Phase transitions in layered systems

TL;DR

This paper proves the existence of phase transitions in a layered Ising model with long-range horizontal interactions and weak vertical couplings, for all inverse temperatures above the mean field critical value, when the interaction range is large.

Contribution

It introduces a layered Ising model with Kac potentials and weak vertical interactions, demonstrating phase transitions for all inverse temperatures above the critical point.

Findings

Phase transition occurs for all $eta$ above critical value

Phase transition persists for small interaction range parameter $\gamma$

Vertical interaction strength $\gamma^A$ does not prevent phase transition

Abstract

We consider the Ising model on the two-dimensional square lattice where on each horizontal line, called "layer", the interaction is given by a ferromagnetic Kac potential with coupling strength $J_\gamma(x,y)=\gamma J(\gamma(x-y))$, where $J(\cdot)$ is smooth and has compact support; we then add a nearest neighbor ferromagnetic vertical interaction of strength $\gamma^{A}$ (where $A\ge 2$ is fixed) and prove that for any $\beta$ (inverse temperature) larger than the mean field critical value there is a phase transition for all $\gamma$ small enough.