Eightfold Degenerate Fermions in Two Dimensions

Multifold degenerate fermions have attracted a lot of research interest in condensed matter physics and materials science, but always lack in two dimensions. In this Letter, from symmetry analysis and lattice model construction, we demonstrate that eightfold degenerate fermions can be realized in two-dimensional systems. In nonmagnetic materials with negligible spin-orbit coupling, the gray magnetic space groups together with SU(2) spin rotation symmetry can protect the two-dimensional eightfold degenerate fermions on a certain high-symmetry axis in the Brillouin zone, no matter whether the system is centrosymmetric or noncentrosymmetric. In antiferromagnetic materials, the eightfold degenerate fermions can also be protected by certain “spin space groups.” Furthermore, by first-principles electronic structure calculations, we predict that the paramagnetic phase of the monolayer ${\mathrm{LaB}}_8$ on a suitable substrate is a two-dimensional eightfold degenerate as well as Dirac node-line semimetal. Especially, the eightfold degenerate points are close to the Fermi level, which makes monolayer ${\mathrm{LaB}}_8$ a good platform to study the exotic physical properties of two-dimensional eightfold degenerate fermions.

Figure 1

(a) The BZ of square lattice. Schematic illustration for generation of (b) a Dirac nodal line and (c) eightfold degenerate point for $P4/mbm$ space group. The “EP” represents eightfold degenerate point.

Figure 2

Energy bands for the lattice model with eightfold degenerate points under the $P4/mbm$ group. (a) A schematic representation of the square lattice with the nearest and next-nearest neighbor hopping integrals. The red square marks the unit cell. (b),(c) The band structures of lattice model along the high-symmetry directions. (d) Energy spectrum for a nanoribbon with 60 unit cells in the $x$ direction and infinite along the $y$ direction. The red (black) bands represent the mirror $M_z$ eigenvalue 1($-1$). The edge states are labeled by blue. The eightfold degenerate points are marked by green dots.

Figure 3

Lattice model and bands structures. (a), (b) Breaking $\mathcal{I}$ symmetry. (c),(d) Breaking $\mathcal{I}$ and $\mathcal{T}$ symmetry with an out-of plane antiferromagnetic order. The red (black) bands represent the mirror $M_z$ eigenvalue 1($-1$). The eightfold degenerate points are marked by green dots. The red arrows represent magnetic moment.

Figure 4

Crystal structure of monolayer ${\mathrm{LaB}}_8$ viewed along (a) [100], (b) [110], and (a) [001] directions. The red and green balls represent La and B atoms, respectively. (d) Phonon spectrum of monolayer ${\mathrm{LaB}}_8$ along the high-symmetry directions. The electronic band structures of monolayer ${\mathrm{LaB}}_8$ along the high-symmetry directions for (e) relaxed crystal structure and (f) compressing the lattice constant by 3%. The “$+$” and “$-$” represent the eigenvalue of mirror $M_z$ with 1 and $-1$, respectively. The eightfold degenerate points are marked by blue dots.