Fermi arc reconstruction at junctions between Weyl semimetals

We analyze junctions between noninteracting fermionic topological phases. A closed-form condition for the existence of localized modes at the interface is derived using the transfer matrix approach. These analytical conditions as well as numerical exact diagonalization are used to study interfaces between Weyl semimetals. We observe a Fermi arc reconstruction at the interface, leading to closed curves of zero energy modes in the interface Brillouin zone. These are stable even in certain cases where the two Weyl semimetals differ only in their Fermi arc connectivities.

Figure 1

A schematic depiction of an interface between WSMs with different Fermi arc connectivities. The hopping between the two surfaces, denoted by a gray arrow, leads to hybridization of the localized modes on the two surfaces, i.e., a Fermi arc reconstruction.

Figure 2

The geometric picture of the Weyl nodes and Fermi arcs for the tight-binding model of Eq. (1) [with $\mathbf{\eta }$ defined by Eq. (58) for $\phi =4\pi /3]$. The Weyl nodes occur in the $k_x=0$ plane at the intersection of the curves defined by ${\eta }_y=0$ and ${\eta }_z=0$, depicted by solid red lines and dashed blue lines, respectively. The surface has localized modes for ${\mathbf{k}}_{\perp }=(k_y,k_z)$ in the shaded region, and the subset of these modes at zero energy, viz. the Fermi arcs, are denoted by the bold red lines.

Figure 3

The basic setup for describing the interface in the transfer matrix formalism. The left and right bulk layers are depicted by green and yellow, while the interface layers are depicted in gray. The matrices $J$'s and $M$'s are generically functions of the transverse momentum.

Figure 4

Schematic of the interfaces discussed in Sec. 3c. (Top) A hard interface with a tunable coupling $\kappa$ between the surfaces. (Bottom) A softer interface with an on-site potential described by ${\tilde{M}}_0$.

Figure 5

A schematic depiction of the system used for the ED. The system is inhomogeneous along $x$ with two interfaces between the two WSMs depicted in orange and green. The transverse direction corresponds to ${\mathbf{k}}_{\perp }\in {\mathbb{T}}^2$.

Figure 6

The analytically computed locii of zero modes on the interface BZ, superimposed on the ED result for the two-node WSM (Sec. 4a) for $\kappa =0,1/7,1/3,1,3,7$ (left to right). The background depicts the lowest conduction band on the surface BZ obtained from ED, with darker shades denoting lower energies. The curves in red and green depict the decaying and growing branches of the curve obtained from the matching condition, while the dashed gray line denotes the remaining solutions. Note that the curves for $\kappa$ and $1/\kappa$ are identical, but the growing and decaying labels have exchanged their roles.

Figure 7

The analytically computed locii of zero modes on the interface BZ, superimposed on the ED result for the four-node WSM (Sec. 4b) for a fixed ${\phi }_0=\frac{5\pi }{4}$, with (top panel) fixed $\kappa =1$ and varying $\mathrm{\Delta }\phi =0.26\pi ,0.35\pi ,0.45\pi ,0.50\pi ,0.60\pi ,0.70\pi$ from left to right, and (bottom panel) fixed $\mathrm{\Delta }\phi =\frac{\pi }{4}$ for $\kappa =0.25,0.35,0.42,0.47,0.60,0.95$ from left to right. The background depicts the lowest conduction band on the surface BZ obtained from ED, with darker shades denoting lower energies. The curves in red and green depict the decaying and growing branches of the curve obtained from the matching condition, while the dashed gray line denotes the remaining solutions. For the variation of $\mathrm{\Delta }\phi$ as well as $\kappa$, as one tunes these parameters beyond the range explored in the plots, the interface modes gap out completely.