Superconducting quantum criticality of topological surface states at three loops

The semimetal-superconductor quantum phase transition on the two-dimensional (2D) surface of a 3D topological insulator is conjectured to exhibit an emergent N=2 supersymmetry, based on a one-loop renormalization group (RG) analysis in the ϵ expansion. We provide additional support for this conjecture by performing a three-loop RG analysis and showing that the supersymmetric fixed point found at this order survives the extrapolation to 2D. We compute critical exponents to order ϵ3, obtaining the more accurate value ν≈0.985 for the correlation length exponent and confirming that the fermion and boson anomalous dimensions remain unchanged beyond one loop, as expected from non-renormalization theorems in supersymmetric theories. We further couple the system to a dynamical U(1) gauge field, and argue that the transition becomes fluctuation-induced first order in an appropriate type-I regime. We discuss implications of this result for quantum phase transitions between certain symmetry-preserving correlated surface states of 3D topological insulators.