Sixfold excitations in electrides

Due to the lack of full rotational symmetry in condensed matter physics, solids exhibit new excitations beyond Dirac and Weyl fermions, of which the sixfold excitations have attracted considerable interest owing to the presence of maximum degeneracy in bosonic systems. Here, we propose that a single linear dispersive sixfold excitation can be found in the electride ${\mathrm{Li}}_{12}{\mathrm{Mg}}_3{\mathrm{Si}}_4$ and its derivatives. The sixfold excitation is formed by the floating bands of elementary band representation $A@12a$ originating from the excess electrons centered at the vacancies (i.e., the $12a$ Wyckoff sites). There exists a unique topological bulk-surface-edge correspondence for the spinless sixfold excitation, resulting in trivial surface “Fermi arcs” but topological hinge arcs. All gapped $k_z$ slices belong to a two-dimensional higher-order topological insulating phase, which is protected by a combined symmetry $\mathcal{T}{\tilde{S}}_{4z}$ and characterized by a quantized fractional corner charge $Q_{\text{corner}}=\frac{3\left|e\right|}{4}$. Consequently, the hinge arcs are obtained in the hinge spectra of the ${\tilde{S}}_{4z}$-symmetric rod structure. The state with a single sixfold excitation, stabilized by both nonsymmorphic crystalline symmetries and time-reversal symmetry, is located at the phase boundary and can be driven into various topologically distinct phases by explicit breaking of symmetries, making these electrides promising platforms for the systematic studies of different topological phases.

Figure 1

Schematics of different types of fermions. (a) Spin-1/2 Weyl fermion with charge $C=+1$. (b) Fourfold Dirac fermion. (c) Spin-1 excitation with charge $C=+2$. (d) Sixfold excitation.

Figure 2

Crystal and electronic structures of ${\mathrm{Li}}_{12}{\mathrm{Mg}}_3{\mathrm{Si}}_4$. (a) Top view of the crystal structure of ${\mathrm{Li}}_{12}{\mathrm{Mg}}_3{\mathrm{Si}}_4$. (b) Bulk BZ and (001) surface BZ with high-symmetry points indicated. (c) PED with an isosurface value of 0.0068 ${\mathrm{bohr}}^{-3}$ for the six blue-colored bands in (d) or (e). The colored balls indicate the positions of the atoms, while the irregular surfaces in blue indicate the charge distribution around the $12a$ sites. (d) and (e) The first-principles band structures of ${\mathrm{Li}}_{12}{\mathrm{Mg}}_3{\mathrm{Si}}_4$ without SOC in different energy ranges. The inset in (d) shows the zoom-in band structure around the sixfold excitation. (f) 2D band structure in the $xoz$ plane around the sixfold excitation of the $H$ point.

Figure 3

The bulk and boundary states of the TB Hamiltonian. (a) The bulk bands, (b) (001) surface dispersions, and (c) [001] hinge dispersions. The yellow shadow denotes the projections of the bulk states. (d) The real-space distributions of the four hinge states for $k_z=\frac{\pi }{a}$ on the structure of a 1D $20\times 20$ supercell.

Figure 4

The bulk band structures (a) without strain, (b) with [111] strain, and (c) with [001] strain.