Surface topological quantum criticality: Conformal manifolds and entanglement

In this article, we propose a possibility of realizing conformal manifolds, smooth manifolds formed by a family of scale-conformal invariant interacting Hamiltonians in two-spatial-dimensional quantum many-body systems. Such phenomena can arise naturally in various interacting systems, including topological surface states or 2D bulks. Based on our earlier work, we show that a conformal manifold emerges as an exact solution when the number of fermion colors, $N_c$, in our models becomes infinite. We identify distinct exact marginal deformation operators uniquely associated with the conformal manifolds. When $N_c$ is set to be finite, strong quantum fluctuations destabilize the conformal manifold, breaking it into more standard isolated fixed points. Remarkably, we find that along the direction of the renormalization group (RG) flow within the manifold, an EPR-like entanglement entropy in the fermion flavor space always increases, and that the infrared-stable fixed points correspond to theories on the conformal manifold where interaction operators are maximally entangled. Our results establish a concrete framework for describing topological quantum critical points in systems with high-dimensional interaction parameter spaces, revealing how entangled conformal operators and their entropy fundamentally shape the universality classes of topological phase transitions in both surface and bulk systems.