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Gapless helical superconductivity on the surface of a three-dimensional topological insulator

Recent angle-resolved photoemission experiments have observed a proximity-induced superconducting gap in the helical surface states of a thin film of the 3D topological insulator Bi$_2$Se$_3$ grown on a superconducting NbSe$_2$ substrate. The superconducting coherence peaks in the electronic density of states are strongly suppressed when the topological insulator is doped with magnetic Mn impurities, which was interpreted as the complete destruction of helical superconductivity in the topological surface states. Motivated by these experiments, we explore a different possibility: gapless helical superconductivity, where a gapless electronic density of states coexists with a nonzero helical superconducting order parameter. We study a model of superconducting Dirac fermions coupled to random magnetic impurities within the Abrikosov-Gor'kov framework, and find finite regions of gapless helical superconductivity in the phase diagram of the system for both proximity-induced and intrinsic superconductivity. For the latter, we derive universal rates of supression of the superconducting transition temperature due to magnetic scattering and, for a Fermi level at the Dirac point, a universal rate of increase of the quantum critical attraction strength.