All-optical spin switching (AOS) is a new phenomenon found in a small group of magnetic media, where a single laser pulse can switch spins from one direction to another, without assistance of a magnetic field, on a time scale much shorter than existing magnetic technology. However, despite intensive efforts over a decade, its underlying working principle remains elusive. Here through manganese-based Heusler ferrimagnets, we show that a group of flat bands around the Fermi level act as gateway states to form efficient channels for spin switching, where their noncentrosymmetry allows us to correlate the spin dynamics to the second-order optical response. To quantify their efficacy, we introduce the third-rank Pancharatnam-Berry tensor (PB tensor), ${\mathit{\eta }}^{\left(3\right)}=\langle i\left|\mathbf{p}\right|m\rangle \langle m\left|\mathbf{p}\right|f\rangle \langle f\left|\mathbf{p}\right|i\rangle ,$ where $|i\rangle$, $|m\rangle$ and $|f\rangle$ are initial, intermediate and final band states, respectively, and $\mathbf{p}$ is the momentum operator. A picture emerges: Those which show AOS, such as the recently discovered ${\mathrm{Mn}}_2\mathrm{RuGa}$, always have a large PB tensor element but have a small sublattice spin moment ratio, consistent with the prior experimental small remanence criterion. This does not only reveal that the delicate balance between the large PB tensor element and the small sublattice spin ratio plays a decisive role in AOS, but also, conceptually, connects the $n\mathrm{th}$-order nonlinear optics to $(n+1)\mathrm{th}$-rank PB tensors in general.

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• Flat bands
• Geometric & topological phases
• Magnetization switching
• Magneto-optical Kerr effect
• Nonlinear optics
• Spin dynamics
• Ultrafast demagnetization
• Ultrafast magnetic effects
• Ferrimagnets
• Heusler alloy
• First-principles calculations
• Time-dependent DFT