Jones Polynomial and Knot Transitions in Hermitian and non-Hermitian Topological Semimetals

Topological nodal line semimetals host stable chained, linked, or knotted line degeneracies in momentum space protected by symmetries. In this Letter, we use the Jones polynomial as a general topological invariant to capture the global knot topology of the oriented nodal lines. We show that every possible change in Jones polynomial is attributed to the local evolutions around every point where two nodal lines touch. As an application of our theory, we show that nodal chain semimetals with four touching points can evolve to a Hopf link. We extend our theory to 3D non-Hermitian multiband exceptional line semimetals. Our work provides a recipe to understand the transition of the knot topology for protected nodal lines.

Figure 1

The global topological invariants can be characterized by the local physics around some special points. The Chern insulators (nodal knot semimetals) can be viewed as generated phases from a critical phase with one or several Dirac points (TPs). The transition of Chern number (Jones polynomial) is attributed to the evolution changes of Berry curvature (local nodal lines) around the Dirac points (TPs) in the presence of perturbations. Panel (b) shows two possible line orientations (types I and II) around the TPs and the arrows indicate the directions of the nodal lines. There are three possible local evolutions for each line orientation.

Figure 2

Critical phase with TPs and generated phases without TPs. Panel (a) shows the nodal chain semimetal with 4 TPs. Panel (b) shows the critical phase, which is equivalent to (a) based on the periodic boundary condition of the 3D BZ. Panels (c)–(e) show several examples of the generated phases after the local evolutions of the 4 TPs.

Figure 3

The classification of the local evolution of two nodal lines with a single TP in the viewpoint of the natural projective plane, which is the plane spanned by the two tangential vectors at the TP of the two nodal lines. The first column shows the low energy theory around TP can be linear (nonvanishing gradient) or quadratic (vanishing gradient) in different cases. Here linear means linear dispersion of ${\gamma }_0^r$ or ${\gamma }_0^i$ along one direction in the BZ. This limits the possible geometry forms of the surface ${\gamma }_0^{r/i}(\mathit{k})=0$ as shown in the second column. The third column shows the possible local evolutions under the constraint low energy theory of TP.

Figure 4

Hopf-link semimetal. Panel (a) shows the physical constraint to the local evolutions of the TPs $T_{1/3}$. Only $L_{0/+}$ of type II are possible in the natural projective plane (blue plane). However, if we rotate the figures in (a), the local evolution becomes $L_{-/0}$ of type I as shown in (b). Panel (c) shows the Hopf-link semimetal can be obtained from the model in Fig. 2 by adding the perturbation $\lambda h_1(\mathit{k},\lambda )=i\lambda \sin 2k_x$.