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It is often thought that emergent phenomena in topological phases of matterare destroyed when tuning to a critical point. In particular, topologicallyprotected edge states supposedly delocalize when the bulk correlation lengthdiverges. We show that this is not true in general. Edge states of topologicalinsulators or superconductors remain exponentially localized---despite avanishing band gap---if the transition increases the topological index. Thisapplies to all classes where the topological classification is larger than$Z_2$, notably including Chern insulators. Moreover, these edge states arestable to disorder, unlike in topological semi-metals. This new phenomenon isexplained by generalizing band (or mass) inversion---a unifying perspective ontopological insulators---to kinetic inversion. In the spirit of thebulk-boundary correspondence, we also identify topological invariants atcriticality, which take half-integer values and separate topologically-distinctuniversality classes by a multi-critical point. This work enlarges the scope oftopological protection and stability by showing that bulk energy gaps can beunnecessary. Experimental probes and stability to interactions are discussed.

Topology and edge states survive quantum criticality between topological insulators