Topological Semimetals carrying Arbitrary Hopf Numbers: Hopf-Link, Solomon’s-Knot, Trefoil-Knot and Other Semimetals

We propose a new type of Hopf semimetals indexed by a pair of numbers (p, q), where the Hopf number is given by pq. The Fermi surface is given by the preimage of the Hopf map, which consists of loops nontrivially linked for a nonzero Hopf number. The Fermi surface forms a torus link, whose examples are the Hopf link indexed by (1, 1), the Solomon’s knot (2, 1), the double Hopf-link (2, 2) and the double trefoil-knot (3, 2). We may choose p or q to be a half integer, where the Fermi surface is a torus knot such as the trefoil knot (3/2, 1). It is even possible to make the Hopf number an arbitrary rational number, where a semimetal whose Fermi surface forms open strings is generated.