Compact localized states, inverse participation ratio, and disorder-induced localization
The kagome lattice has a perfectly flat band at E = −2t arising from destructive interference of hoppings around hexagons — each eigenstate is a compact localized state (CLS) supported on just 6 sites. In the presence of disorder W, the flat band acquires finite width and the CLSs hybridize, but the states remain strongly localized with localization length ξ ∼ t²/W². The inverse participation ratio IPR = Σ|ψ_i|⁴ signals single-site localization as W → ∞.