Chern Semimetal and the Quantized Anomalous Hall Effect in ${\mathrm{HgCr}}_2{\mathrm{Se}}_4$

In 3D momentum space, a topological phase boundary separating the Chern insulating layers from normal insulating layers may exist, where the gap must be closed, resulting in a “Chern semimetal” state with topologically unavoidable band crossings at the Fermi level. This state is a condensed-matter realization of Weyl fermions in $(3+1)\mathrm{D}$, and should exhibit remarkable features, such as magnetic monopoles and Fermi arcs. Here we predict, based on first principles calculations, that such a novel quantum state can be realized in a known ferromagnetic compound ${\mathrm{HgCr}}_2{\mathrm{Se}}_4$, with a single pair of Weyl fermions separated in momentum space. The quantum Hall effect without an external magnetic field can be achieved in its quantum-well structure.

Figure 1

Electronic structures of ${\mathrm{HgCr}}_2{\mathrm{Se}}_4$. (a) The total and partial density of states (DOS); (b) The band structures without SOC (showing the up and down spin parts separately); (c) The schematic understanding for the band inversion, where the $|S⟩$ state is lower than the $|P⟩$ states in the down spin channel; (d) The band structure after including SOC [with majority spin aligning to the (001) direction]. The low energy states at $\Gamma$ are indicated as explained in the main text.

Figure 2

Weyl nodes and gauge flux in ${\mathrm{HgCr}}_2{\mathrm{Se}}_4$ (a) The band crossing points in the $\overrightarrow{k}$ space; (b) The Chern number as function of $k_z$; (c) The schematic plot of the band dispersion around the Weyl nodes in the $k_z=\pm k_z^c$ plane, the inset shows the chiral spin texture; (d) The gauge flux evaluated as Berry curvature in the $(k_x,k_z)$ plane.

Figure 3

Edge states and Fermi arcs of ${\mathrm{HgCr}}_2{\mathrm{Se}}_4$ [30]. (a) The edge states for the plane with $k_z=0.06\pi$. A ribbon with two edges is used, and there are two edge states for each edge (because $C=2$). The states located at different edges are indicated by different line types. (b) The calculated Fermi arcs for the $(k_y,k_z)$ side surface.

Figure 4

Quantized anomalous Hall effect in ${\mathrm{HgCr}}_2{\mathrm{Se}}_4$ thin film [30]. (a) The subband energy levels at $\Gamma$ point as a function of film thickness. (b) The Hall conductance as a function of film thickness.