading order propagator D − in the noncompact model behaves very differently as a function of the lattice distance than D + : while the latter shows exponential decay in the thermodynamic limit, the former decreases only with a power law (for d > 2) or increases logarithmically with the distance (for d = 2).…”

“…In the latter case the relation to the compact model cannot be formulated directly in the thermodynamic limit but one can take term by term the thermodynamic limit on both sides of the correspondence. This somewhat tricky limit is studied in an accompanying paper [9] in the two-dimensional systems for a number of physically interesting invariant quantities. Ultimately we expect the fundamental differences between the compact and the noncompact models to be rooted in two facts: the presence of a infinite volume mass gap and the absence of long range order (for d ≤ 2) in the compact models and the opposite characteristics in the noncompact models for all d ≥ 1.…”