Pyrochlore Spin Ice — Emergent Magnetic Monopoles

Two-in Two-out Ice Rules · Dirac Strings · Monopole–Antimonopole Pairs

Parameters

Legend

Spin pointing IN
Spin pointing OUT
Magnetic monopole (+)
Antimonopole (−)
Dirac string
Ice Rule: Each tetrahedron obeys "2-in, 2-out". Violations create emergent monopoles (+3-in or 3-out) carrying magnetic charge Q=±2μ/a². They are connected by Dirac strings — chains of reversed spins.

Key: Monopoles interact via Coulomb law V(r)=μ₀Q²/4πr, effectively free charges despite no fundamental monopoles.