  Topological phase transitions track changes in topological properties of a
system and occur in real materials as well as quantum engineered systems, all
of which differ greatly in terms of dimensionality, symmetries, interactions,
and driving, and hence require a variety of techniques and concepts to describe
their topological properties. For instance, depending on the system, topology
may be accessed from single-particle Bloch wave functions, Green's functions,
or many-body wave functions. We demonstrate that despite this diversity, all
topological phase transitions display a universal feature: namely, a divergence
of the curvature function that composes the topological invariant at the
critical point. This feature can be exploited via a renormalization-group-like
methodology to describe topological phase transitions. This approach serves to
extend notions of correlation function, critical exponents, scaling laws and
universality classes used in Landau theory to characterize topological phase
transitions in a unified manner.
