Phase transition in ferromagnetic Ising model with a cell-board external field

TL;DR

This paper demonstrates a first-order phase transition in a ferromagnetic Ising model on a 2D lattice with a periodic cell-board external magnetic field, using reflection positivity and chessboard estimates.

Contribution

It establishes the occurrence of a first-order phase transition in the Ising model with a periodic external field, a novel result for such configurations.

Findings

Phase transition occurs when h<2J/L1 + 2J/L2.

First-order transition proven using reflection positivity.

Application of chessboard estimate in the proof.

Abstract

We show the presence of a first-order phase transition for a ferromagnetic Ising model on $\mathbb{Z}^2$ with a periodical external magnetic field. The external field takes two values $h$ and $-h$, where $h>0$. The sites associated with positive and negative values of external field form a cell-board configuration with rectangular cells of sides $L_1\times L_2$ sites, such that the total value of the external field is zero. The phase transition holds if $h<\frac{2J}{L_1}+ \frac{2J}{L_2}$, where $J$ is an interaction constant. We prove a first-order phase transition using the reflection positivity (RP) method. We apply a key inequality which is usually referred to as the chessboard estimate.