Topological band evolution between Lieb and kagome lattices

Abstract

Among two-dimensional lattices, both kagome and Lieb lattices have been extensively studied, showing unique physics related to their exotic flat and Dirac bands. Interestingly, we realize that the two lattices are in fact interconvertible by applying strains along the diagonal direction, as they share the same structural configuration in the unit cell, i.e., one corner-site and two edge-center states. We study phase transitions between the two lattices using the tight-binding approach and propose one experimental realization of the transitions using photonic devices. The evolution of the band structure demonstrates a continuous evolution of the flat band from the middle of the Lieb band to the top/bottom of the kagome band. Though the flat band is destroyed during the transition, the topological features are conserved due to the retained inversion symmetry, as confirmed by Berry curvature, Wannier charge center, and edge state calculations. Meanwhile, the triply degenerate Dirac point ($M$) in the Lieb lattice transforms into two doubly degenerate Dirac points, one of which moves along $M\ensuremath{-}\mathrm{\ensuremath{\Gamma}}$ and the other moves along $M\ensuremath{-}K/{K}^{\ensuremath{'}}$ directions that form the kagome band eventually. Interestingly, the Dirac cones in the transition states are strongly tilted, showing a coexistence of type-I and type-II Dirac points. We finally show that these transitions can be experimentally realized in photonic lattices using waveguide arrays.