Tunable line node semimetals

Weyl semimetals are examples of a new class of topological states of matter, which are gapless in the bulk with protected surface states. Their low-energy sector is characterized by massless chiral fermions, which are robust against translationally invariant perturbations. A variant of these systems has two nondegenerate bands touching along lines rather than points. A proposal to realize such a phase involves alternating layers of topological insulators and magnetic insulators, where the magnetization lies perpendicular to the symmetry axis of the heterostructure. The shape, size, and the dispersion in the vicinity of the nodal lines varies with the strength of the magnetization, offering a new knob to control the properties of the system. In this paper, we map out the evolution of the nodal lines and the dependence of the conductivity on magnetization and identify signatures of the low-energy sector in quantum oscillation measurements.

Figure 1

An example of the nodal curve for the closed region of parameters. The parameters chosen are $m=0.9,{\Delta }_S=0.6,\text{and}{\Delta }_D=0.4$ (with $d=v_F=1$).

Figure 2

The dispersion surfaces $\pm {\varepsilon }_{-}(k_z,k_y)$, showing the nodal curve where the top and bottom surfaces touch along ${\varepsilon }_{-}=0$. The parameter values are the same as for Fig. 1.

Figure 3

The variation of energy as a function of momentum perpendicular to the nodal line at four representative points is shown. Parameters remain as in Fig. 1.

Figure 4

The shape of the closed line node in Fig. 1 is plotted as the field parameter $m$ is changed, while keeping ${\Delta }_S$ and ${\Delta }_D$ fixed. The arrow shows the direction of increasing $m$. The closed curves have $m<{\Delta }_S+{\Delta }_D$, while the open curve violates this condition.

Figure 5

The volume enclosed by a surface of energy $\varepsilon$ in momentum space $N\left(\varepsilon \right)$ is plotted while varying the field parameter $m$. A perfect torus is obtained for a particular value of $m$. For smaller $m$, the torus gets squeezed, while for larger values the torus gets stretched (along the $k_z$ axis).

Figure 6

The slope of the linear (low-energy) density of states is plotted as a function of $m$. The points are obtained from numerical evaluation of the integral in Eq. (11). The black line is the analytical result from Ref. [20], while the dashed line shows the asymptotic value.

Figure 7

For large values of $m$, where the nodal line is open, the conductivity is roughly constant. Here, we plot the conductivity divided by the asymptotic value as a function of $m$. These normalized curves are isotropic but the asymptotic values themselves are different for parallel and perpendicular directions with respect to the growth axis of the multilayer.

Figure 8

We plot $N\left(\varepsilon \right)$ for two different values of $m$. The top figure is for the case where we have a closed nodal line where the quadratic dependence is evident reflecting a linear density of states. For large $m$, for open line nodes, the quadratic dependence crosses over to a linear behavior very quickly as one departs from the node. This means that a linear DoS is trustworthy only for very low energies when $m\gg {\Delta }_S+{\Delta }_D$ (see Fig. 9).

Figure 9

Constant energy surfaces for the same parameters as in Fig. 6, but with $m=1.15\mathrm{eV}>{\Delta }_S+{\Delta }_D$, and a larger $m=1.28\mathrm{eV}\gg {\Delta }_S+{\Delta }_D$. As $m$ increases the topology changes to disconnected surfaces.

Figure 10

Frequency for quantum oscillations for magnetic fields applied along the $x$ direction is plotted as a function of energy. The top curve shows the result corresponding to the maximal area, while the bottom corresponds to the minimal area. Dashed curves are obtained in the approximation of treating the surfaces as a circular torus. The parameters are the same as above with $m=0.5\mathrm{eV}$, taking $v_F=e4\mathrm{m}/\mathrm{s}$ and $d=100\mathrm{n}/\mathrm{m}$.

Figure 11

Fequency for magnetic fields applied along the $z$ direction as a function of the energy. The solid curve shows the result corresponding to the only extremal orbit in this case. Note the orders of magnitude difference compared to Fig. 10 arising from the shape of the torus.

Figure 12

As $m$ is varied, the density at which the jump in frequency occurs is modified. Here we plot the variation and note that it occurs for small densities when $m$ is not too much larger than the difference $|{\Delta }_S-{\Delta }_D|$.