Superconducting quantum criticality of topological surface states at three loops

Understand

The semimetal-superconductor quantum phase transition on the two-dimensional (2D) surface of a 3D topological insulator is conjectured to exhibit an emergent $\mathcal{N}=2$ supersymmetry, based on a renormalization group (RG) analysis at one-loop order in the $\epsilon$ 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 $\epsilon^3$, obtaining the more accurate value $\nu\approx 0.985$ 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.