Hopf-chain networks evolved from triple points

TL;DR

This paper introduces topological Hopf-chain networks in momentum space that evolve from triple-points in certain crystals, revealing new topological phases with distinctive electronic properties.

Contribution

It demonstrates the existence of Hopf-chain networks in momentum space derived from a four-band model and identifies real materials where these networks can be realized and manipulated.

Findings

Hopf-chain networks can be formed from triple-points by symmetry variation.

These networks are found in Sc3XC crystals and can be induced by strain.

Hopf networks exhibit unique Landau levels and magneto-transport phenomena.

Abstract

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.