Hopf-chain networks

Exotic links and chains attract interests across various disciplines including mathematics, biology, chemistry and physics. Here, we propose that topological Hopf-chain networks, consisting of one-, two- and three-dimensional (3D) Hopf chains, can be found in the momentum space. These networks can be evolved from a 3D triple-points phase by varying symmetries of a four-band model. Moreover, we identify that the Hopf-chain networks exist in a family of crystals Sc3XC (X = Al, Ga, In, Tl). The crystals are 3D triple-points metals, and transit to topological metals with Hopf-chain networks under strains. These novel Hopf networks exhibit unique Landau levels and magneto-transport properties.