Flat bands and higher-order topology in polymerized triptycene: Tight-binding analysis on decorated star lattices

In a class of carbon-based materials called polymerized triptycene, which consist of triptycene molecules and phenyls, exotic electronic structures such as Dirac cones and flat bands arise from the kagome-type network. In this paper, we theoretically investigate the tight-binding models for polymerized triptycene, focusing on the origin of flat bands and the topological properties. The mechanism of the existence of the flat bands is elucidated by using the “molecular-orbital” representation, which we have developed in the prior works. Further, we propose that the present material is a promising candidate to realize the two-dimensional second-order topological insulator, which is characterized by the boundary states localized at the corners of the sample. To be concrete, we propose two methods to realize the second-order topological insulator and elucidate the topological properties of the corresponding models by calculating the corner states as well as the bulk topological invariant, namely, the ${\mathbb{Z}}_3$ Berry phase.

Figure 1

(a) The structure of a tripticene molecule. Red spheres denote the $s{p}^3$ carbon atoms. (b) Polyermized triptycene containing one phenyl between neighboring triptycene molecules. Blue triangles denote triptycene molecules. (c) Schematic figure for the polymerized triptycene and its kagome network obtained by reducing chains of ${\mathrm{C}}_6$ rings as a single site.

Figure 2

The tight-binding model of Eq. (2). The primitive vectors and the sublattices are indicated in the figure.

Figure 3

The band structures for $t=1$ and (a) $t^{\prime }=2$, (b) $t^{\prime }=0.5$, (c) $t^{\prime }=\sqrt{\frac{3}{2}}$, and (d) $t^{\prime }=\sqrt{3}$. The high-symmetry points in momentum space are $\mathrm{\Gamma }=(0,0),K=(\frac{4\pi }{3},0)$, and $M=(\pi ,\frac{\pi }{\sqrt{3}})$.

Figure 4

(a) Schematic picture of the reduction of three sites connected by $t^{\prime }$-bonds to a single site, and the resulting kagome lattice. (b) Schematic picture of the real-space distribution of the three-site wave functions ${\mathit{\phi }}_1\text{−}{\mathit{\phi }}_3$. (c) The band structure for the Hamiltonian of Eq. (2) with $(t,t^{\prime })=(1,2)$ (blue solid lines) and the approximate band structure (red dashed lines).

Figure 5

(a) The tight-binding model of Eq. (15). The primitive vectors and the sublattices are indicated in the figure. (b) The band structure for $t=1$, $t^{\prime }=2$. (c) The tight-binding model of Eq. (17). (d) The band structure for $(t,t^{\prime },t^{\prime \prime })=(1,3,1.5)$.

Figure 6

(a) Schematic picture of the reduction of four sites connected by $t^{\prime }$ or $t^{\prime \prime }$ bonds to a single site, and the resulting breathing kagome lattice. (b) Schematic picture of the real-space distribution of the four-site wave functions ${\mathit{\phi }}_1\text{−}{\mathit{\phi }}_4$. Dashed lines denote the case for $t^{\prime }=t^{\prime \prime }$, where the inversion symmetry is restored. (c) The band structure for the Hamiltonian of Eq. (17) with $(t,t^{\prime },t^{\prime \prime })=(1,3,1.5)$ (blue solid lines) and the approximate band structure (red dashed lines).

Figure 7

(a) The tight-binding model for Eq. (19). (b) Schematic figure of the finite system used to investigate the corner state. (c) The energy spectrum for the finite system with $(t,\tilde{t},t^{\prime })=(0.5,0.1,3)$ and $L=12$ (the number of sites is 1491). Two in-gap states are indicated by light-blue circles; here, I is 313th state and II is 1323th state. Insets are energy levels near the in-gap states. The amplitude of the wave functions in the real space for (d) the state I and (e) the state II.

Figure 8

(a) Schematic figure of the finite system used to investigate the corner state in polymerized triptycene containing biphenyl. Orange circles denote the sites at bottom-left and top-right corners. (b) The energy spectrum for the finite system with $(t,t^{\prime },t^{\prime \prime })=(1,3,1.5)$ and $L=12$ (the number of sites is 1972). Six in-gap states are indicated by light-blue circles; here I is 313th state, II is 818th state, III is 1323th state, IV is 1804th state, and V and VI, which are degenerate, are 963th and 964th states, respectively. [(c)–(h)] The amplitude of the wave functions in the real space. (c), (d), (e), (f), (g), and (h) correspond to I, II, III, IV, V, and VI, respectively.

Figure 9

(a) Schematic figure for the twist introduced in Eq. (23). (b) The paths of the contour integral in Eq. (24) in the parameter space. [(c) and (d)] The minimal cluster for the Hamiltonian of Eq. (19). (c) is obtained by setting $\tilde{t}=0$, while (d) is obtained by setting $t=0$. (e) $\tilde{t}/t$ dependence of the ${\mathbb{Z}}_3$ Berry phase for the model of Eq. (19) [i.e., case (i)] with $2/9$-filling. The system size used for the computation is $12\times 12$ unit cells. (f) $t^{\prime }/t^{\prime \prime }$ dependence of the ${\mathbb{Z}}_3$ Berry phase for the model of Eq. (18) [i.e., case (ii)] with $5/12$-filling. The system size used for the computation is $8\times 8$ unit cells. The insets in (e) and (f) represent the minimal clusters to which the ground states are connected adiabatically.

Figure 10

Schematic figure for the $q+2$-site chain (in a gray box) described by the Hamiltonian of (A8). Blue triangles at the ends correspond to $t{h}^{△}$ and $t{h}_{\mathit{k}}^{▽}$, operation of which gives rise to the on-site energy $-t$ at the end sites [see Eqs. (A5) and (A6)].