Abstract: Are the occupied bands of a solid fully filled?

The answer to this ``yes/no" question separates band insulators from metals, seemingly with no room left for a middle option. Topological insulators (TIs) are fascinating precisely because they break this basic dichotomy: due to the topologically non-trivial properties of their band structure, the bulk of a TI is insulating, but its boundary is metallic. As a consequence, TIs push band structure theory beyond its foundations in Bloch theorem by the need to break translation invariance in order to observe the metallic boundary states. In this talk I will address this problem by way of a diagonalization method, built on a generalization of Bloch's theorem, designed for obtaining closed-form solutions of tight-binding models subjected to arbitrary boundary conditions. Building on the analytical insight derived from this method, I will investigate thermodynamic signatures of metallic edge modes in terms of Hill's non-extensive thermodynamics, since the traditional thermodynamic limit of infinite size erases the operational distinction between TIs and ordinary insulators (well, there is no boundary in the thermodynamic limit!).