Abstract
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 (D) 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 (0D) 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 σ bonds. We conducted an ab inito 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 2D Graphene were topologically protected upon the introduction of the Hopf links, and low-energy excitations are described by 1D, 2D, and 3D gapless Fermions.
Highlights
Mass of an elementary particle is related to a broken symmetry of a vacuum, and the energy-momentum relationship is linked to the symmetries of the vacuum1
We have considered the concept of topological materials8–10 in materials with hard covalent bonds. 1D Carbon-Nano-Chains, Hopf-linked bilayer-Graphene (2D), and 3D Hopfene have been proposed as examples of new topological carbon allotropes, which will be useful to examine fundamental physics of massless Dirac Fermions4, 23, 31, 45, 55 in topologically nontrivial geometries
The unique aspect of this approach is a topological link to bind various sheets strongly together without forming a proper chemical bond. This configuration is topologically different from the simple stacking weakly bound together by van der Waals force72
Summary
Mass of an elementary particle is related to a broken symmetry of a vacuum, and the energy-momentum relationship is linked to the symmetries of the vacuum. Materials with certain symmetries can be designed, and carbon allotropes are useful due to their richness in families with various translational and rotational symmetries of crystals, such as a cage like a Fullerene , a tube called a carbon-nano-tube3, 4 , and a sheet including a Graphene5–7 These materials have topologically trivial crystal structures without having a link nor a knot. If we restricted ourselves to investigate materials, which have perfect crystalline structures only, Graphene would not be discovered, since the absence of the long-range order in 2D was rather rigorously proved theoretically by Mermin-Wager and Hohenberg37 It is the existence of a ripple , which stabilises Graphene at finite temperature, such that strictly rigid long-range order as a crystal is absent. The unique features of these Chains and Chainmail are their flexibility with the rigidity ensured by the σ bonds
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