  Topological insulators (superconductors) are materials that host
symmetry-protected metallic edge states in an insulating (superconducting)
bulk. Although they are well understood, a thermodynamic description of these
materials remained elusive, firstly because the edges yield a non-extensive
contribution to the thermodynamic potential, and secondly because topological
field theories involve non-local order parameters, and cannot be captured by
the Ginzburg-Landau formalism. Recently, this challenge has been overcome: by
using Hill thermodynamics to describe the Bernevig-Hughes-Zhang model in two
dimensions, it was shown that at the topological phase change the thermodynamic
potential does not scale extensively due to boundary effects. Here, we extend
this approach to different topological models in various dimensions (the Kitaev
chain and Su-Schrieffer-Heeger model in one dimension, the Kane-Mele model in
two dimensions and the Bernevig-Hughes-Zhang model in three dimensions) at zero
temperature. Surprisingly, all models exhibit the same universal behavior in
the order of the topological-phase transition, depending on the dimension.
Moreover, we derive the topological phase diagram at finite temperature using
this thermodynamic description, and show that it displays a good agreement with
the one calculated from the Uhlmann phase. Our work reveals unexpected
universalities and opens the path to a thermodynamic description of systems
with a non-local order parameter.
