Fig. 2: Measurements of the Euler class.
a The experimentally measured vectors n(k) for the Euler Hamiltonian H(k) in Eq. (1) with m = 1 (corresponding to a nontrivial Hamiltonian) over a 20 × 20 discretized Brillouin zone and b their distribution on the sphere S2. c The experimentally measured vectors n(k) for the Euler Hamiltonian H(k) in Eq. (1) with m = 3 (corresponding to a trivial Hamiltonian) over a 20 × 20 discretized Brillouin zone and d their distribution on the sphere S2. In the nontrivial case, the vectors n(k) exhibit a nontrivial skyrmion structure and cover the entire sphere once, yielding a nonzero Euler class ξ = 2, whereas in the trivial case, the vectors cover parts of the sphere, yielding a zero Euler class.
