Surface topological quantum criticality: Conformal manifolds and discrete strong-coupling fixed points

In this article, we study quantum critical phenomena in surfaces of symmetry-protected topological matter, i.e., surface topological quantum criticality. A generic phase boundary of gapless surfaces in a symmetry-protected state shall be a codimension one manifold in an interaction parameter space of dimension ${D}_{p}$ (where $p$ refers to the parameter space) where the value of ${D}_{p}$ further depends on bulk topologies. In the context of fermionic topological insulators that we focus on, ${D}_{p}$ depends on the number of half-Dirac cones $\mathcal{N}$. We construct such manifolds explicitly for a few interaction parameter spaces with various ${D}_{p}$ values. Most importantly, we further illustrate that in cases with ${D}_{p}=3$ and 6, there are submanifolds of fixed points that dictate the universalities of surface topological quantum criticality. These infrared stable manifolds are associated with emergent symmetries in the renormalization-group-equation flow naturally appearing in the loop expansion. Unlike in the usual order-disorder quantum critical phenomena, typically governed by an isolated Wilson-Fisher fixed point, we find in the one-loop approximation surface topological quantum criticalities are naturally captured by conformal manifolds where the number of marginal operators uniquely determines their codimensions. Isolated strong-coupling fixed points also appear, usually as the endpoints in the phase boundary of surface topological quantum phases. However, their extreme infrared instabilities along multiple directions suggest that they shall be related to multicritical surface topological quantum critical phenomena rather than generic surface topological quantum criticality. We also discuss and classify higher-loop symmetry-breaking effects, which can either distort the conformal manifolds or further break the conformal manifolds down to a few distinct fixed points.