We propose a theory for the microscopic origin of the multiple superconducting and magnetic phases observed in ${\mathrm{CeRh}}_2{\mathrm{As}}_2$ based on the existence of Van Hove singularities near the Fermi energy. The nonsymmorphic symmetry of this material implies that these singularities are located away from high-symmetry momenta; i.e., they have so-called type-II character. This allows us to include the significant Rashba spin-orbit coupling in ${\mathrm{CeRh}}_2{\mathrm{As}}_2$ in a parquet renormalization group approach. When Fermi-surface nesting is strong, our analysis reveals two closely competing superconducting states with opposite parities, as well as an instability toward spin-density wave states that support both of them, consistent with the phase diagram of ${\mathrm{CeRh}}_2{\mathrm{As}}_2$. Type-II Van Hove singularities are generic to nonsymmorphic space groups, and so our theory implies that many other compounds may support closely competing even- and odd-parity superconductivity.

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• Rashba coupling
• Spin density waves
• Superconductivity
• Superconductors
• Van Hove singularity
• Renormalization group