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

TL;DR

This paper develops a detailed Grassmann representation and analyzes the infinite volume limit of non-integrable 2D Ising models in cylindrical geometries, advancing the multiscale analysis techniques for models with boundaries.

Contribution

It provides a new Grassmann formulation, precise asymptotic estimates, and improved multiscale construction methods for non-integrable Ising models in cylindrical domains.

Findings

Derived the Grassmann representation of the model.

Proved asymptotic estimates of fermionic Green's functions.

Introduced simplifications in the multiscale analysis for boundary domains.

Abstract

In this paper, meant as a companion to arXiv:2006.04458, we consider a class of non-integrable $2D$ Ising models in cylindrical domains, and we discuss two key aspects of the multiscale construction of their scaling limit. In particular, we provide a detailed derivation of the Grassmann representation of the model, including a self-contained presentation of the exact solution of the nearest neighbor model in the cylinder. Moreover, we prove precise asymptotic estimates of the fermionic Green's function in the cylinder, required for the multiscale analysis of the model. We also review the multiscale construction of the effective potentials in the infinite volume limit, in a form suitable for the generalization to finite cylinders. Compared to previous works, we introduce a few important simplifications in the localization procedure and in the iterative bounds on the kernels of the effective potentials, which are crucial for the adaptation of the construction to domains with boundaries.