Electrons possess both wave- and particle-like quantum mechanical properties. In 1958 Anderson proposed that strong disorder causes electron waves to interfere and become trapped, turning a metal abruptly into an insulator; this is Anderson localization. Mott showed that intense Coulomb repulsion between electrons can split the conducting band and drive a metal–insulator transition now called the Mott transition. Van Vleck built a theory in which delocalized and localized electrons interact through exchange interaction to create magnetization in crystals. These concepts underlie semiconductors, magnetic memory and spin-tronics that power modern information technology. Moreover, studies of disorder and correlation guide the search for high-temperature superconductors and quantum-computer materials.

Anderson’s localization theory, using phase-invariant perturbation and Green’s functions, demonstrates that in three dimensions a quantum phase transition separates metallic and insulating phases at a critical disorder strength. Mott criticized the limits of conventional band theory and, for strongly correlated systems, introduced the Mott insulator where electrons split by Hubbard U open a gap even at half-filling. He also devised the variable-range-hopping (VRH) model to account for low-temperature conduction. Van Vleck’s exchange-interaction theory became the foundation for deriving spin Hamiltonians, later generalized to molecular-field approximations and superexchange mechanisms. Together these theories supply a framework for understanding transport in ferromagnetic semiconductors, transition-metal oxides, graphene and other systems dominated by disorder or correlation. Insights into critical exponents and universality connect to statistical physics and influence today’s topological-matter research.

Anderson (Phys. Rev. 109, 1492, 1958) employed a self-consistent perturbative approach and derived the scaling of localization length ξ, anticipating a β-function sign change in three dimensions; the ensuing single-parameter scaling theory now universally describes localization–delocalization transitions in non-interacting systems. Mott (Proc. Roy. Soc. A62, 416, 1949) discussed band splitting from strong correlations and, in Rev. Mod. Phys. 40, 677 (1968), introduced the VRH law σ(T)=σ0exp[−(T0/T)1/4], later refining the exponent to 1/3 (Phys. Rev. Lett. 35, 1293, 1975). Van Vleck analyzed crystal-field effects on the g-factor and susceptibility and formalized superexchange via J = −2t^2/U. These results underpin DGPEA, DMFT, critical σ(E)~E^{d/2−1} scaling and spin-liquid theories; experimentally they guide ARPES, STM and μSR studies. Recently, cold-atom quantum simulators of the Anderson–Hubbard model have observed dynamic criticality, reaffirming the contemporary relevance of the Nobel-winning work.