Random Field Ising Model — Imry-Ma Argument

The Imry-Ma argument (1975): in dimension d, a random field of strength h disorder breaks long-range order when the gain in field energy (~h·L^(d/2)) exceeds the domain wall cost (~L^(d-1)). For d≤2 this always wins → no ferromagnetic order. In d=3 order survives for weak disorder, destroyed above a critical h.

2D RFIM spin configuration. White=+1, black=-1. Random fields break large domains.

Disorder strength h: 0.5 Dimension d: 2 Exchange J: 1.0 Temperature T: 1.0

Magnetization ⟨m⟩: —

Domain size L*: —

Ordered state: —

L* = (J/h)^(2/(2-d)) for d<2

⟨|m|⟩ vs disorder h for d=2,3

Imry-Ma energy balance: domain wall cost J·L^(d-1) vs field gain h·L^(d/2). Crossover L* marks optimal domain size.

Domain size L* vs h for current d.