Chiral topological insulating phases from three-dimensional nodal loop semimetals

We identify a topological $\mathbb{Z}$ index for three-dimensional chiral insulators with $P*T$ symmetry where two Hamiltonian terms define a nodal loop. Such systems may belong in the AIII or DIII symmetry class. The $\mathbb{Z}$ invariant is a winding number assigned to the nodal loop and has a correspondence to the geometric relation between the nodal loop and the zeros of the gap terms. Dirac cone edge states under open boundary conditions are in correspondence with the winding numbers assigned to the nodal loops. We verify our method with the low-energy effective Hamiltonian of a three-dimensional material of topological insulators in the ${\mathrm{Bi}}_2{\mathrm{Te}}_3$ family.

Figure 1

A sketch of the winding number of a NL. The blue circle indicates the NL given by $H_0=0$, the red lines indicate the 1D solution of $|{\mathbf{h}}_y|=0$ for models of the chiral class, and each line gives a singularity in the integrand of (3) in the NL plane. The arrows around each singularity show the direction of ${\mathbf{h}}_y$ near this point, and the corresponding winding numbers of these points are labeled in the sketch. The winding number of the NL, ${\nu }_{\text{NL}}$, is given by the summation of winding numbers of every singularity within the NL, as the integral path can be smoothly transformed into circles around each singularity.

Figure 2

The NLs and the 1D lines given by $|{\mathbf{h}}_y|=0$, represented by blue and red lines, respectively. For the latter, we only show the lines with $k_y=0$, as the ones for $k_y=\pi$ lie outside the NLs. (a) $\mu =0.2, {\mu }_s=0.8$, and $t_z=0$; (b) $\mu =1, {\mu }_s=0.8$, and $t_z=0$; (c) $\mu =1.8, {\mu }_s=0.8$, and $t_z=0$; (d) $\mu =1.6, {\mu }_s=0.5$, and $t_z=0.2$; (e) $\mu =1.6, {\mu }_s=0.5$, and $t_z=0.4$; (f) $\mu =1.6, {\mu }_s=0.5$, and $t_z=0.6$.

Figure 3

Phase diagram with $t_z=0$ (left) and $\mu =1$ (right). The number of Dirac cones in edge states is equal to the total winding number of the model, ${\nu }_{\text{sum}}={\nu }_{\text{NL}}^{+}+{\nu }_{\text{NL}}^{-}$. In the left panel, the yellow region with ${\nu }_{\text{NL}}^{+}={\nu }_{\text{NL}}^{-}=0$ includes three different situations: (i) Two lines are enclosed by the NLs when ${\mu }_s<1$ and $\mu <1$; (ii) all the lines are out of the NLs when ${\mu }_s<1$ and $\mu >1$; and (iii) no line exists when ${\mu }_s>1$.

Figure 4

The spectra of the two doublet bands nearest to $E=0$, (a)–(c) for open boundary conditions along $x$, and (d)–(f) for period boundary conditions. The three columns are for three topologically different phases, while the left two are topologically nontrivial. The parameters are $\mu =1$ and (a), (d) ${\mu }_s=t_z=0$; (b), (e) ${\mu }_s=t_z=1$; and (c), (f) ${\mu }_s=2, t_z=0$.