Random-Field Ising Model

Quenched disorder destroys long-range order below d=4 (Imry-Ma 1975). Domains form to reduce local field energy at the cost of wall energy.

T=0.5
Δ=1.0
J=1.5
|M|: —
Domains: —
MCS: 0
Imry & Ma (1975): in d≤2, an arbitrarily weak random field destroys the ordered phase. The argument: a domain of size L gains energy ~Δ·L^(d/2) from aligning with local fields (random walk), but costs ~J·L^(d-1) in wall energy. For d<2, field always wins; at d=2, it's marginal. In dilute antiferromagnets (e.g., Fe_xZn_{1-x}F₂), the applied uniform field creates effective random staggered fields — experimental realization of RFIM. The lower critical dimension is d_lc=2.