Systematic construction of topological flat-band models by molecular-orbital representation

On the basis of the “molecular-orbital” representation which describes generic flat-band models, we propose a systematic way to construct a class of flat-band models with finite-range hoppings that have topological natures. In these models, the topological natures are encoded not into the flat band itself but into the dispersive bands touching the flat band. Such a band structure may become a source of exotic phenomena arising from the combination of flat bands, topology, and correlations.

Figure 1

Schematic figure for the construction of the tight-binding models considered in this paper.

Figure 2

The energy spectrum for the Hamiltonian of Eq. (13). A red line represents the degenerate zero-energy states, i.e., flat bands.

Figure 3

(a) The magnetic Brillouin zone for $\varphi =1/3$. We set the magnetic lattice vectors as ${\tilde{\mathit{a}}}_1=3{\mathit{a}}_1$ and ${\tilde{\mathit{a}}}_2={\mathit{a}}_2$. The corresponding reciprocal lattice vectors, ${\tilde{\mathit{b}}}_1$ and ${\tilde{\mathit{b}}}_2$, are denoted by blue arrows. The coordinates of the corners of the first Brillouin zones are ${\tilde{\mathit{k}}}_1=\frac{7}{27}(3{\tilde{\mathit{b}}}_1+2{\tilde{\mathit{b}}}_2)$ and ${\tilde{\mathit{k}}}_2=\frac{1}{2}({\tilde{\mathit{b}}}_1+{\tilde{\mathit{b}}}_2)$. (b) The band structure for $\varphi =1/3$. The Chern numbers of the bands are indicated in the figure. (c) A Dirac point formed by the third and the seventh band. At the Dirac point, they are also degenerated with three flat bands.

Figure 4

(a) The band structure of the MO Haldane model for $(t,t^{\prime },M,\varphi ,\gamma )=(-1,-0.3,0,\frac{\pi }{2},1)$. The high-symmetry points in the Brillouin zone are $\mathrm{\Gamma }=(0,0)$, $K=(\frac{4\pi }{3},0)$, and $M=(\pi ,\frac{\pi }{\sqrt{3}})$. (b) The band structure for the same parameter on a cylinder geometry. (c) A $M\text{−}\varphi$ phase diagram of the MO Haldane model at $t^{\prime }=-0.3$. The orange, brown, and blue regions correspond to the Chern numbers of the lowest band 1, 0, and $-1$, respectively. In the yellow region, the lowest dispersive band overlaps with the flat band.

Figure 5

The band structure of the model of (21) with $(t,t^{\prime },M,M^{\prime },\varphi )=(-0.6,-0.3,0,2,\frac{\pi }{2})$.

Figure 6

(a) The band structure of the MO Kane-Mele model for $(t,t^{\prime },t^{″},\varphi ,M)=(-1,-0.06,-0.05,\frac{\pi }{2},-0.1)$; (b) the same parameter on a cylinder geometry. (c) The $t^{″}\text{−}M$ phase diagram of the MO Kane-Mele model at $t^{\prime }=-0.06$.

Figure 7

(a) The pyrochlore lattice. Orange arrows represent the lattice vectors: ${\mathit{a}}_1=(0,1/2,1/2)$, ${\mathit{a}}_2=(1/2,0,1/2)$, and ${\mathit{a}}_3=(1/2,1/2,0)$. Blue and red spheres represent the sites of the diamond lattice on which the MOs for the upward and the downward tetrahedra are defined, respectively. (b) The band structure of the MO Fu-Kane-Mele model for $(t,t^{\prime },\delta t)=(1,0.125,0.4)$. The high-symmetry points in the Brillouin zone are $\mathrm{\Gamma }=(0,0,0)$, $X=(2\pi ,0,0)$, $W=\left(2\pi ,\pi ,0\right)$, $K=\left(\frac{3\pi }{2},\frac{3\pi }{2},0\right)$, and $L=\left(\pi ,\pi ,\pi \right)$. (c) The band structure for the same parameters with a slab geometry. The surface is perpendicular to the $\left[1\overline{1}\overline{1}\right]$ direction. The high-symmetry points in the surface Brillouin zone are $\overline{\mathrm{\Gamma }}=(0,0)$, $M_1=(\sqrt{2}\pi ,\frac{\sqrt{2}\pi }{\sqrt{3}})$, $M_2=(0,\frac{2\sqrt{2}\pi }{\sqrt{3}})$, and $M_3=(-\sqrt{2}\pi ,\frac{\sqrt{2}\pi }{\sqrt{3}})$.