Weyl semimetal with broken time reversal and inversion symmetries

Weyl semimetal is a new topological state of matter, characterized by the presence of nondegenerate band-touching nodes, separated in momentum space, in its band structure. Here we discuss a particular realization of a Weyl semimetal: a superlattice heterostructure, made of alternating layers of topological insulator and normal insulator material, introduced by one of us before. The Weyl node splitting is achieved most easily in this system by breaking time reversal symmetry, for example by magnetic doping. If, however, spatial inversion (I) symmetry remains, the Weyl nodes will occur at the same energy, making it possible to align the Fermi energy simultaneously with both nodes. The goal of this work is to explore the consequences of breaking the I symmetry in this system. We demonstrate that, while this generally moves the Weyl nodes to different energies, thus eliminating nodal semimetal and producing a state with electron and hole Fermi surfaces, the topological properties of the Weyl semimetal state, i.e., the chiral edge states and the corresponding Hall conductivity, survive for moderate I symmetry breaking. Moreover, we demonstrate that a new topological phenomenon arises in this case, if an external magnetic field along the growth direction of the heterostructure is applied. Namely, this leads to an equilibrium dissipationless current, flowing along the direction of the field, whose magnitude is proportional to the energy difference between the Weyl nodes and to the magnetic field, with a universal coefficient, given by a combination of fundamental constants.

Figure 1

A plot of the band dispersion along the $k_x=k_y=0$ line for a moderate magnitude of the inversion symmetry breaking, corresponding to $\lambda /{\Delta }_S=0.3$. The Weyl band-touching nodes are visibly shifted in opposite directions in energy, but are otherwise intact. At charge neutrality the Fermi energy is pinned at ${\epsilon }_F=0$, and there are equal-volume electron and hole Fermi surfaces.

Figure 2

Band dispersion along the $k_x=k_y=0$ line for $\lambda ={\lambda }_{c1}=\sqrt{m^2-{({\Delta }_S-{\Delta }_D)}^2}$. The electron and hole Fermi surfaces touch and the Weyl nodes disappear.

Figure 3

Band dispersion along the $k_x=k_y=0$ line for $\lambda ={\lambda }_{c2}=\sqrt{{(m+{\Delta }_D)}^2-{\Delta }_S^2}$. The Fermi surfaces disappear entirely and the semimetallic state gives way to a fully gapped insulator.

Figure 4

Phase diagram of the multilayer structure in the $m-\lambda$ plane for ${\Delta }_S>{\Delta }_D$. There are five distinct phases: (1) Insulator (INS), outside of the ${\lambda }_{c2}$ line. (2) Nodal semimetal (NSM), which exists along the interval $\lambda =0$, $m_{c1}<m<m_{c2}$. (3) Semimetal with electron and hole pockets, but with Weyl nodes preserved (FSSM 1). This is bounded by the ${\lambda }_{c1}=\pm \sqrt{m^2-m_{c1}^2}$ and ${\lambda }_{c3}=\pm \sqrt{{(m-{\Delta }_D)}^2-{\Delta }_S^2}$ lines. (4) Semimetal without Weyl nodes (FSSM 2). This is bounded by the ${\lambda }_{c2}=\pm \sqrt{{(m+{\Delta }_D)}^2-{\Delta }_S^2}$ and the ${\lambda }_{c1}$ lines. (5) QAH insulator phase, enclosed by the ${\lambda }_{c3}$ line. The anomalous Hall conductivity is nonzero in all semimetallic phases and the QAH phase.

Figure 5

Same as in Fig. 4, but for ${\Delta }_D>{\Delta }_S$. There is a additional transition in this case, which occurs as $\lambda$ is increased along the $m=0$ line. The $\left|\lambda \right|<|{\lambda }_{c2}|$ phase is a TI, while the $\left|\lambda \right|>|{\lambda }_{c2}|$ phase is an ordinary insulator.