A local characterization of the insulating state

An insulator differs from a metal because of a different organization of the electrons in their ground state. This feature can be probed by means of the quantum metric tensor: a geometrical property, inspired by---and closely related to---the modern theory of polarization. Such tensor addresses the system as a whole, and it is therefore limited to macroscopically homogeneous samples. Here we show that the same approach leads to a local marker, which can detect the metallic vs. insulating character of a given sample region using as sole ingredient the ground state many-body wavefunction. Simulations on paradigmatic systems validate our theory.