Topological Triply Degenerate Points Induced by Spin-Tensor-Momentum Couplings

The recent discovery of triply degenerate points (TDPs) in topological materials has opened a new perspective toward the realization of novel quasiparticles without counterparts in quantum field theory. The emergence of such protected nodes is often attributed to spin-vector-momentum couplings. We show that the interplay between spin-tensor- and spin-vector-momentum couplings can induce three types of TDPs, classified by different monopole charges ($\mathcal{C}=\pm 2$, $\pm 1$, 0). A Zeeman field can lift them into Weyl points with distinct numbers and charges. Different TDPs of the same type are connected by intriguing Fermi arcs at surfaces, and transitions between different types are accompanied by level crossings along high-symmetry lines. We further propose an experimental scheme to realize such TDPs in cold-atom optical lattices. Our results provide a framework for studying spin-tensor-momentum coupling-induced TDPs and other exotic quasiparticles.

Figure 1

(a)–(c) Band structures of three types of TDPs in the $k_y=0$ plane for model (2). (a) Type I with $\alpha =1$ and $\beta =0$. (b) Type II with $\alpha =1$, $\beta =2$, and $N_{ij}$ is chosen as $F_z^2$. (c) Type III with $\alpha =1$, $\beta =3$, and $N_{ij}=N_{xz}$. (d)–(f) Splittings of three types of TDPs due to a Zeeman perturbation $ϵF_z$ with $ϵ=0.05$. (d) Type I splits into two linear Weyl points with $\mathcal{C}=1$ and one double-Weyl point [57] with $\mathcal{C}=2$; note that $\beta =0.5$ is used instead of 0. (e) Type II splits into two linear Weyl points with $\mathcal{C}=\pm 1$, and one double-Weyl point [57] with $\mathcal{C}=2$. (f) Type III splits into four linear Weyl points with $\mathcal{C}=\pm 1$.

Figure 2

Phase transitions between type-I and type-II TDPs by tuning $\alpha$ while fixing $\beta =1$ in Eq. (2). (a) Chern numbers as functions of $\alpha$ for the lower (dashed blue), middle (dotted green), and upper (solid red) bands. (b)–(e) Band structures along the $k_x=k_y=0$ line with $\alpha =2$, 0.5, $-0.5$, and $-2$, respectively. We have labeled the Chern contributions of each branch: $+1$ (the solid magenta lines), $-1$ (the dashed cyan lines), and 0 (the dotted black lines).

Figure 3

Bulk band structures with TDPs and (110) surface spectral densities with Fermi arcs of model (3). (a),(c) Type I with $\mathcal{C}=\pm 2$ and two surface arcs. (b),(d) Type II with $\mathcal{C}=\pm 1$ and only one surface arc. In both cases, two TDPs appear at $(0,0,\pm \pi /3)$, and each projected node is marked by its monopole charge. In our calculation, $\gamma =-0.5$, $\omega =0.25$, and $t_0=0.5$ are used; $\beta =0.5$ is used in (a) and (c), and $\beta =1.5$ in (b) and (d).

Figure 4

Phase transitions between type-I and type-III TDPs by tuning $\beta$ while fixing $\alpha =1$ in Eq. (2). (a) Chern numbers as functions of $\beta$ for the lower (dashed blue), middle (dotted green), and upper (solid red) bands. (b) Band crossing at the two transition lines with $\beta =2$.