We study the instabilities to the conformal critical point of an exactly solvable family of Gross-Neveu-Yukawa models. Using conformal field theory techniques, we construct the zero-temperature phase diagram and identify the superconducting and charge neutral ordered phases that destabilize the critical point. Both instabilities appear only when the fermions are strongly renormalized, above a critical anomalous dimension. A higher fermion anomalous dimension also raises the critical degree of time-reversal-symmetry breaking required to suppress superconductivity, indicating that pairing becomes more robust with stronger renormalization.

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• Quantum criticality
• Superconductivity
• Dirac semimetal
• Strongly correlated systems
• Bethe-Salpeter equation
• Conformal field theory
• Gross-Neveu model