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

TL;DR

This paper proves that in non-integrable 2D Ising models with weak multi-spin interactions, energy correlations in cylindrical domains converge to those of the nearest-neighbor model with renormalized couplings, using RG analysis.

Contribution

It introduces a systematic RG approach to analyze boundary effects in non-integrable Ising models, showing boundary operators have improved scaling dimensions.

Findings

Energy correlations converge to the same limit as the integrable model

Boundary operators have better scaling dimensions than bulk operators

A novel cancellation mechanism controls boundary RG flow

Abstract

We consider a class of non-integrable 2D Ising models, whose Hamiltonian, in addition to the nearest neighbor couplings, includes weak multi-spin interactions, even under spin flip. We study the model in cylindrical domains of arbitrary aspect ratio and prove that, in the scaling limit, the multipoint energy correlations converge to the same limiting correlations as those of the nearest-neighbor Ising model in the cylinder with renormalized couplings, up to an overall multiplicative constant, independent of the shape and size 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. A cancellation mechanism based on an approximate image rule for the fermionic Green's function is of crucial importance for controlling the RG flow of the marginal boundary terms.