Dirac points emerging from flat bands in Lieb-kagome lattices

The energy spectra for the tight-binding models on the Lieb and kagome lattices both exhibit a flat band. We study a model which continuously interpolates between these two limits. The flat band located in the middle of the three-band spectrum for the Lieb lattice is distorted, generating two pairs of Dirac points. While the upper pair evolves into graphenelike Dirac cones in the kagome limit, the low-energy pair evolves until it merges, producing the band-bottom flat band. The topological characterization of the Dirac points is achieved by projecting the Hamiltonian on the two relevant bands in order to obtain an effective Dirac Hamiltonian. The low-energy pair of Dirac points is particularly interesting: when they emerge, they have opposite winding numbers, but as they merge, they have the same winding number. This apparent paradox is due to a continuous rotation of their states in pseudospin space, characterized by a winding vector. This simple, but quite rich model, suggests a way to a systematic characterization of two-band contact points in multiband systems.