On the band topology of the breathing kagome lattice

A two-dimensional second-order topological insulator exhibits topologically protected zero-energy states at its corners. In the literature, the breathing kagome lattice with nearest-neighbor hopping is often mentioned as an example of a two-dimensional second-order topological insulator. Here we show by explicit construction that the corner states of the breathing kagome lattice can be removed by a continuous change of the hopping parameters, without breaking any of the model's symmetries, without closing bulk and boundary gaps, and without introducing hopping terms not present in the original model. Furthermore, we topologically classify all three-band lattice models with the same crystalline symmetries as the breathing kagome lattice and show that though none of the phases have protected zero-energy corner states, some of the phases are obstructed atomic limits which exhibit a filling anomaly.