Conversion Rules for Weyl Points and Nodal Lines in Topological Media

According to a widely held paradigm, a pair of Weyl points with opposite chirality mutually annihilate when brought together. In contrast, we show that such a process is strictly forbidden for Weyl points related by a mirror symmetry, provided that an effective two-band description exists in terms of orbitals with opposite mirror eigenvalue. Instead, such a pair of Weyl points convert into a nodal loop inside a symmetric plane upon the collision. Similar constraints are identified for systems with multiple mirrors, facilitating previously unreported nodal-line and nodal-chain semimetals that exhibit both Fermi-arc and drumhead surface states. We further find that Weyl points in systems symmetric under a $\pi$ rotation composed with time reversal are characterized by an additional integer charge that we call helicity. A pair of Weyl points with opposite chirality can annihilate only if their helicities also cancel out. We base our predictions on topological crystalline invariants derived from relative homotopy theory, and we test our predictions on simple tight-binding models. The outlined homotopy description can be directly generalized to systems with multiple bands and other choices of symmetry.

Figure 1

Conversion of mirror ($m_z$)-related Weyl points [red and blue blobs in (a)] into a nodal loop [yellow line in (b)] inside an $m_z$-invariant plane $\mathrm{\Pi }$ (cyan). The conversion follows from studying the winding $n_W$ of the pseudospin $\widehat{\mathbf{d}}(\mathbf{k})$ (green arrows) on a surrounding sphere $S^2$ [composed of the upper (red) and the lower (blue) hemisphere, meeting at the equator $S^1$ (black)]. By the $m_z$ symmetry, $n_W$ has opposite sign on the two hemispheres. Since the winding is the same in both panels, the conversion from (a) to (b) is permitted.

Figure 2

Nodal lines (NLs) exhibiting both surface Fermi arcs (SFAs) and drumhead states. (a) A model [86] exhibiting a NL with charge $|n_W|=1$ in each of the $k_z\in \{0,\pi \}$ planes. The density of states in the (100) surface Brillouin zone (SBZ) reveals two mirror ($m_z$-)related SFAs, while (b) the (001) SBZ exhibits drumhead states on an annulus bounded by the projection of the NLs. (c)–(d) Adjusting the model parameters converts the NL in the $k_z=\pi$ plane to two Weyl points. The remaining NL connects with a SFA to both WPs inside the (100) SBZ.

Figure 3

Nodal chain with surface Fermi arcs (SFAa). (a) An $m_{x,y}$-symmetric model [86] exhibiting two nodal lines (NLs) with $|n_W|=1$ in both the $k_x=0$ and the $k_y=\pi$ plane, each creating the dipolar $\widehat{\mathbf{d}}(\mathbf{k})$ of Fig. 1. The winding number on quarter cylinders projecting onto the green solid-dashed quarter circles in the (001) SBZ is $n_{W2}=\pm 1$, compatible with the observed connectivity of the SFAs. (b) Adjusting the model parameters produces a nodal-chain phase with four SFAs. (c),(d) The nodal structures inside the bulk BZ for parameters producing the surface spectra in panels (a) and (b).

Figure 4

Conversions of Weyl points (WPs, red blobs) in systems with $C_{2z}\mathcal{T}$ symmetry conserve two integer charges defined on the surrounding sphere [composed of two hemispheres (red and blue) separated at the $k_z=0$ plane ($\mathrm{\Pi }$, cyan) by the equator (black)]. (a) Two out-of-plane WPs related by $C_{2z}\mathcal{T}$ carry the same chirality $n_C$, as manifested by the same winding of $\widehat{\mathbf{d}}(\mathbf{k})$ in the $k_y=0$ plane (magenta projection, left), and induce zero helicity $n_H$ of $\widehat{\mathbf{d}}(\mathbf{k})$ in plane $\mathrm{\Pi }$ (green projection, bottom). (b) Conversion to in-plane WPs [86] creates helical $\widehat{\mathbf{d}}(\mathbf{k})$ in plane $\mathrm{\Pi }$. The conservation of $n_H$ requires the obtained WPs to carry opposite helicity, as can be checked by following $\widehat{\mathbf{d}}(\mathbf{k})$ along the equator projection (gray loop). As $n_C$ is also unchanged, the conversion from (a) to (b) is permitted.