How does one distinguish a metal from an insulator? One signature might be the presence of a charge gap at the chemical potential.
Is there a broken symmetry associated with a metal-insulator phase transition?
A key concept emphasized by Anderson is that distinct phases of matter are associated with the rigidity of their order parameter. For example, for a superconductor the superfluid density is associated with the phase stiffness of the ground state wave function.
It turns out that the Drude weight and the charge compressibility are both zero in a Mott insulator and non-zero in a metal. This is discussed in an interesting article by Imada.
in dynamical mean-field theory the Mott transition at half filling is first order.
Kohn (and later Thouless) emphasized that for a metal one could calculate the Drude weight from the size and twist dependence of the ground state energy in the presence of twisted boundary conditions.