Wikinventia — Atlas of discoveries and inventions · Global Age

The block-spin transformation — Kadanoff's physical justification of critical scaling

1966 AD · Transmission: Global
PhysicsTheoryNorth American

Leo Kadanoff, at the University of Illinois at Urbana-Champaign, publishes in 1966 "Scaling Laws for Ising Models near Tc" in the short-lived journal Physics, founded by Philip Anderson and Bernd Matthias. Kadanoff proposes dividing the Ising model — a simplified model of magnetism consisting of a lattice of spins that can point up or down — into microscopically large but much smaller-than-correlation-length cells or "blocks", and treating the total magnetization of each block as a new collective variable. Iterating this "block spin" transformation repeatedly, the system becomes self-similar: it looks the same at different observational scales near the critical point. This argument provides the first physical — not merely empirical — justification of the scaling hypothesis Benjamin Widom had proposed the previous year (1965): Kadanoff's calculation exactly reproduces Widom's relation among the critical exponents. Furthermore, by assuming that the block process converges to a single "fixed point" independent of the lattice's microscopic detail, Kadanoff's argument explains why very different physical systems (a magnet, a fluid, a binary alloy) share the same critical exponents near their respective phase transitions — the phenomenon known as universality. Kadanoff himself called the subdivisions of his lattice "cells", not "blocks"; the name "block spin" would become popular later. This idea becomes the direct basis on which, five years later, Kenneth Wilson builds the formal renormalization group (1971).

InstitutionUniversity of Illinois at Urbana-Champaign
Historical regionUnited States
Primary sourceKadanoff, L. P. — "Scaling Laws for Ising Models near Tc" (Physics Physique Fizika, 2, 263-272, June 1, 1966). DOI: 10.1103/PhysicsPhysiqueFizika.2.263
Secondary sourcePhysics Today — obituary of Leo Kadanoff; Journal of Statistical Physics — "In Memory of Leo P. Kadanoff"; Wikipedia — biography of Leo Kadanoff; arXiv:2208.00864 — "100 Years of the (Critical) Ising Model"
Original languageEnglish
View this entry in the interactive atlas → View in graph →