Extended Data Fig. 4: Multi-gap topology in kagome models.
From: Experimental observation of non-Abelian topological acoustic semimetals and their phase transitions
a, Taking \((\epsilon _{{{\mathrm{A}}}},\epsilon _{{{\mathrm{B}}}},\epsilon _{{{\mathrm{C}}}})\) = (1, 0, −1) and \((t,t\prime ) = (0,0)\) gives crossing Dirac strings (DS) in both gaps (blue for first gap, red for second gap). b and c, As the next step, turning off the onsite potentials and switching on the hopping terms induces band nodes. The nodes in the second gap (filled/empty red circles indicating ± topological charges) cross the DS in the first gap, forming a stable pair as the double node at Γ (brown circle) in (c) which has finite patch Euler class ξ = 1. Meanwhile, the first gap features nodes at K points (triangles). c-i, Braiding process and transfer of band nodes from one gap to another through triple points. Band nodes and DS strings evolve such that the degeneracy at K in the first gap (blue triangles) is tuned into a double node configuration that has finite patch Euler class in the second gap (brown circles).
