Interfacial Fermi Loops from Interfacial Symmetries

We propose a concept of interfacial symmetries such as interfacial particle-hole symmetry and interfacial time-reversal symmetry, which appear in interfaces between two regions related to each other by particle-hole or time-reversal transformations. These symmetries result in novel dispersion of interface states. In particular, for the interfacial particle-hole symmetry, the gap closes along a loop (“Fermi loop”) at the interface. We numerically demonstrate this for the Fu-Kane-Mele tight-binding model. We show that the Fermi loop originates from a sign change of a Pfaffian of a product between the Hamiltonian and a constant matrix.

Figure 1

(a) Schematic illustration of the interface between two regions ($\alpha$ and $\beta$) of the FKM model. (b-1),(b-2) Interface energy bands near the Fermi energy ($E=0$) in the interface BZ for ${\lambda }_{\alpha }=0.15t_0$, and ${\lambda }_{\beta }=-0.1t_0$. (b-1) For $t_1=t_0$ and $t_2=1.4t_0$, the system does not have the rotational symmetry. There are two Dirac cones. (b-2) For $t_2=t_0$, $t_1=1.4t_0$, there are six Dirac cones due to ${\mathrm{C}}_3$ symmetry.

Figure 2

(a-1),(b-1) Dispersion and the Fermi surface for the interface states with IPHS. Thickness of each region $\alpha$, $\beta$ is 24 unit cells along the [111] axis. The parameters are (a) $(t_1,t_2)=(1.4,1)t_0$, (b) $(t_1,t_2)=(1,0.6)t_0$, and ${\lambda }_{\alpha }=-{\lambda }_{\beta }=0.125t_0$ for (a) and (b). In the BZ in each panel, the Fermi surfaces at $E=0$ are shown as blue curves. (a-2),(b-2) Sign of the Pfaffian $\mathrm{sgn}[\mathrm{Pf}(\tilde{U}H)]$ in the interface BZ. The shaded regions and white regions represent the negative and positive signs, respectively. Schematic illustrations of the Fermi loops of a closed orbit (a-3) and open orbits (b-3) are shown on the torus of the interface BZ.