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

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.

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