The insulating versus conducting behavior of condensed matter is commonly addressed in terms of electronic excitations and/or conductivity. At variance with such wisdom, W. Kohn hinted in 1964 that the insulating state of matter reflects a peculiar organization of the electrons in their ground state, and does not require an energy gap.

Kohn’s theory of the insulating state got a fresh restart in 1999; at the root of these developments is the modern theory of polarization, developed in the early 1990s, and based on a geometrical concept (Berry phase). Since insulators and metals polarize in a qualitatively different way, quantum geometry also discriminates insulators from conductors. A common geometrical “marker”, based on the quantum metric, caracterizes all insulators (band insulators, Anderson insulators, Mott insulators, quantum Hall insulators...); such marker diverges in conductors.