The scaling limit of the energy correlations in non integrable Ising models

TL;DR

This paper derives explicit formulas for energy correlations in a non-integrable 2D Ising model near criticality, showing they simplify to the integrable case in the scaling limit with controlled convergence.

Contribution

It provides a detailed analysis of non-integrable Ising models using fermionic mapping and multiscale methods, extending understanding of their critical behavior.

Findings

Explicit expressions for multipoint energy correlations in the scaling limit.

Correlations simplify to those of the integrable Ising model with parameter renormalization.

Convergence bounds to the scaling limit are established.

Abstract

We obtain an explicit expression for the multipoint energy correlations of a non solvable two-dimensional Ising models with nearest neighbor ferromagnetic interactions plus a weak finite range interaction of strength $\lambda$, in a scaling limit in which we send the lattice spacing to zero and the temperature to the critical one. Our analysis is based on an exact mapping of the model into an interacting lattice fermionic theory, which generalizes the one originally used by Schultz, Mattis and Lieb for the nearest neighbor Ising model. The interacting model is then analyzed by a multiscale method first proposed by Pinson and Spencer. If the lattice spacing is finite, then the correlations cannot be computed in closed form: rather, they are expressed in terms of infinite, convergent, power series in $\lambda$. In the scaling limit, these infinite expansions radically simplify and reduce to the limiting energy correlations of the integrable Ising model, up to a finite renormalization of the parameters. Explicit bounds on the speed of convergence to the scaling limit are derived.