Topological nodal semimetals

We present a study of “nodal-semimetal” phases in which nondegenerate conduction and valence bands touch at points (the “Weyl semimetal”) or lines (the “line-node semimetal”) in three-dimensional momentum space. We discuss a general approach to such states by perturbation of the critical point between a normal insulator (NI) and a topological insulator (TI), breaking either time-reversal (TR) or inversion symmetry. We give an explicit model realization of both types of states in a NI-TI superlattice structure with broken TR symmetry. Both the Weyl and the line-node semimetals are characterized by topologically protected surface states, although in the line-node case, some additional symmetries must be imposed to retain this topological protection. The edge states have the form of “Fermi arcs” in the case of the Weyl semimetal: these are chiral gapless edge states, which exist in a finite region in momentum space, determined by the momentum-space separation of the bulk Weyl nodes. The chiral character of the edge states leads to a finite Hall conductivity. In contrast, the edge states of the line-node semimetal are “flat bands”: these states are approximately dispersionless in a subset of the two-dimensional edge Brillouin zone, given by the projection of the line node onto the plane of the edge. We discuss unusual transport properties of the nodal semimetals and, in particular, point out quantum critical-like scaling of the dc and optical conductivities of the Weyl semimetal and similarities to the conductivity of graphene in the line-node case.

Figure 1

The $k_z=5\pi /6d$ section of the eigenstate spectrum for a sample of finite size in the $x$ direction with ${\Delta }_D/{\Delta }_S=0.8$, and $b/{\Delta }_S=1$. $k_y$ is in units of ${\Delta }_S/v_F$. The intensity of gray is a function of the degree of surface localization of a given eigenstate, measured by an inverse participation ratio of its wave function. The surface-state dispersion is black, while bulk states are lighter gray.

Figure 2

The $k_z=5\pi /6d$ section of the eigenstate spectrum for a sample of finite size in the $x$ direction with ${\Delta }_D/{\Delta }_S=0.8$, and $b/{\Delta }_S=1$ in the presence of a particle-hole asymmetry in the TI surface-state spectrum of the form $(k_x^2+k_y^2)/2m^{*}$, with $1/m^{*}=0.3v_F^2/{\Delta }_S$. $k_y$ is in units of ${\Delta }_S/v_F$. The surface state has acquired a dispersion due to the particle-hole asymmetry.

Figure 3

The “Fermi line” transforms into a finite-volume Fermi surface with equal-volume electron (top and bottom) and hole (left and right) pockets due to the particle-hole asymmetry of the TI surface states. Note that the line contact of the conduction and valence bands survives, although it is not a constant energy curve. This line contact threads through the middle of the chain of small Fermi surfaces, and as a consequence the topological surface state remains (see Fig. 2). The parameters here are taken to be the same as in Fig. 2.