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While the internal structure of Abelian topological order is well understood,how to characterize the non-Abelian topological order is an outstanding issue.We propose a distinctive scheme based on the many-body Chern number matrix tocharacterize non-Abelian multicomponent fractional quantum Hall states. As aconcrete example, we study the many-body ground state of two-component bosonsat the filling faction $\nu=4/3$ in topological flat band models. Utilizingdensity-matrix renormalization group and exact diagonalization calculations, wedemonstrate the emergence of non-Abelian spin-singlet fractional quantum Halleffect under three-body interaction, whose topological nature is classified bysix-fold degenerate ground states and a fractionally quantized Chern numbermatrix.

Chern number matrix of the non-Abelian spin-singlet fractional quantum Hall effect