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By means of finite size exact diagonalization we theoretically study theelectronic many-body effects on the nearly flat-band structure withtime-reversal symmetry in a checkerboard lattice model and identify thetopological nature of two quantum phases, with ninefold and threefolddegeneracy, that appear, respectively, at small and large values $\lambda$ of anearest neighbor spin dependent interaction. Numerical evidences from theevolution of low-lying energy spectra and Berry phases with bothspin-independent and spin-dependent twisted boundary conditions reveal thatthese two different ground states share the same topological spin Chern number.Quantum phase transition between these two states by tuning $\lambda$ isconfirmed by evaluating the energy spectra and quasispin excitation spectraclosing. At last, the counting rules of spin excitation spectra aredemonstrated as the fingerprints of the fractionalized quantum spin Hallstates.

Interaction driven quantum phase transition in fractional quantum spin Hall effects