Energy correlations of non-integrable Ising models: The scaling limit in the cylinder

Abstract

We consider a class of non-integrable, or non-planar, Ising models in two dimensions, whose Hamiltonian, in addition to the standard nearest neighbor couplings, includes additional weak multi-spin interactions which are even under spin flip. We study the model in cylindrical domains of arbitrary aspect ratio and compute the multipoint energy correlations at the critical temperature in terms of a multiscale expansion, uniformly convergent in the domain size and in the lattice spacing. We prove that, in the scaling limit, the multipoint energy correlations converge to the same limiting correlations as those of the planar Ising model in a finite cylinder with renormalized horizontal and vertical couplings, up to an overall multiplicative constant independent of the shape of the domain. The proof is based on a representation of the generating function of correlations in terms of a non-Gaussian Grassmann integral, and a constructive Renormalization Group (RG) analysis thereof.A key technical novelty compared with previous works is a systematic analysis of the effect of the boundary corrections to the RG flow, in particular a proof that the scaling dimension of boundary operators is better by one dimension than their bulk counterparts. In addition, a cancellation mechanism based on an approximate image rule for the fermionic Green's function is of crucial importance for controlling the flow of the (superficially) marginal boundary terms under RG iterations.

Non-integrable Ising models in cylindrical geometry: Grassmann representation and infinite volume limit

“…In this article, which is a companion to [AGG20], we consider a class of non-integrable perturbation of the 2D nearest-neighbor Ising model in cylindrical geometry, and discuss some of the key ingredients required in the multiscale construction of the scaling limit of the energy correlations in finite domains. The material presented here generalizes and simplifies the approach proposed by two of the authors in [GGM12], where a similar problem in the translationally invariant setting was investigated.…”

“…This is a serious obstacle in the program of proving conformal invariance of the scaling limit of statistical mechanics models [Giu]; the goal would be to prove results comparable to the remarkable ones obtained for the nearest neighbor 2D Ising model [Smi10; CS09; CHI15], but for a class of non-integrable models, such as perturbed Ising [Aiz+19] or dimer models [GMT20] in two dimensions, via methods that do not rely on the exact solvability of the microscopic model. In this paper and in its companion [AGG20], we attack this program by constructing the scaling limit of the energy correlations of a class of non-integrable perturbations of the standard 2D Ising model in the simplest possible finite domain with boundary, that is, a finite cylinder.…”

“…The other ingredients, including most of the novel aspects of the RG construction in finite volume, such as the definition of the localization procedure in finite volume, the norm bounds on the edge part of the effective potentials and the asymptotically sharp estimates on the correlation functions in the cylinder, are deferred to [AGG20]; see the end of [AGG20, Section 1] for a detailed summary and roadmap of the proof of Theorem 1.1.…”

“…In reference with the second item, let us remark that the exponential decay needed (and proved below) for the propagator between two points z, z ′ ∈ Λ, is in terms of the 'right' distance between z and z ′ , namely: the standard Euclidean distance on the cylinder between z and z ′ in the case of the bulk-part of the single-scale propagator; the Euclidean distance on the cylinder between z, z ′ and the boundary of Λ, in the case of the edge-part of the single-scale propagator. In particular, the exponential decay of the edge-part of the single-scale propagator in the distance of z, z ′ from the boundary of Λ is of crucial importance for proving improved dimensional bounds on the finitesize corrections to the thermodynamic and correlation functions of the interacting model, which are systematically used in the conclusion of the proof of Theorem 1.1 in the companion paper [AGG20], see [AGG20,Section 4].…”

“…As a corollary of Lemma 3.2, we additionally prove that the 'kernels' of W(A) and V (1) (φ, ξ, A) (i.e., the coefficients of their expansions in A, φ, ξ, thought of as functions of the positions of the components of A, φ, ξ on the cylinder) can be naturally decomposed into sums of a 'bulk' part (equal, essentially, to their infinite plane limit restricted to the cylinder, with the appropriate boundary conditions) plus an 'edge' part (their boundary corrections), exponentially decaying in the appropriate distances. In particular, the edge part of the kernels decays exponentially (on the lattice scale) away from the boundary, a fact that will play a major role in the control of the boundary corrections to the correlation functions in [AGG20].…”