High-degeneracy points protected by site-permutation symmetries

Space-group symmetries dictate the energy degeneracy of quasiparticles (e.g., electronic, photonic) in crystalline structures. For spinless systems, there can only be double or triple degeneracies protected by these symmetries, while other degeneracies are usually taken as accidental. In this Rapid Communication we show that it is possible to design higher degeneracies exploring site permutation symmetries. These design principles are shown to be satisfied in previously studied lattices, and new structures are proposed with three, four, and five degeneracy points for spinless systems. The results are general and apply to different quasiparticle models. Here, we focus on a tight-binding approach for the electronic case as a proof of principle. The resulting high-degeneracy points are protected by the site-permutation symmetries, yielding pseudospin-1 and -2 Dirac fermions. The strategy proposed here can be used to design lattices with high-degeneracy points in electronic (e.g., metal-organic frameworks), photonic, phononic, magnonic, and cold-atom systems.

Figure 1

Example of lattices fulfilling the minimum ingredients for HD points, that is, the $c_1$ coordination criteria, namely, (a) kagome, (b) triangular-rectangular, (c) square-octagonal, (d) pyramidal, (e) snub-hexagonal, (f) crossed-tetrahedral, (g) pyrochlore, and (h) cubic-octahedra lattices. The blue circles indicate the sites positions, black lines connect first neighbors, and red lines define the unit cell.

Figure 2

TB model, within 1NN interaction ($\alpha =30/d_{1\mathrm{NN}}$), for the (a) kagome, (b) triangular-rectangular, (c) square-octagonal, (d) pyramidal, (e) snub-hexagonal, (f) crossed-tetrahedral, (g) pyrochlore, and (h) cubic-octahedra lattices. The energy scale is $t/n_1$, where the 1NN hopping strength is $t=-1$, and $n_1$ is the number of equivalent NNs.

Figure 3

Band structure close to the $\mathrm{\Gamma }$ point in the snub-hexagonal lattice with broken $P^{\left(1k\right)}$ site-permutation symmetries. (a) Lattice with breaking of $k=2$, (b) $2\le k\le 3$, (c) $2\le k\le 4$, and (d) $2\le k\le 5$ site-permutation symmetries. (e) Polar lattice with broken $P^{\left(ij\right)}$ for $i=1,2$, and 3 and $j=4,5$, and 6. (f) Sublattice enumeration, with dashed line indicating the Wigner-Seitz cell.