Topological phase transitions and flat bands on an islamic lattice.

We construct an islamic lattice by considering the nearest-neighbor hoppings with staggered magnetic fluxes and the next-nearest-neighbor hoppings. This model supports abundant quantum phases for various values of filling fractions. At $1/4$ filling, there exist Chern insulator (CI) phases with Chern numbers $C=\pm 1,~-2$ and a zero-Chern-number topological insulator (ZCNTI) phase. At $3/8$ filling, several CI phases with Chern numbers $C= \pm1,~3$ and the ZCNTI phase are obtained. For the filling fraction 3/4, CI phases with Chern numbers $C=\pm 1,~2$ and two ZCNTI phase areas appear. Interestingly, these ZCNTI phases host both robust corner states and gapless edge states which can be characterized by the quantized polarization and quadrupole moment. We further find that staggered magnetic fluxes can give rise to the ZCNTI state at $1/4$ and $3/4$ fillings. Phase diagrams for filling fractions $1/8$, $1/2$, $5/8$ and $7/8$ are presented as well. In addition, flat bands are obtained for various filling fractions by tuning the hopping parameters. At 1/8 filling, a best topological flat band (TFB) with flatness ratio about 12 appears. Several trivial flat bands but with total Chern number $|C|=1$ emerge in this model and exactly flat band is found at 3/8 filling. We further investigate $\nu=1/2$ fractional Chern insulate state when hard-core bosons fill into this TFB model.