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Local topological phase transitions in periodic condensed matter systems

Topological properties of a periodic condensed matter system are global features of its Brillouin zone (BZ). In contrast, the validity of effective low energy theories is usually limited to the vicinity of a high symmetry point in the BZ. We derive a general criterion under which the control parameter of a topological phase transition localizes the topological defect in an arbitrarily small neighbourhood of a single point in $k$-space upon approaching its critical value. Such a local phase transition is associated with a Dirac-like gap closing point, whereas a flat band transition is not localized in $k$-space. This mechanism and its limitations are illustrated with the help of experimentally relevant examples such as HgTe/CdTe quantum wells and bilayer graphene nanostructures.