Double Dirac nodal lines enforced by multiple nonsymmorphic symmetries

The topological quantum states of matter can be characterized by the geometric form and number of symmetry-enforced band degeneracy, such as nodal points, lines, and surfaces from twofold up to eightfold degeneracy. Here, we report the observation of double Dirac (fourfold) nodal lines stabilized by multiple nonsymmorphic symmetries in puckered honeycomb systems, black phosphorus, and metal monochalcogenides. By angle-resolved photoemission spectroscopy, we found a fourfold (Dirac) nodal line running along the armchair zone boundary of black phosphorus, reproduced by first-principles band calculations and proven by our symmetry analysis to be protected by space-time inversion and glide-mirror symmetries. The presence of the multiple glide-mirror symmetries in its binary counterpart, GeS, diversifies into five Dirac nodal lines, two of which come closer at the zigzag zone boundary to form a nearly eightfold nodal line. Our results demonstrate the correspondence between nonsymmorphicity and band degeneracy with puckered honeycomb systems.

Figure 1

Ball-and-stick model of BP [panels (a) and (b)] and $MX$ [panels (c) and (d)], viewed from the top [panels (a) and (c)] and side [panels (b) and (d)]. Gray, black, and yellow balls represent P, $M$ (Ge or Sn), and $X$ (S or Se) atoms, respectively. The black rectangles show the primitive unit cell. The light red, green, and blue lines are the plane of mirror (or glide-mirror) symmetries in $x, y$, and $z$, respectively, with respect to the inversion center marked by the circled red dot. The two sublattices of BP in a sublayer are marked by the circled A and B. (e)–(h) Brillouin zone of BP [panels (e) and (f)] and $\mathit{MX}$ [panels (g) and (h)] plotted for $k_x >$ 0, $k_y >$ 0, and $k_z >$ 0 (octant). The band degeneracy of BP and MX is obtained by our symmetry analysis without considering spin-orbit coupling (w/o SOC) [panels (e) and (g)] and with spin-orbit coupling (w/ SOC) [panels (f) and (h)]. As for w/o SOC in panels (e) and (g), the spin up and spin down are doubly counted. The number of band degeneracy at vertices, edges, and faces of the Brillouin zone is indicated by markers defined at the bottom of panels (e) and (g).

Figure 2

(a)–(d) Theoretical electronic structure of bulk BP, calculated by DFT in the absence of spin-orbit coupling (w/o SOC) and shown in high-symmetry directions introduced in Fig. 1. The orange circle in panel (a) marks the crossing of C1 and C2 bands, and orange arrows in panels (b) and (c) indicate fourfold (Dirac) degenerate nodal lines. (e)–(h) Experimental electronic structure of BP, taken by ARPES and shown in high-symmetry directions corresponding to those in panels (a)–(d). Data shown in panel (e) are symmetrized with respect to the $\mathrm{\Gamma }$ point. Experimental conditions in the order of photon energy, polarization, and scattering plane are as follows: (e) 145 eV, $p$-pol, $yz$; (f) 63–83 eV, $p$-pol, $yz$; (g) 145 eV, $p$-pol, $yz$; and (h) 145 eV, $s$-pol, $yz$. The sample temperature was kept at $16\pm 4$ K during measurements.

Figure 3

(a)–(d) Theoretical electronic structure of bulk GeS, calculated by DFT in the absence of spin-orbit coupling (w/o SOC) and shown in high-symmetry directions introduced in Fig. 1. Orange and red circles indicate fourfold and nearly eightfold degenerate nodes, respectively, and the orange arrows indicate fourfold nodal lines. (e)–(h) Experimental band structure of GeS taken by ARPES and shown in high-symmetry directions corresponding to those in panels (a)–(d). Data shown in panel (e) are symmetrized with respect to the $\mathrm{\Gamma }$ and X points. Experimental conditions in the order of photon energy, polarization, and scattering plane are as follows: (e) 106 eV, $s$-pol, $yz$; (f) 129–141 eV, $s$-pol, $yz$ for XU, 124 eV, $p$-pol, $yz$ for UR, 198–213 eV, $p$-pol, $sz$ for RS; (g) 200–213 eV, $p$-pol, $yz$ for $\mathrm{\Gamma }\mathrm{Z}$; and (h) 119 eV, $s$-pol, $yz$ for ZT, 97–108 eV, $s$-pol, $xz$ for TY. The sample temperature was kept at $25\pm 4$ K during measurements.

Figure 4

(a) Band structure of BP (black) and GeS (yellow) obtained by DFT in the presence of spin-orbit coupling (w/ SOC). The C1 and C2 bands of BP in Fig. 2 and C3– C6 bands of GeS in Fig. 3 are simultaneously plotted with respect to the Dirac energy (${\mathit{E}}_{\mathrm{D}}$) and X points. (b) Band structure of BP obtained by DFT (w/ SOC) and plotted along L-X-S. The inset in panel (b) is the magnified view of the band structure indicated by the red box. (c) Band structure of GeS obtained by DFT (w/ SOC) and plotted along X-U-R-T-Y. The inset in panel (c) shows the magnified view of the band structures indicated by the red box. The orange arrows in panels (b) and (c) show DNLs.