As a prototype altermagnetic candidate, ${\mathrm{RuO}}_2$ has attracted abundant attention due to its exotic properties and potential applications in electronic devices. However, its altermagnetic nature is hotly debated primarily because theoretically the appearance of altermagnetism depends on the Hubbard $U$, a quantity that is difficult to measure experimentally. Here, we use high harmonic generation as a way to gauge the strength of the Hubbard $U$ in ${\mathrm{RuO}}_2$ through first-principles calculations. Without the Hubbard $U$, ${\mathrm{RuO}}_2$ is nonmagnetic and has the Kramers degeneracy. Our orientation-dependent harmonic yield exhibits a dumbbell-like shape. With the Hubbard $U$, altermagnetism appears and the Kramers degeneracy is lifted in ${\mathrm{RuO}}_2$. The orientation-dependent harmonic yield now has a clover shape. Microscopically, electronic excitations are altered by the interplay between the orbital ordering and the crystal field. By comparing with existing experiments, we conclude that the dumbbell-shape anisotropy in the orientation-dependent harmonic yield is likely to be intrinsic, indicating weak altermagnetism in ${\mathrm{RuO}}_2$. Our study represents another perspective for future research in this field.

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• Electronic structure
• High-order harmonic generation
• Light-matter interaction
• Semiconductor Bloch equations
• Altermagnets
• Strongly correlated systems
• Density functional theory