Triple degenerate point in three dimensions: Theory and realization

Dirac and Weyl points in condensed matters could resemble the behaviors of their counterparts, Dirac and Weyl fermions, in high-energy physics. However, some fermions have no analog in fundamental fermions. They are expected to render novel physical properties. In this work we study an important fermion in condensed matters which is threefold degenerate. Such a fermion has no counterpart in high-energy physics and is enforced by crystalline symmetry. We perform a systematic search over all 230 space groups with time-reversal symmetry but without spin-orbital coupling (SOC). We find that a triple degenerate point (TDP) intrinsically appears in a three-dimensional irreducible representation. Based on the order of their energy dispersion, TDPs can be classified into two types, namely linear and quadratic TDPs. Furthermore, the linear TDP can be classified into two categories depending on the definition of topological charge. To distinguish topologies of linear and quadratic TDPs, we study their Landau spectrum under a magnetic field. Notably, these TDPs present dramatically different properties. There exists two chiral Landau levels (LLs) for linear TDPs with nonzero topological charge, whereas no chiral LL appears for the quadratic TDPs. Given the novel physical properties of TDPs, we provide concrete materials to support our discoveries. Consequently, our work provides a platform to study the triple degenerate points in condensed matters.

Figure 1

Schematic figures for TDPs: (a) TDP appears on a high symmetry path. TDPs lie at the high symmetry point, (b) TDP carries a nonzero topological charge, (c) TDP with a double degenerate line, and (d) a quadratic TDP.

Figure 2

(a)–(c) Energy spectrum along $k_z$ for Weyl point (a), a linear TDP (b), and a quadratic TDP (c). The light blue lines refer to the analytical results and circle dots stand for the numerical results. The pink lines represent the zeroth Landau level. In the calculation we take $B=5$ T. (d) The Landau spectrum calculated from the tight-binding model of SG 198, the red arrows point out the CLLs around $\mathrm{\Gamma }$ point.

Figure 3

(a) Crystal structure for GeRh (SG 198). (b) The electronic band structures for GeRh without SOC. The red dot indicates the TDP at $\mathrm{\Gamma }$ point. (c) The enlarged view around the TDP. (d) Calculated constant energy slice (at $E=-0.19$ eV) for the (001) surface indicates two Fermi arcs emitted from the projection of the TDP (marked by a cyan dot).

Figure 4

(a) Crystal structure for ${\mathrm{Mg}}_3{\mathrm{Ru}}_2$ (SG 213). (b) The electronic band structures for ${\mathrm{Mg}}_3{\mathrm{Ru}}_2$ without SOC. (c) The enlarged view of TDP at the $\mathrm{\Gamma }$ point. (d) The constant energy slice (at $E=0.28$ eV energy) for the (001) surface. White arrows in the panel indicate the Fermi arcs, and cyan dots denote the TDP points.

Figure 5

(a) Crystal structure for ${\mathrm{CoAs}}_3$ (SG 204). (b) The electronic band structures for ${\mathrm{CoAs}}_3$ without SOC. (c) The enlarged view of the corresponding triple degenerate point at the $\mathrm{\Gamma }$ point. (d) Surface spectrum on (111) for ${\mathrm{CoAs}}_3$. The cyan dot indicates TDP, and the white arrow points to the corresponding surface state.

Figure 6

(a) Crystal structure for ${\mathrm{DyRh}}_3\mathrm{C}$ (SG 221). (b) The electronic band structure for ${\mathrm{DyRh}}_3\mathrm{C}$ without SOC. (c) The enlarged view for TDP at the $\mathrm{\Gamma }$ point. (d) Surface spectrum on (001) for ${\mathrm{DyRh}}_3\mathrm{C}$. The cyan dot indicates TDP, and the white arrow points to the corresponding surface state.

Figure 7

(a) Crystal structure for ${\mathrm{Na}}_3\left(\mathrm{GePt}\right){}_4$ (SG 217). (b) The electronic band structures for ${\mathrm{Na}}_3\left(\mathrm{GePt}\right){}_4$ without SOC. (c) The enlarged view of a corresponding triple degenerate point at the $\mathrm{\Gamma }$ point. (d) Surface spectrum on a (111) surface shows the existence of Fermi arcs (pointed by a white arrow) emerging from the projection (marked by a cyan dot) of TDP at $\mathrm{\Gamma }$ point.

Figure 8

The electronic band structures in the presence of SOC: (a) GeRh, (b) ${\mathrm{Mg}}_3{\mathrm{Ru}}_2$, (c) ${\mathrm{CoAs}}_3$, (d) ${\mathrm{DyRh}}_3\mathrm{C}$, and (e) ${\mathrm{Na}}_3\left(\mathrm{GePt}\right){}_4$. (f) Indicates that there indeed exists a Weyl nodal line on path $\mathrm{\Gamma }-R$ for ${\mathrm{Na}}_3\left(\mathrm{GePt}\right){}_4$.

Figure 9

Orbital characteristic weights of (a) GeRh, (b) ${\mathrm{Mg}}_3{\mathrm{Ru}}_2$, (c) ${\mathrm{CoAs}}_3$, (d) ${\mathrm{DyRh}}_3\mathrm{C}$, and (e) ${\mathrm{Na}}_3{\left(\mathrm{GePt}\right)}_4$, respectively. The projected density of states (PDOS) is shown on the right sides of the panels. The size of the circle indicates the characteristic weight of the orbital.

Figure 10

The calculated band structure for SG 198 based on the tight-binding model, the red dot points out the TDP at $\mathrm{\Gamma }$ point. The parameters are taken as $M_0=1.2,c_0=0.2,c_1=1.2,c_2=0.6$.

Figure 11

Energy contours for quadratic TDPs at $\mathrm{\Gamma }$ point belonging to SGs (a) 204, (b) 217, and (c) 221, at $E=0.05$ eV

Figure 12

Surface spectrum with SOC for (a) GeRh and (b) ${\mathrm{CoAs}}_3$, respectively. White arrows in the panel indicate the Fermi arcs, and cyan dots denote the DP points.

Figure 13

The 3D band structure for the Weyl nodal line along $\mathrm{\Gamma }-P$ in ${\mathrm{Na}}_3{\left(\mathrm{GePt}\right)}_4$.