Hopf-chain networks evolved from triple points

Exotic links and chains attract interest 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. Moreover, we identify that the Hopf-chain networks exist in a family of crystals Sc₃XC0.28em0ex(X=Al,0.28em0exGa,0.28em0exIn,0.28em0exTl). The crystals are 3D triple-points metals, and transit to topological metals with Hopf-chain networks under strains. These Hopf networks will exhibit unique magnetotransport properties.