Nodal-knot semimetals

Topological nodal-line semimetals are characterized by one-dimensional lines of band crossing in the Brillouin zone. In contrast to nodal points, nodal lines can be in topologically nontrivial configurations. In this Rapid Communication, we introduce the concept of “nodal-knot semimetals,” whose nodal lines form topologically nontrivial knots in the Brillouin zone. We introduce a generic construction of nodal-knot semimetals, which yields the simplest trefoil nodal knot and other more complicated nodal knots. The knotted-unknotted transitions by nodal-line reconnections are also studied. Our work brings the knot theory to the subject of topological semimetals.

Figure 1

Schematic illustration of four types of nodal line: (a) ordinary nodal ring, (b) nodal chain, (c) nodal link, and (d) nodal knot. A nodal knot is a single nodal line entangled with itself.

Figure 2

The trefoil nodal knot of the continuum model in Eq. (10). The parameter $m$ in Eq. (10) is $m=0.5$.

Figure 3

(a)–(c) Nodal knots projected to the surface Brillouin zone (red lines), and the regions of surface bands (light-blue regions have one band, while dark-blue regions have two bands). (d)–(f) The surface-state energy dispersions plotted along the dashed lines in (a)–(c), respectively. The Bloch Hamiltonian is Eq. (11) with $m_0=2.8$.

Figure 4

The evolution of nodal knot as a function of $m_z$. Here, $m_0=2.8$ is fixed.

Figure 5

Nodal knots and nodal links for several values of $(p,q)$. The Hamiltonian is given by the ansatz Eq. (6); $m_0=2.5$. (a) $(p,q)=(5,3)$; (b) $(p,q)=(4,3)$; (c) $(p,q)=(5,2)$; (d) $(p,q)=(4,2)$; (e) $(p,q)=(6,2)$; (f) $(p,q)=(6,3)$. In (a)–(c) we have a nodal knot (a single knotted nodal line), while in (d)–(f) we have a nodal link. In the cases (a)–(c), $p$ and $q$ are relatively prime; in (d)–(f), they are not.