Higher-order topological semimetals and nodal superconductors with an order-two crystalline symmetry

Using a systematic relation between topological gapless phases in three dimensions and topological gapped phases in two dimensions, we identify four types of higher-order topological semimetals or nodal superconductors (HOTS), hosting (i) flat zero-energy “Fermi arcs” at crystal hinges, (ii) flat zero-energy hinge arcs coexisting with surface Dirac cones, (iii) chiral or helical hinge modes, or (iv) flat zero-energy hinge arcs connecting nodes only at finite momentum. Bulk-boundary correspondence relates the hinge states to the bulk topology protecting the nodal point or loop. We classify all HOTS for all tenfold-way classes with an order-two crystalline (anti)symmetry, such as mirror, twofold rotation, or inversion.

Figure 1

(a) Rhombic pillar geometry and local density of states at energy $E=0$ of model (1). (b) Quasi-one-dimensional band structures in the rhombic pillar geometry for the models of Eqs. (1), (4), (7), and (9) (from left to right). The full and dashed black curves indicate the bulk and surface excitation gaps, respectively. In models (1), (4), and (7), we set $t=1$ and $k_0=0.6\pi$. In model (9), we used $m=0$, $t=1$, $t^{\prime }=2$.

Figure 2

Bulk band structures (top) and nodal lines in the three-dimensional Brillouin zone (bottom) for the representative models of Eqs. (1), (4), (7), and (9) (from left to right) from the main text. The same parameters were used as in Fig. 1. In the depictions of the nodal lines, we include a color gradient proportional to the $k_z$ coordinate of the nodal line.

Figure 3

Band structures in a rhombic pillar geometry for the example systems as defined in Eqs. (B1), (B3), (B6), (B9), and (B11) (from left to right, top) and bulk band structures (middle) and nodal line (bottom). In the pillar geometry (top), periodic boundary conditions are applied in the $z$ direction, which is along the pillar axis. The full and dashed black curves denote the excitation gap in the bulk and on the surface, respectively. Symmetry-preserving perturbations have been added to the models given in the text to turn accidentally degenerate nodal points into nodal loops and to gap out nonanomalous surface states, as desribed the text. In the models of Eqs. (B1), (B3), (B6), and (B11), we took the parameter values $t=1$, $k_0=0.6\pi$; for the model of Eq. (B9), we took the parameter values $m=0$, $t=1$, $t^{\prime }=2$.