When it comes to elemental metals and insulators, a simple signature to distinguish them is the temperature dependence of the electrical resistivity.
Metals have a resistivity that increases monotonically with increasing temperature.
Insulators (and semiconductors) have a resistivity that decreases monotonically with increasing temperature.
Furthermore, for metals the temperature dependence is a power law and for insulators it is activated/exponential/Arrhenius.
This distinction reflects the presence or absence of a charge gap, i.e., energy gap in the charge excitation spectrum.
Metals are characterised by a non-zero value of the charge compressibility
However, in strongly correlated electron materials the temperature dependence of the resistivity is not a definitive signature of a metal versus an insulator.
The figure below shows the measured temperature dependence of the resistivity (on a logarithmic scale) of the organic charge transfer salt kappa-(ET)2Cu2(CN3) for several different pressures. At low pressures it is a Mott insulator and for high pressures a metal (and a superconductor at low temperatures).
The data is taken from this paper.
Note that at intermediate pressures the resistivity is a non-monotonic function of temperature, becoming a Fermi liquid and then a superconductor at low temperatures.
Some might say that the system undergoes an insulator-metal transition as the temperature is lowered.
The system undergoes a smooth crossover from a bad metal to a Fermi liquid as the temperature is lowered. It is always a metal, i.e. there is no charge gap.
This view is clearly supported by the theoretical results shown below (taken from this preprint) and earlier related work based on dynamical mean-field theory (DMFT).
The graph below is the calculated temperature dependence of the resistivity for a Hubbard model, for a range of interaction strengths U [in units of W, the half band width].
Note, the non-monotonic temperature dependence for values of U for which the system becomes a bad metal, i.e. the resistivity exceeds the Mott-Ioffe-Regel limit.
But, for the temperature and U range shown the system is always a metal.
This is clearly seen in the calculated non-zero value of the charge compressibility, shown below.
I thank Nandan Pakhira for emphasising this point to me and encouraging me to write this post.