Extended Data Fig. 1: Topological invariants in physics.

a, An example of an order parameter winding in real space: a magnetic vortex^{2,5,6,7,50,51}. In this case, the order parameter is the local magnetization m(x), confined to a magnetic easy plane in real space (x, y). It may happen that m(x) winds around a point in real space, forming a magnetic vortex characterized by a winding number topological invariant, in this example given by w = 1. b, An example of a quantum wavefunction winding in momentum space: the one-dimensional topological insulator (Su-Schrieffer-Heeger model)^{3,4,8,9,10,11,12,13,52,53,54,55,56}. This phase is described by Bloch Hamiltonian h(k) = d(k) ⋅ σ, where k is the one-dimensional crystal momentum, σ refers to the Pauli matrices and d(k) is a two-component object confined to the (d_x, d_y) plane. The normalized quantity \(\widehat{{\bf{d}}}(k)\equiv {\bf{d}}(k)/|{\bf{d}}(k)|\) (orange arrow) moves around the unit circle (dotted blue) as k varies. The topological invariant is related to how many times \(\widehat{{\bf{d}}}(k)\) winds around the origin as k scans through the one-dimensional Brillouin zone. c, Node loops linking in momentum space^{17,18,19,20,57}: a three-dimensional electronic structure may exhibit multiple node loops (cyan and purple), characterized by k_n(θ), where n indexes the loops and θ parametrizes the loop trajectory in momentum space. The loops may link one another, encoding a linking number topological invariant. This example shows a Hopf link. (See also Supplementary Information.).