Minimal models for topological Weyl semimetals

Topological Weyl semimetals (TWS) can be classified as type I TWS, in which the density of states vanishes at the Weyl nodes, and type II TWS, in which an electron pocket and a hole pocket meet at a singular point of momentum space, allowing for distinct topological properties. We consider various minimal lattice models for type II TWS. The simplest time-reversal-breaking band structure, with a pair of Weyl nodes sharing a single electron pocket and a single hole pocket (hydrogen model), exhibits relics of surface Fermi arc states only away from the Fermi energy, with no topological protection. Topologically protected Fermi arcs can be restored by an additional term (hydrogen model) that produces a bulk structure where the electron and hole pockets of each Weyl point are disjoint. In time-reversal-symmetric but inversion-breaking models, we identify nontopological surface track states that arise out of the topological Fermi arc states at the transition from type I to type II and persist in the type II TWS. The distinctions among these minimal models can aid in distinguishing between generic and model-dependent behavior in studies of superconductivity, magnetism, and quantum oscillations of type II Weyl semimetals.

Figure 1

Surface and bulk dispersions in the minimal time-reversal-breaking models. Panels (a)–(c): Bulk Fermi pockets (black) and surface Fermi arcs (red and blue), shown at $E=-0.2t$, for three models: (a) the simplest type I case [Eq. (8) with $\gamma =0$], (b) the simple two-pocket type II case [Eq. (8) with $\gamma =3t$], and (c) the two-node TRB case with isolated pockets surrounding each node ($\gamma =1.5t$). The thin green and purple lines correspond to the cuts shown in panels (d)–(f) and (g)–(i) respectively. Panels (d)–(f): Bulk surface dispersions with $k_x$ at $k_z=0$. Surface states at the top and the bottom surfaces are degenerate and shown in purple. Panels (g)–(i): Bulk surface dispersions with $k_x$ at $k_z=0.2$. Surface states at the top and the bottom surfaces are nondegenerate and shown in red and blue respectively.

Figure 2

Bulk band structure for the hydrogen atom of type I and type II Weyl semimetal. Panels (a)–(c): The bulk band structure for the Hamiltonian in Eq. (8). Electron pockets shown in red and hole pockets shown in blue merge at the Weyl nodes shown in green. Here we have chosen parameters $k_y=0$ with parameters $k_0=\pi /2, t_x=t, m=2t$ for (a) type I Weyl semimetal with $\gamma =0$, (b) the critical point between type I and type II Weyl semimetal with $\gamma =2t$, and (c) type II Weyl semimetal with $\gamma =3t$. The cones comprising the Weyl nodes develop a characteristic tilt of the type II TWS as $\gamma$ is increased. Panels (d)–(f): Cuts through the Weyl nodes at $k_y=k_z=0$ for the same parameters as panels (a)–(c). Panels (g)–(i): Constant energy cuts through the nodal energy ($E=0$) for the same parameters as panels (a)–(c). We see that for a type I TWS, there are no states at the Fermi energy. At the critical point between a type I and type II TWS, we see lines of bulk states appearing between the nodes. These lines open into bulk hole and electron pockets (in the repeated zone scheme) when the system becomes a type II TWS.

Figure 3

Fermi surface and arc configuration for the hydrogen atom of type I and type II TWS. Panels (a)–(c): Bulk Fermi surfaces and surface Fermi arcs for a type I TWS with the same bulk parameters as in Figs. 2, 2, and 2 calculated in a slab geometry with $L=50$ layers in the $y$ direction. The slab calculations are done at the following constant energy: (a) $E=-0.2t$, (b) $E=0$, and (c) $E=0.2t$. We color the states which are exponentially localized to the $y=1$ ($y=L$) surface red (blue) and note that such surface states form topological arcs connecting the two Weyl nodes (shown as green dots and marked with pink arrows). We note that at $E=0$ the two Fermi arcs are degenerate along $k_z=0$ and we color them purple to signify this. Panels (d)–(f): Bulk Fermi surfaces and surface Fermi arcs for a type II TWS with the same bulk parameters as in Figs. 2, 2, and 2 calculated in a slab geometry with $L=50$ layers in the $y$ direction. The slab calculations are done at the same constant energies as above: (d) $E=-0.2t$, (e) $E=0$, and (f) $E=0.2t$.

Figure 4

Bulk band structure for type I and type II TRB model with separate pockets (the helium atom). Panels (a)–(c): The bulk band structure for the Hamiltonian in Eq. (9). Electron pockets shown in red and hole pockets shown in blue merge at the Weyl nodes shown in green. Here we have chosen parameters $k_y=0$ with the parameters $k_0=\pi /2, t_x=t, m=2t$, and ${\gamma }_x=t/2$ for (a) type I TWS with $\gamma =0$, (b) type II TWS with $\gamma =t$, and (c) type II TWS with $\gamma =1.5t$. The cones comprising the Weyl nodes again develop a characteristic tilt of the type II TWS as $\gamma$ is increased. Panels (d)–(f): Cuts through the Weyl nodes at $k_y=k_z=0$ for the same parameters as panels (a)–(c). Panels (g)–(i): Constant energy cuts through the nodal energy ($E=0$) for the same parameters as panels (a)–(c). Note that for a type I TWS, there are no states at the Fermi energy while in the type II regime, there are two sets of electron and hole pockets on either side of the Weyl nodes. We see that unlike the hydrogen-atom model, the helium-atom model has disjoint pairs of electron and hole pockets and a pair of each meet at the two Weyl nodes.

Figure 5

Fermi surface and Fermi arc configuration for type I and type II time-reversal-breaking model with separate pockets (the helium atom). Panels (a)–(c): Bulk Fermi surfaces and surface Fermi arcs for a type II Weyl semimetal given by Eq. (9) with the same bulk parameters as in Figs. 4, 4, and 4 calculated in a slab geometry with $L=50$ layers in the $y$ direction. The slab calculations are done at the constant energies: (a) $E=-0.2t$, (b) $E=0$, and (c) $E=0.2t$. As in Fig. 3, we color the states that are exponentially localized to the $y=1$ ($y=L$) surface red (blue) and note that such surface states form topological arcs connecting the two Weyl nodes (shown as green dots). We note unlike in Fig. 3, each node is isolated in its own hole (a) or electron (c) pocket when the chemical potential is away from $E=0$. These pockets are connected by arcs confined to the surface in the $y$ direction. However, in this type II TWS the Fermi pockets enclosing a Weyl node can be quite extended, unlike a type I TWS, and the arcs can terminate on a pocket quite far away from the projection of the nodes. Panels (d)–(f): Bulk Fermi surfaces and surface Fermi arcs for a type II TWS with the same bulk parameters as in Figs. 4, 4, and 4 calculated in a slab geometry with $L=50$ layers in the $y$ direction. The slab calculations are done at the same constant energies as above: (d) $E=-0.2t$, (e) $E=0$, and (f) $E=0.2t$. We see that as the tilt grows, so do the pockets enclosing the nodes. We note that a trivial electron pocket appears around the $(k_x,k_z)=(\pi ,\pi )$ point. This pocket encloses no Weyl nodes and so is not connected via Fermi arcs to any other pockets.

Figure 6

Bulk band structure for type I and type II inversion breaking TWS. Panels (a)–(c): The bulk band structure for the Hamiltonian in Eq. (11). Electron pockets shown in red and hole pockets shown in blue merge at the Weyl nodes shown in green. Here we have chosen parameters $k_y=0$ with the parameters $k_0=\pi /2, t_x=t/2, m=2t$ for (a) type I TWS with $\gamma =0$, (b) the critical point between a type I and a type II TWS with $\gamma =2t$, and (c) type II TWS with $\gamma =2.4t$. The cones comprising the Weyl nodes develop a characteristic tilt of the type II Weyl node as $\gamma$ is increased. Panels (d)–(f): Cuts through the Weyl nodes at $k_y=0$ and $k_z=-\pi /2$ for the same parameters as panels (a)–(c). These cuts are shown as the green lines in panels (g)–(i). Panels (g)–(i): Constant energy cuts through the nodal energy ($E=0$) for the same parameters as panels (a)–(c). We see that for a type I Weyl semimetal, there are no states at the Fermi energy. At the critical point between a type I and type II TWS, the density of states still vanishes. In the type II regime, electron and hole pockets form near the Weyl nodes. These pockets enclose the projections of the Weyl nodes when the chemical potential is shifted away from $E=0$. Trivial pockets also appear at $\mathbf{k}=(0,0,0)$ and $\mathbf{k}=(0,0,\pi )$.

Figure 7

Fermi surface and Fermi arc configuration for type I and type II inversion-breaking Weyl semimetal. Panels (a) and (b): The Fermi surface and Fermi arc configuration for the Hamiltonian given in Eq. (11) in the type I limit ($\gamma =0$) calculated in a slab geometry with $L=50$ layers and with bulk parameters the same as in Figs. 6, 6, and 6. We show this calculation at constant energies: $E=-0.25t$ (a) and $E=0.25t$ (b). Here we see that Weyl nodes located at $(k_x,k_z)=(\pm \pi /2,\pm \pi /2)$ are connected by surface states (red and blue lines) to one of opposite chirality across the Brillouin zone in the $k_x$ direction. Panels (c) and (d): The Fermi surface and Fermi arc configuration for the Hamiltonian given in Eq. (11) in the type II limit ($\gamma =2.4t$) calculated in a slab geometry with $L=50$ layers and with bulk parameters the same as in Figs. 6, 6, and 6. We show these for the same constant energies as above: $E=-0.25t$ (c) and $E=0.25t$ (d). The locations of the Weyl nodes are marked with pink arrows. We term the exponentially localized surface states that form closed-loop track states. Fermi arcs are shown as bold lines and connect Weyl nodes in the $k_z$ direction.

Figure 8

Evolution of Fermi surface and Fermi arc configuration for inversion-breaking Weyl semimetal as a function of $\gamma$. Panels (a)–(d): The evolution of the Fermi surface and Fermi arc configuration in a slab geometry for Eq. (11). Bulk states are down in black; surface states are shown in red and blue. We have chosen the parameters $k_0=\pi /2, t_x=t/2, m=2t$. The calculations are done at constant energy $E=-0.25t$ for $\gamma =0$ (a), $\gamma =0.8t$ (b), $\gamma =1.4t$ (c), and $\gamma =2$ (d) shown in an extended Brillouin zone where both $k_x$ and $k_z$ range from $-1.5\pi$ to $1.5\pi$. We see that at the critical point between type I and type II (d), the Fermi arcs that previously connected Fermi pockets in the $k_x$ direction now connect Fermi pockets in the $k_z$ direction and track states have formed on the bottom surface (blue) around the $(k_x,k_z)=(\pi ,\pi )$ point.

Figure 9

Sketch of the three types of surface states in a topological Weyl semimetal. (a) Two type I Weyl nodes of opposite chirality connected by a Fermi arc on the top (red) and bottom (blue) surfaces. In an arbitrary type II TWS at an energy away from the Weyl energy, these arcs would connect Fermi pockets instead of nodes. (b) A single Fermi pocket enclosing two nodes of opposite chirality. Since no Gaussian surface can be constructed in a region that is both gapped and encloses only one node, the only possible surface states are trivial ones, shown in red and blue at the boundary of the pocket that hybridize with bulk states due to lack of topological protection. (c) Pairs of Weyl nodes, two of each chirality with each node surrounded by a Fermi pocket. The pockets are connected by Fermi arcs (thinner red and blue contours) as well as track states (thicker blue lines) on the bottom surface. Note that states on opposite sides of a given loop of track states will disperse in opposite directions and so a Gaussian surface enclosing a given Fermi pocket will still have one net surface state of each chirality.