The flaw was that mean-field theory ignored fluctuations at all length scales. Near a critical point, the correlation length (\xi) (the distance over which spins are correlated) diverges to infinity. The system becomes scale-invariant: a magnet looks statistically the same whether you zoom in or out.
At the heart of theoretical physics lies a tension: microscopic laws are often simple, yet macroscopic behavior is rich and complex. The Renormalization Group (RG) is the formalism that bridges this gap. Conceived initially in quantum field theory (Stueckelberg, Petermann, 1953; Gell-Mann, Low, 1954), RG found its most intuitive physical grounding in the study of continuous phase transitions (Wilson, 1971). Later, in a remarkable synthesis, Kenneth Wilson applied the same RG philosophy to the Kondo problem, a seemingly narrow issue of a single magnetic atom in a non-magnetic metal, which had resisted decades of perturbative attempts. The flaw was that mean-field theory ignored fluctuations
Kenneth G. Wilson’s seminal 1975 paper, "The Renormalization Group: Critical Phenomena and the Kondo Problem," applied the Renormalization Group (RG) strategy to solve complex physical problems by addressing multiple length and energy scales. The work, which earned Wilson a Nobel Prize, explains universality in phase transitions and uses numerical renormalization group methods to solve the Kondo divergence in magnetic impurities. Read the official publication from APS Journals or Wilson's Nobel Lecture Institut "Jožef Stefan" The renormalization group and critical phenomena At the heart of theoretical physics lies a
Wilson took the RG idea further. He discretized the Kondo model in logarithmic energy intervals and applied an iterative numerical transformation. His paper "The Renormalization Group: Critical Phenomena and the Kondo Problem" (Rev. Mod. Phys. 1975) is a landmark; it contains the first exact calculation of the resistivity through (T_K), the specific heat, and the susceptibility. Later, in a remarkable synthesis, Kenneth Wilson applied