Kagome bands disguised in a coloring-triangle lattice

The kagome bands hosting exotic quantum phases generally and understandably pertain only to a kagome lattice. This has severely hampered the research of kagome physics due to the lack of real kagome-lattice materials. Interestingly, we discover that a coloring-triangle (CT) lattice, named after color-triangle tiling, also hosts kagome bands. We demonstrate first theoretically the equivalency between the kagome and CT lattices, and then computationally in photonic (waveguide lattice) and electronic (Au overlayer on electride $\mathrm{C}{\mathrm{a}}_2\mathrm{N}$ surface) systems by first-principles calculations. The theory can be generalized to even distorted kagome and CT lattices to exhibit ideal kagome bands. Our findings open an avenue to explore the alluding kagome physics.

Figure 1

Kagome bands along high-symmetry $k$ paths, with (a) flat band on top and (b) flat band at bottom. The location of the flat band depends on the sign of the hopping integral. (c) The kagome lattice. (d) The CT lattice. The dashed red rhombus in (c) and (d) denotes the three-site unit cell.

Figure 2

Illustration of a (a) kagome lattice and (b) CT lattice formed by patterning a triangular lattice. The red arrows denote the patterned unit-cell basis vectors. (c) Line graph of a kagome lattice (blue) constructed from a parent hexagonal graph (lattice, green dots) with straight edges (red). (d) Line graph of a CT lattice (blue) constructed from a parent hexagonal graph (green dots) with curved edges (red). (e) Band variation in a √3 × √3 supercell of triangular lattice with one-third of the NN hoppings tuned from $t_2=0$ to $t_2=t_1$ (bands corresponding to three typical $t_2/t_1$ ratios, namely, 0, 0.5, and 1, are presented). Note that for the $t_2=0$ case, the three bands are just folded from a single band of perfect triangular lattice. The inset shows the first Brillouin zone of the supercell (green) folded from that of the unit cell of the triangular lattice (blue). (f) The kagome bands realized in the CT lattice for $t_2=t_1/4$, without (green) and with (red) SOC $(\lambda =0.01t_1)$.

Figure 3

Unit cell of photonic (a) kagome and (b) CT lattices formed by weakly coupled waveguide arrays (dark blue circles). The silica medium is colored light blue, and the air holes in the CT lattice are colored white. (c), (d) Band structure corresponding to the photonic kagome and CT lattices, respectively.

Figure 4

(a) Top and side view of a triangular Au lattice deposited on the surface of a $\mathrm{C}{\mathrm{a}}_2\mathrm{N}$ monolayer. The blue and red rhombus denotes the unit cell and √3 × √3 supercell of the pristine $\mathrm{C}{\mathrm{a}}_2\mathrm{N}$ monolayer, respectively. (b) Charge density distribution in the plane 1 shown in the side view of (a), without and with the Au lattice patterning. (c) Electronic band structure of the Au-patterned $\mathrm{C}{\mathrm{a}}_2\mathrm{N}$ monolayer, with band composition displayed in different colors. (d) The DFT band structure near the Fermi level (red) and the interpolated bands from MLWFs with Ca-$s$ characteristics (green).