Three-Dimensional Higher-Order Topological Insulator Protected by Cubic Symmetry

Recently discovered photonic higher-order topological insulators enable unprecedented flexibility in the robust localization of light in structures of different dimensionality. While the potential of the two-dimensional systems is currently under active investigation, only a few studies explore the physics of the three-dimensional higher-order topological insulators. Here we propose a three-dimensional structure with cubic symmetry exhibiting vanishing bulk polarization but nonzero corner charge and hosting a zero-dimensional corner state mediated by the long-range interactions. We trace the evolution of the corner state with the next-nearest-neighbor coupling strength and prove the topological origin of the corner mode, calculating the associated topological invariants. Our results thus reveal the potential of long-range couplings for the formation of three-dimensional higher-order topological phases.

Figure 1

Geometry of the proposed three-dimensional higher-order topological insulator. Nearest-neighbor couplings $J$ and $K$, both positive, are shown by the single and double green lines, respectively. Two additional types of next-nearest-neighbor coupling with amplitudes $M$ and $V$ are shown by the red dashed and blue dotted lines, respectively. (a) Single face of the unit cell. (b) Three-dimensional unit cell and enumeration of the sites used to construct the Bloch Hamiltonian given in Eq. (1). (c) A finite sample of the three-dimensional structure. Dashed circles indicate two possible choices of the unit cell with strong and weak links inside. A corner highlighted by red hosts the topological corner state.

Figure 2

Calculated band structure of the proposed model with nearest-neighbor couplings $J=1$, $K=6$ and long-range interactions $M=4$ and $V=-3$. The complete bandgap that opens at energies close to zero is shaded in green. Inset shows the first Brillouin zone and the associated high-symmetry points.

Figure 3

Eigenmodes of a finite 3D structure. (a) Mode energies for a $9\times 9\times 9$ structure with coupling constants $J=1$, $K=6$, $M=2.7$ versus long-range coupling $V$, $-15\le V\le 15$. (b) Mode energies for a $9\times 9\times 9$ structure with coupling constants $J=1$, $K=6$, $V=-3$ versus long-range coupling $M$, $0\le M\le 5$. Color encodes the logarithm of the inverse participation ratio for the eigenmodes. (c) Eigenmode profile corresponding to the in-gap corner state in a $13\times 13\times 13$ structure with coupling constants $J=1$, $K=6$, $M=4$, $V=-3$. Color shows the absolute value of the mode amplitude at a given site of the lattice.

Figure 4

Colormap showing the inverse participation ratio ($\mathfrak{I}$) of the corner mode as a function of the next-nearest-neighbor couplings $M$ and $V$ calculated for a $9\times 9\times 9$ system with $J=1$ and $K=6$, when the corner state exists inside the bandgap.

Figure 5

Unit-cell geometry for the proposed three-dimensional higher-order topological insulator. Nearest-neighbor couplings $J$ and $K$, both positive, are shown by the single and double green lines, respectively. Two additional types of next-nearest-neighbor coupling with amplitudes $M$ and $V$ are shown by the red dashed and blue dotted lines, respectively. (a) Single face of the unit cell with the weak links $J$ inside. (b) Three-dimensional unit cell with the weak links $J$ inside and enumeration of the sites used to construct the Bloch Hamiltonian given in Eq. (A1).

Figure 6

Illustration of the corner charge calculation. (a) Maximal Wyckoff positions in the cubic unit cell possessing $O_h$ symmetry. Because of symmetry, Wannier centers can appear only in the specified points. (b) Semi-infinite sample of the considered structure with $O_h$ symmetry and illustration of the corner charge calculation.

Figure 7

Local density of states calculated at the weak link corner of a $9\times 9\times 9$ system with $J=1$, $K=6$, $M=4$, and $V=3$, when the corner mode appears in the continuum. Auxiliary parameter $\gamma$ responsible for the width of the peak is set to $0.2$.

Figure 8

Localization of the topological corner mode at the interface between the two domains with different dimerizations. (a) Sketch of the considered structure. Red dot highlights the position of the topological interface mode. (b)–(d) Eigenmode profile at the faces numbered 1–3 calculated for $J=1$, $K=6$, $M=4$, $V=-3$.

Figure 9

Eigenmodes of a finite 3D $9\times 9\times 9$ structure versus the disorder strength $\sigma$ averaged over ten random realizations. The disorder is introduced in the coupling constants as (a) $J=J_0+\sigma \mathrm{\Delta }J$, $K=K_0+\sigma \mathrm{\Delta }K$; (b) $J=J_0+\sigma \mathrm{\Delta }J$, $K=K_0+\sigma \mathrm{\Delta }K$, $M=M_0+\sigma \mathrm{\Delta }M$, $V=V_0+\sigma \mathrm{\Delta }V$, where coupling constants $J_0=1$, $K_0=6$, $M_0=4$, $V_0=-2.5$. $\mathrm{\Delta }K$, $\mathrm{\Delta }J$, $\mathrm{\Delta }M$, $\mathrm{\Delta }V$ are uniformly distributed random numbers in the interval $[0,1]$. Color shows the logarithm of the inverse participation ratio. Orange dots correspond to the corner state.