Theory of the insulating state

All undergraduates learn that Bloch theorem and band structure explain the insulating/metallic behavior of most crystalline solids across the periodic table. By the late 1950s it became clear that there exist classes of materials to which band theory does not apply: Mott insulators, where the insulating character is due to electron correlation, and Anderson insulators, where it is due to lattice disorder; other kinds of insulators followed. In a visionary 1964 paper W. Kohn stated that the insulating character of a material stems from a different organization of the electrons in their ground state, and does not require an energy gap. Kohn's approach remained somewhat incomplete, and little visited for many years. The theory of the insulating state got a fresh restart from 1999 onwards: we present here a comprehensive state-of-the-art account. The modern theory, rooted in geometrical concepts, addresses all kinds of insulators on a common formal and computational basis. © Società Italiana di Fisica