Fig. 3. Euler class calculations.

a Patches in the Brillouin zone and their corresponding Euler class for 7% strained silicate under an electric field of 1.0 eV/Å. The Euler class χ₁₀ is calculated from the phonon eigenvectors of ${\mathcal{B}}_{10}$ and ${\mathcal{B}}_{11}$ within the patches. The resultant patch Euler class in the yellow (patch 1), green (patch 2) and magenta (patch 3) areas are 0, 0.5 and 0, respectively. The large blue circle with dashed boundary at Γ is a quadratic node with the patch Euler class of 1. b 3D band structures in the grey areas of (a), with two linear nodes —one (cyan triangle) in Δ₁₁ along Γ-M and one (dark red circle) in Δ₁₀ at K, as well as a quadratic node (blue circle) in Δ₁₀ at Γ. These dispersions root in the stability of Euler class, meaning that a stable double node produces a quadratic dispersion. c Patches in the Brillouin zone to calculate the Euler class for 7% strained silicate under an electric field of 1.8 eV/Å. The patch Euler class in the blue (patch 1) and purple (patch 2) patches are both 0. The large blue circle with dashed boundary at Γ and the large dark red circles with dashed boundary at K are quadratic nodes with the patch Euler class of 1. d 3D band structures in the grey areas of (c), with one linear node (cyan triangle) in Δ₁₁ along Γ–M and two quadratic nodes in Δ₁₀ at Γ (blue circle) and K (dark red circle).