Random Field Ising Model — Imry-Ma

Quenched random magnetic fields compete with ferromagnetic coupling. Imry-Ma argument: in d≤2, arbitrarily weak random fields destroy long-range order by domain formation.

Parameters

Temperature T/J: 0.5 Random field h/J: 0.0 MC steps/frame: 5

Imry-Ma: For d≤2, long-range order destroyed for any h≠0. For d≥3, order survives small h.

m = -- | E = --

Random Field Ising Model: H = −J Σ⟨ij⟩ sᵢsⱼ − Σᵢ hᵢsᵢ where hᵢ ~ N(0,h²). Imry-Ma (1975): domain wall energy ∝ R^(d-1), random field energy gain ∝ h R^(d/2). Balance gives optimal domain size R* ∝ (J/h)^(2/(4-d)) (d<4).