In this article, we propose a possibility of realizing a distinct family of scale-conformal invariant interacting Hamiltonians in two-dimensional quantum many-body systems. Such phenomena can occur in various interacting systems, including topological surfaces or two-dimensional bulks. Based on a previous study [S. Vijayan and F. Zhou, Phys. Rev. B 111, 174504 (2025) ], here we further demonstrate that a semiclassical conformal manifold can emerge 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 semiclassical conformal manifolds. When $N_c$ is set to be finite but large, we also show that quantum fluctuations that induce fermion field renormalization can result in mildly infrared relevant or irrelevant renormalization-group (RG) flow within a semiclassical conformal manifold. These quantum effects lead to standard, isolated, infrared-stable Wilson-Fisher fixed points. These stable fixed points, along with ultraviolet stable fixed points, form a discrete manifold, due to the spontaneous symmetry breaking of an emergent $\mathrm{SO}\left(\mathcal{N}\right)$ dynamical symmetry in the RG flow as $N_c\to \infty$. In addition, we find that along the direction of the RG flow within the manifold, an EPR-like entanglement entropy in the fermion flavor space always increases. The infrared-stable Wilson-Fisher fixed points, induced by quantum fluctuations, are always linked to theories on the manifold where interaction operators are maximally entangled. Our studies provide an effective framework for addressing topological quantum critical points with high-dimensional interaction parameter spaces that potentially host an overwhelmingly large number, often an exponentially large number of fixed points of complex stabilities. They also highlight the central role of entangled conformal operators and their entropy in shaping the universality classes of topological quantum phase transitions on surfaces or in bulk. We conclude our studies with open questions and possible future directions.

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• Dirac fermions
• Quantum criticality
• Superconductivity
• Conformal field theory
• Large-N expansion in field theory