Probing spin-split bands in altermagnets through second harmonic generation

Time-reversal symmetry breaking without magnetization gives rise to spin-split bands in altermagnets, where a rotational transformation instead of a translation or inversion connects the opposite-spin sublattices. However, this spin splitting is momentum dependent, and the direct spectroscopic probe remains untapped. Here we report that momentum-resolved second harmonic generation (SHG) can do this job. In ${\mathrm{RuO}}_{2}$, the first peak in SHG arises from the electronic transition touching the bands near the Fermi level and can identify its band gap. The other peaks are referred to the deeper transitions far away from the Fermi level. All of them are rooted in the third-order Pancharatnam-Berry tensor that manifests as triple multiplied Berry connections and represents an intrinsic band geometric property. More importantly, the energy shifts of the peaks in SHGs from the spin-up channel relative to the spin-down one indicate the spin splittings at different energy levels. Our findings are also true for ${\mathrm{CuF}}_{2}$ and MnTe. This study uncovers the second-order nonlinearity of momentum-resolved SHG as a local probe for the spin-split bands in altermagnets.