Surface topological quantum criticality: Conformal manifolds and Discrete Strong Coupling Fixed Points

We study the quantum critical phenomena of the attractively interacting gapless surface states of a 3D topological insulator with $\mathcal{N}$ surface Dirac cones. A cubic lattice is employed to model the 3D TI. For $\mathcal{N}=2,3$, we observe a continuum manifold of interacting fixed points in the multi-dimensional parameter space in addition to the expected isolated fixed points. These conformal submanifolds are embedded within the phase boundary separating the gapless and the superconducting phases. The phase boundary forms a $D_p-1$ dimensional manifold in the $D_p$-dimensional parameter space. As a result, the conformal submanifolds describe the universality class of the phase boundary.

The existence of these conformal manifolds arises from an emergent $SO(\mathcal{N})$ symmetry in the renormalization group equation, expanded to one-loop order. Unlike in conventional order-disorder quantum critical phenomena, typically governed by an isolated infrared-stable Wilson-Fisher fixed point, we find that surface topological quantum criticalities in the one-loop approximation are more naturally captured by the conformal manifolds, uniquely determined by the number of marginal operators. We also discuss higher-loop symmetry-breaking effects, which may either distort the conformal manifolds or reduce them to a few distinct fixed points.