Superconducting quantum criticality of topological surface states at three loops

TL;DR

This paper investigates the quantum critical behavior of surface states in 3D topological insulators, providing a three-loop renormalization group analysis that supports emergent supersymmetry and explores the nature of the phase transition.

Contribution

It extends previous one-loop studies by performing a three-loop RG analysis, confirming the supersymmetric fixed point and calculating more precise critical exponents.

Findings

The supersymmetric fixed point persists at three loops.

The correlation length exponent is approximately 0.985.

Coupling to a gauge field suggests the transition is fluctuation-induced first order.

Abstract

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. We discuss implications of this result for quantum phase transitions between certain symmetry-preserving correlated surface states of 3D topological insulators.