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We study symmetry breaking quantum phase transitions in topologicalinsulators and superconductors where the single electron gap remains open inthe bulk. Specifically, we consider spontaneous breaking of the symmetry thatprotects the gapless boundary modes, so that in the ordered phase these modesare gapped. Here we determine the fate of the topological boundary modes rightat the transition where they are coupled to the strongly fluctuating orderparameter field. Using a combination of exact solutions and renormalizationgroup techniques, we find that the surface fermionic modes either decouple fromthe bulk fluctuations, or flow to a strongly coupled fixed point which remainsgapless. In addition, we study transitions where the critical fluctuations areconfined only to the surface and find that in several cases the critical pointis naturally supersymmetric. This allows a determination of critical exponentsand points to an underlying connection between band topology and supersymmetry.Finally, we study the fate of gapless Majorana modes localized on point andline defects in topological superconductors at bulk criticality, which isanalogous to a quantum impurity problem. Again, an interplay of topology andstrong correlations causes these modes to remain gapless but in a stronglycoupled state. Experimental candidates for realizing these phenomena arediscussed.

Quantum Criticality in Topological Insulators and Superconductors: Emergence of Strongly Coupled Majoranas and Supersymmetry