Beschreibung:
<jats:title>Abstract</jats:title>
<jats:p>Carbon allotropes such as diamond, nano-tube, Fullerene, and Graphene were discovered and revolutionised material sciences. These structures have unique translational and rotational symmetries, described by a crystallographic group theory, and the atoms are arranged at specific rigid positions in 3-dimensional (<jats:italic>D</jats:italic>) space. Regardless of these exotic molecular structures, the structures of materials are topologically trivial in a mathematical sense, that their bonds are connected without a link nor a knot. These days, the progress on the synthetic chemistry is significant to make various topologically non-trivial molecular structures. Topological molecules (0<jats:italic>D</jats:italic>) including Trefoil knots, a Hopf-link, a Möbius strip, and Borromean rings, were already realised. However, their potentially exotic electronic properties have not been sufficiently explored. Here, we propose a new 3D carbon allotrope, named Hopfene, which has periodic arrays of Hopf-links to knit horizontal Graphene sheets into vertical ones without connecting by <jats:italic>σ</jats:italic> bonds. We conducted an <jats:italic>ab inito</jats:italic> band structure calculation using a Density-Functional-Theory (DFT) for Hopfene, and found that it is well-described by a tight-binding model. We confirmed the original Dirac points of 2<jats:italic>D</jats:italic> Graphene were topologically protected upon the introduction of the Hopf links, and low-energy excitations are described by 1<jats:italic>D</jats:italic>, 2<jats:italic>D</jats:italic>, and 3<jats:italic>D</jats:italic> gapless Fermions.</jats:p>