Interaction driven quantum phase transition in fractional quantum spin   Hall effects
  By means of finite size exact diagonalization we theoretically study the electronic many-body effects on the nearly flat-band structure with time-reversal symmetry in a checkerboard lattice model and identify the topological nature of two quantum phases, with ninefold and threefold degeneracy, that appear, respectively, at small and large values $\lambda$ of a nearest neighbor spin dependent interaction. Numerical evidences from the evolution of low-lying energy spectra and Berry phases with both spin-independent and spin-dependent twisted boundary conditions reveal that these two different ground states share the same topological spin Chern number. Quantum phase transition between these two states by tuning $\lambda$ is confirmed by evaluating the energy spectra and quasispin excitation spectra closing. At last, the counting rules of spin excitation spectra are demonstrated as the fingerprints of the fractionalized quantum spin Hall states. 
Wei Li, D. N. Sheng, C. S. Ting, and Yan Chen
cond-mat.str-el