Geometric distinctions between metals and insulators

Raffaele Resta has a very helpful review article The insulating state of matter: a geometrical theory.
It elegantly connects what to me are a rather disparate range of topics:
Kohn's classic 1964 characterisation of insulators, Berry's phase, Chern numbers, periodic versus open boundary conditions, quantum metrics, Anderson localization, Mott insulators, Quantum Hall insulators, ....

A central quantity defines the "localisation" associated with the ground state wavefunction.
In a 1999 PRL Resta and Sorella introduced this quantity and showed how it diverged at the quantum phase transition between a band insulator and a Mott insulator in a one-dimensional ionic Hubbard model.
In a 2006 J. Chem. Phys. paper Resta connects this localisation to Boy's localisation in quantum chemistry.

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