Topological Quantum Phase Transitions On The Two-Dimensional Kagome Lattice And Square-Octagon Lattice

Exploring topological quantum phase transitions and finding new topological states of matter have been widely concerned in the past several years. The original model to explore the topological quantum phase transitions in the time-reversal symmetry breaking lattice systems is the well-known honeycomb-lattice Haldane model. The anomalous quantum Hall effect appears at the half-filled Haldane model, with a non-zero Chern numbers C=±1while in the absence of Landau levels, distinguishing from the C=0normal insulating state. This has been arousing great recent interests of finding new topological insulating states with the time-reversal symmetry. Meanwhile, topological bands with nonzero Chern numbers, more or less similar to the quantum anomalous Hall states in the honeycomb-lattice Haldane model, have also been obtained in other two-dimensional lattice models. Topological quantum phase transitions are characterized by the change of the topological invariants that identify the system. When the topological quantum phase transitions happen in a time-reversal symmetry breaking system, the Chern numbers of the corresponding bands may change accordingly. Due to the topological protection, these can only occur when the band gap closes. Such topological bands and related topological quantum phase transitions are also currently explored by manipulating ultra-cold atoms in optical lattice systems with artifical gauge potentials.In this article, we mainly investigate the topological quantum phase transitions in the two representative2D multi-band lattice models with superimposed staggered fluxes:the three-band kagome lattice model and the four-band square-octagon lattice model. First, the research background and involved basic theory of the kagome lattice model and square-octagon lattice model will be introduced. Then, cases of tight-binding electrons on the kagome lattice and square-octagon lattice with no superimposed fluxes are respectively illustrated. Finally, topological quantum phase transitions of the kagome lattice and square-octagon lattice with superimposed staggered fluxes are analysed. By tuning the staggered flux parameter φ on the nearest-neighbor hopping bonds and the hopping integral t2of the next-nearest-neighbor bonds, we show rich topological quantum phase transitions of both lattices. We have obtained the full rich phase diagram of the kagome lattice in the t2-φ parameter space and illustrated a series of topological quantum phase transitions of the square-octagon lattice with the parameter φ being fixed, in which topological bands with high Chern numbers have also been observed. Besides, we have also found interesting topological flat bands on the square-octagon lattice, especially when the next-next-nearest-neighbor hopping integra t3is included. The flatness ratio (the ratio of the band gap over the bandwidth) of the topological flat bands can reach the high value of about43, which might provides another ideal platform to realize the intriguing fractional quantum Hall effect (FQHE) without Landau levels in these topological flat bands.(J. Phys.:Condens.Matter, the first author).