Compatibility relationships in van der Waals quasicrystals

The incommensurate twist structure and the interlayer coupling induce van der Waals quasicrystals (vdW-QCs). Replacing conventional band theory requiring translational symmetry, the resonant coupling Hamiltonian endowing the quasiband structure in $\mathit{k}$ space is adopted to describe electronic properties of vdW-QCs [Moon et al., Phys. Rev. B 99, 165430 (2019)]. Here we investigate the symmetries of the resonant coupling Hamiltonians in dodecagonal and octagonal vdW-QCs. Through symmetry analyses we derive compatibility relationships (CRs) between $\mathrm{\Gamma }$ point and other irreducible pathways and predict the symmetry changes and band splits. Especially, we find that from $\mathrm{\Gamma }$ point to Brillouin zone corner points of monolayers, arbitrary twofold degenerate states are split into one $A^{\prime }$ and one $A^{″}$ state, and from $\mathrm{\Gamma }$ point to the intersection points of two Brillouin zones of monolayers, arbitrary twofold degenerate states are split into one $A$ and one $B$ state. Instead of projection operation analyses, we discuss the CRs of different point groups between the coupled bilayers and uncoupled monolayers to construct the interlayer hybridization selection rules (IHSRs) [Yu et al., Phys. Rev. B 105, 125403 (2022)], which govern how the interlayer states interact with each other in the resonant coupling systems of dodecagonal and octagonal vdW-QCs. These derived IHSRs indicate that the first two main resonant couplings allow the nonequivalent hybridizations only between $B_1$ and $B_2$ states and the equivalent hybridizations for $A$, $A_i$, $A^{\prime }$, $A^{″}$, $E$, or $E_i$ states.

Figure 1

Top views of lattice structures for ${\mathbb{D}}_{6d}$ dodecagonal vdW-QC in (a) and ${\mathbb{D}}_{4d}$ octagonal vdW-QC in (b). Their corresponding BZs for two component layers in (c) and (d). The green solid and gray dashed lines in (c) and (d) stand for the reflection mirrors and the axes of twofold rotations in reciprocal spaces, respectively.

Figure 2

Wave vectors ${\mathit{Q}}_i$ for the (first, second) strongest resonant coupling in dodecagonal vdW-QC for (a) and (d) ${\mathit{k}}_0$ shifting away from $\mathrm{\Gamma }$ along $-{\mathit{k}}_x$ direction, (b) and (e) ${\mathit{k}}_0$ at $\mathrm{\Gamma }$, and (c) and (f) ${\mathit{k}}_0$ shifting equivalently away from $\mathrm{\Gamma }$ along $\mathrm{\Gamma }{O}_0$ direction [also see Fig. 1]. Wave vectors ${\mathit{Q}}_i$ with odd and even $i$ belong to the bottom and top layers, respectively. The reflection mirrors for ${\mathbb{C}}_s$ point group symmetry in (a) and (d) are denoted by green solid lines. The twofold axes for ${\mathbb{C}}_2$ point group symmetry in (c) and (f) are denoted by gray dashed lines. In (b) and (e), the interlayer coupling conditions between ${\mathit{Q}}_0$ and ${\mathit{Q}}_1$ are depicted, namely ${\mathit{Q}}_0+{\mathit{G}}^t={\mathit{Q}}_1+{\mathit{G}}^b$. Blue and red hexagon networks are the BZs of the bottom and top layers of dodecagonal vdW-QCs in each subplot.

Figure 3

Wave vectors ${\mathit{Q}}_i$ for the (first, second) strongest resonant coupling in octagonal vdW-QC for (a) and (d) ${\mathit{k}}_0$ shifting away from $\mathrm{\Gamma }$ along $-{\mathit{k}}_x$ direction, (b) and (e) ${\mathit{k}}_0$ at $\mathrm{\Gamma }$, and (c) and (f) ${\mathit{k}}_0$ shifting equivalently away from $\mathrm{\Gamma }$ along $\mathrm{\Gamma }{O}_0$ direction [also see Fig. 1]. Wave vectors ${\mathit{Q}}_i$ with odd and even $i$ belong to the bottom and top layers, respectively. The reflection mirrors ${\sigma }_x$ for ${\mathbb{C}}_s$ point group in (a) and (d) are denoted by green solid lines. The twofold axes $\mathrm{\Gamma }{O}_0$ of $C_{2,\mathrm{\Gamma }O0}$ for ${\mathbb{C}}_2$ point group in (c) and (f) are denoted by gray dashed lines. Blue and red hexagon networks are the BZs of the bottom and top layers of dodecagonal vdW-QCs in each subplot. These ${\mathit{Q}}_i$ points connected by ${\sigma }_x$ in (a) and (d) and these ${\mathit{Q}}_i$ points connected by $C_{2,\mathrm{\Gamma }O0}$ are denoted as curved lines with double arrows.

Figure 4

(a) and (d) The quasiband structures of the dodecagonal vdW-QC for the (first, second) strongest resonant couplings with Dirac point $E_D$ as energy reference. (b) and (c) Zoom-in images of (a) with irreducible representations in negative and positive energy regions around $\mathrm{\Gamma }$ point. (e) and (f) Zoom-in images of (d) with irreducible representations in negative and positive energy regions around $\mathrm{\Gamma }$ point.

Figure 5

The quasiband structures of the octagonal vdW-QC around $\mathrm{\Gamma }$ point with irreducible representations for (a) the first and (b) second strongest resonant couplings.

Figure 6

(a) and (b) The classified eigenenergy spectra for the (first, second) strongest resonant coupling Hamiltonians at $\mathrm{\Gamma }$ point under ($C_6$, $C_2$) operations in dodecagonal vdW-QC. (c) and (d) The classified eigenenergy spectra for the (first, second) strongest resonant coupling Hamiltonians at $\mathrm{\Gamma }$ point under ($C_4$, $C_4$) operations in octagonal vdW-QC. In each subplot, the left and right blue lines stand for the energy levels of the bottom and top layers and the middle red lines for the coupled bilayer. The value of $\theta$ corresponds to the eigenvalue $e^{i\theta }$ of $C_n$ acting on the eigenstates of the bottom and top layers.

Figure 7

$\left|\langle {\phi }_{ir,\theta }^b\left|U\right|{\phi }_{i{r}^{\prime },{\theta }^{\prime }}^t\rangle \right|$ in unit of eV in (a) and (b) for the (first, second) strongest resonant couplings under ($C_6$, $C_2$) operations in dodecagonal vdW-QC. $\left|\langle {\phi }_{ir,\theta }^b\left|U\right|{\phi }_{i{r}^{\prime },{\theta }^{\prime }}^t\rangle \right|$ in unit of eV in (c) and (d) for the (first, second) strongest resonant couplings under ($C_4$, $C_4$) operations in octagonal vdW-QC. The color map stands for the value of $\left|\langle {\phi }_{ir,\theta }^b\left|U\right|{\phi }_{i{r}^{\prime },{\theta }^{\prime }}^t\rangle \right|$, where $|{\phi }_{ir,\theta }^b\rangle$ and $|{\phi }_{i{r}^{\prime },{\theta }^{\prime }}^t\rangle$ are the eigenstates of the bottom and top layers. The value of $\theta$ corresponds to the eigenvalue $e^{i\theta }$ of $C_n$ acting on the eigenstates of the bottom and top layers.