Fractional corner charges in spin-orbit coupled crystals

We study two-dimensional spinful insulating phases of matter that are protected by time-reversal and crystalline symmetries. To characterize these phases, we employ the concept of corner charge fractionalization: corners can carry charges that are fractions of even multiples of the electric charge. The charges are quantized and topologically stable as long as all symmetries are preserved. We classify the different corner charge configurations for all point groups, and match them with the corresponding bulk topology. For this we employ symmetry indicators and (nested) Wilson loop invariants. We provide formulas that allow for a convenient calculation of the corner charge from Bloch wave functions and illustrate our results using the example of arsenic and antimony monolayers. Depending on the degree of structural buckling, these materials can exhibit two distinct obstructed atomic limits. We present density functional theory calculations for open flakes to support our findings.

Figure 1

Corner charge fractionalization due to $C_3$ rotational symmetry. (a) The finite system ${\mathrm{\Lambda }}_{\mathrm{F}}$ on which $H_{\mathrm{F}}$ is defined. (b) The boundary regions ${\mathrm{c}}_1,{\mathrm{c}}_2,{\mathrm{c}}_3\in \mathrm{C}$. Due to the $C_3$ symmetry in ${\mathcal{G}}_{\mathrm{F}}$, we have that $Q_{{\mathrm{c}}_1}=Q_{{\mathrm{c}}_2}=Q_{{\mathrm{c}}_3}$. Together with $Q_{{\mathrm{c}}_1}+Q_{{\mathrm{c}}_2}+Q_{{\mathrm{c}}_3}\in 2\mathbb{Z}$, this implies a corner charge fractionalization in even multiples of $\frac{1}{3}$. (c) A 1D edge addition, modeled by the Hamiltonian $H_{1\mathrm{D}}$. We prove in Sec. 2b that the corner charges $Q_{{\mathrm{c}}_i}$ are only changed by even integers.

Figure 2

Brillouin zones of crystals with $C_2, C_4, C_3$, and $C_6$ symmetries and their rotation invariant points. In $C_2$-symmetric systems there are three twofold HSPs: $\mathit{X}, \mathit{Y}$, and $\mathit{M}$. In $C_4$-symmetric systems there are two twofold HSPs: $\mathit{X}$ and ${\mathit{X}}^{\prime }$, and one fourfold HSP: $\mathit{M}$. In $C_3$-symmetric systems there are only two threefold HSPs: $\mathit{K}$ and ${\mathit{K}}^{\prime }$. Finally, in $C_6$-symmetric systems there are three twofold HSPs: $\mathit{M}, {\mathit{M}}^{\prime }$, and ${\mathit{M}}^{\prime \prime }$, as well as two threefold HSPs: $\mathit{K}$ and ${\mathit{K}}^{\prime }$.

Figure 3

Sets of allowed eigenvalues for spinful rotational symmetries. (a) $C_2$ symmetry, (b) $C_4$ symmetry, (c) $C_3$ symmetry. The possible eigenvalues of $C_6$ symmetry are not shown since they do not allow for the definition of symmetry indicators (there is at most one $C_6$-symmetric point in any two-dimensional Brillouin zone).

Figure 4

Maximal Wyckoff positions for unit cells with rotational symmetry. (a) $C_2$ symmetry, (b) $C_4$ symmetry, (c) $C_3$ symmetry, (d) $C_6$ symmetry or $C_3+\mathcal{I}$ symmetry. Boundary charges arise when the Wyckoff positions which Wannier centers are located at are cut through by the crystal termination.

Figure 5

Energy gap as a function of the buckling parameter $d_z$ for (a) bismuth, (b) antimony, and (c) arsenic monolayers. The black line (circles) indicates the indirect gap, while the green line (triangles) indicates the direct gap. Top and side views of the lattice structure are illustrated in (d), together with the Wyckoff positions of the space group 164. (e) Low-energy spectrum of a finite armchair-terminated flake of the $3c$ OAL. The inset presents the energies around the Fermi level, with filled states in orange. (f) The electronic densities of the corner states with color scale proportional to the normalized square modulus $|{\psi }_i{|}^2$ of the eigenstates (normalized with respect to the largest $|{\psi }_i{|}^2$). The tellurium atoms used for edge passivation are shown as stars.