A phase transition and critical phenomenon for the two-dimensional random field Ising model

TL;DR

This paper investigates the phase transition in the two-dimensional random field Ising model, revealing how boundary influence and correlation length behave under varying disorder strength near the critical temperature.

Contribution

It establishes a precise phase transition at the critical temperature, quantifying how disorder affects boundary influence and correlation length in the 2D RFIM.

Findings

Boundary influence decays as N^{-1/8} when disorder is weak.

Disorder significantly accelerates decay of boundary influence beyond a threshold.

Correlation length scales as psilon^{-8/7} at T_c and exponentially for T < T_c.

Abstract

We study the random field Ising model in a two-dimensional box with side length $N$ where the external field is given by independent normal variables with mean $0$ and variance $\epsilon^2$. Our primary result is the following phase transition at $T = T_c$: for $\epsilon \ll N^{-7/8}$ the boundary influence (i.e., the difference between the spin averages at the center of the box with the plus and the minus boundary conditions) decays as $N^{-1/8}$ and thus the disorder essentially has no effect on the boundary influence; for $\epsilon \gg N^{-7/8}$, the boundary influence decays as $N^{-\frac{1}{8}}e^{-\Theta(\epsilon^{8/7}\, N)}$ (i.e., the disorder contributes a factor of $e^{-\Theta(\epsilon^{8/7}\, N)}$ to the decay rate). For a natural notion of the correlation length, i.e., the minimal size of the box where the boundary influence shrinks by a factor of $2$ from that with no external field, we also prove the following: as $\epsilon\downarrow 0$ the correlation length transits from $\Theta(\epsilon^{-8/7})$ at $T_c$ to $e^{\Theta(\epsilon^{-4/3}\,\,)}$ for $T < T_c$.