Band touching from real-space topology in frustrated hopping models

We study “frustrated” hopping models, in which at least one energy band—at the maximum or minimum of the spectrum—is dispersionless. The states of the flat band(s) can be represented in a basis, which is fully localized, having support on a vanishing fraction of the system in the thermodynamic limit. In the majority of examples, a dispersive band touches the flat band(s) at a number of discrete points in momentum space. We demonstrate that this band touching is related to states which exhibit nontrivial topology in real-space. Specifically, these states have support on one-dimensional loops which wind around the entire system (with periodic boundary conditions). A counting argument is given that determines, in each case, whether there is band touching or none, in precise correspondence to the result of straightforward diagonalization. When they are present, the topological structure protects the band touchings in the sense that they can only be removed by perturbations, which also split the degeneracy of the flat band.

Figure 1

(Color online) Conventions for the shortest length Bravais lattice vectors for the kagome lattice (brown arrows).

Figure 2

(Color online) Depiction of localized eigenstates on the boundary of a single and triple plaquette. Those sites with nonzero weight are denoted by a full (red) circle. The magnitude of the weights is always the same but the phases alternate between $\pm 1$. The phases are denoted by $\pm$ signs next to the relevant lattice sites.

Figure 3

(Color online) The localized states are exact eigenstates due to destructive interference between the hopping amplitudes from sites with nonzero weight (filled circles) to sites outside the boundary (empty circle). The lattice sites with nonzero weight are contained in a finite area within a boundary marked by the dashed line.

Figure 4

(Color online) The two noncontractible loop states around the handles of the torus. One loop consists of the sites marked by full (blue) circles and the other by the empty (red) circles. As the other eigenstates in the flat band, the wave function has an alternating $\pm$ phase on the sites along the loops.

Figure 5

(Color online) Pyrochlore volume enclosed by four plaquettes. Two of the four plaquettes are highlighted by thick (red and green) lines. Each one of the two plaquettes supports a localized eigenstate.

Figure 6

(Color online) Dice lattice localized eigenstate denoted by thick (red) closed loop surrounding the site labeled by $R$. One of two nontrivial loops is indicated by a thick straight (blue) line. All sites with some particle weight on them are indicated by a filled circle. The amplitude is indicated on every site with nonzero weight. From this picture we can understand why the localized states are eigenstates, again invoking the picture of destructive interference. Consider the sites marked 1 and 2. Hopping from these sites can occur to either $\mathbf{R}$ or the site right outside the boundary of the localized state. in both cases, the hopping amplitude from sites 1 and 2 cancels out. Similar considerations for the other sites of the localized states yield the same result. For the noncontractible loop state, every site neighboring the sites with nonzero weight has three hopping amplitudes contributing $1+\omega +{\omega }^2$. As long as this sum vanishes, this is an exact eigenstate.

Figure 7

(Color online) $p$-band honeycomb local eigenstate and noncontractible loop state. The orbital states are denoted by a vector (in the present case the vector coordinates can be chosen real). The orbital vectors are always perpendicular to one link emanating from the site and with the special form of hopping assumed in the model. The hopping amplitude on this link vanishes in this state.

Figure 8

(Color online) Type I $p$-band dice lattice local eigenstate.

Figure 9

(Color online) Type II $p$-band dice lattice local eigenstate.

Figure 10

(Color online) Nonoverlapping set of area units. Summing over type I states on these area units adds up to zero.

Figure 11

(Color online) Noncontractible loop states. Only those sites with arrows on them have nonzero occupation.

Figure 12

(Color online) Sum of type II states (marked by filled circles at their centers) on a noncontractible strip of area units. Only those sites with arrows on them have nonzero occupation.

Figure 13

(Color online) Tight-binding model with equal hopping amplitudes on three different link types.

Figure 14

(Color online) Illustration of the local constraint in the kagome-3 hopping model. Localized eigenstates living on the bowtie plaquettes are summed over the six plaquettes bordering a hexagonal plaquette in the kagome lattice. The figure shows two of these localized states with the correct relative phase needed for the summation to vanish. Only those sites with nonzero weight are denoted by (red) circles (filled for $+1$ and unfilled for $-1$).

Figure 15

(Color online) Localized eigenstates of the kagome-3 hopping model. The states have nonzero weight only on the four sites surrounding a bowtie plaquette (the encircled regions). The sites with nonzero weight are marked by filled (red) circles and with their relative signs indicated next to them. There are a number of different flavors of bowtie plaquettes (and localized states) $-3$ per unit cell. However there are only two independent bowtie states per unit cell. Here we show one choice of two independent bowtie states marked as $A$ and $B$.

Figure 16

(Color online) Noncontractible loop states of the kagome-3 model. There are two types of the noncontractible loops. One kind is the exactly same states appearing in the kagome model and is denoted by a continuous (red) path between the sites with nonzero weight. The other kind of noncontractible loop states are denoted by dashed line and consist of two sites with nonzero weight and alternating sign on each hexagonal plaquette in a chain of plaquettes.