Single Nodal Loop of Accidental Degeneracies in Minimal Symmetry: Triclinic ${\mathrm{CaAs}}_3$

The existence of closed loops of degeneracies in crystals has been intimately connected with associated crystal symmetries, raising the following question: What is the minimum symmetry required for topological character, and can one find an example? Triclinic ${\mathrm{CaAs}}_3$, in the space group $P\overline{1}$ with only a center of inversion, has been found to display, without need for tuning, a nodal loop of accidental degeneracies with topological character, centered on one face of the Brillouin zone that is otherwise fully gapped. The small loop is very flat in energy, yet is cut four times by the Fermi energy, a condition that results in an intricate repeated touching of inversion related pairs of Fermi surfaces at Weyl points. Spin-orbit coupling lifts the fourfold degeneracy along the loop, leaving trivial Kramers pairs. With its single nodal loop that emerges without protection from any point group symmetry, ${\mathrm{CaAs}}_3$ represents the primal “hydrogen atom” of nodal loop systems.

Figure 1

Crystal structure of ${\mathrm{CaAs}}_3$, viewed in the $a\text{−}b$ plane. Arsenic atoms (yellow) form two-dimensional chains similar to black phosphorus. The center of inversion lies midway between neighboring Ca ions (shown in red). The solid lines outline the primitive cell.

Figure 2

Band structure of ${\mathrm{CaAs}}_3$ along a few special directions, from a $\mathrm{GGA}+\mathrm{mBJ}+\mathrm{SOC}$ calculation, and (right panel) the density of states. The region of interest lies near the $Y={\overrightarrow{b}}^{*}/2$ zone boundary (ISIM) point. Band inversion at $Y$ can be easily imagined by ignoring the mixing that causes anticrossing along the $X-Y$ direction. Even without SOC, a gap of $\sim 10\mathrm{meV}$ separates occupied and unoccupied states along the $Y\text{−}\mathrm{\Gamma }$ direction (see inset).

Figure 3

Left panel: Brillouin zone of ${\mathrm{CaAs}}_3$, showing the nodal loop centered at $Y$ on the top (and bottom) face of this view of the zone. Right panels: two perspective views of the nodal line enclosed within the Fermi surfaces, with electron and hole surfaces denoted by different colors. The notations $a^{*}$, $b^{*}$, $c^{*}$ of the reciprocal lattice vectors denote direction only.

Figure 4

Edge states (peaks in the spectral density) of ${\mathrm{CaAs}}_3$ calculated using the MLWF tight-binding representation truncated at the (001) surface. The panels compare spectra before (left) and after (right) inclusion of SOC. SOC hardly affects the projected bulk bands while altering the surface (bright color) bands strongly. The notation “$X(20)$,” for example, indicates that the end point is 20% of the distance toward $X$.

Figure 5

Nodal lines of accidental degeneracies for the model Hamiltonian; where a loop passes into a neighboring Brillouin zone. For $m=1.44$ on the left, a single loop is centered on the ISIM point ${\overrightarrow{b}}^{*}/2$. The $m=0$ case is shown on the right, with two pairs of inversion symmetry related lines threading from zone to zone.