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The non-Abelian topological order has attracted a lot of attention for itsfundamental importance and exciting prospect of topological quantumcomputation. However, explicit demonstration or identification of thenon-Abelian states and the associated statistics in a microscopic model is verychallenging. Here, based on density-matrix renormalization group calculation,we provide a complete characterization of the universal properties of bosonicMoore-Read state on Haldane honeycomb lattice model at filling number $\nu=1$for larger systems, including both the edge spectrum and the bulk anyonicquasiparticle (QP) statistics. We first demonstrate that there are threedegenerating ground states, for each of which there is a definite anyonic fluxthreading through the cylinder. We identify the nontrivial countings for theentanglement spectrum in accordance with the corresponding conformal fieldtheory. Through inserting the $U(1)$ charge flux, it is found that two of theground states can be adiabatically connected through a fermioniccharge-$\textit{e}$ QP being pumped from one edge to the other, while theground state in Ising anyon sector evolves back to itself. Furthermore, wecalculate the modular matrices $\mathcal{S}$ and $\mathcal{U}$, which containall the information for the anyonic QPs. In particular, the extracted quantumdimensions, fusion rule and topological spins from modular matrices positivelyidentify the emergence of non-Abelian statistics following the $SU(2)_2$Chern-Simons theory.

Topological Characterization of Non-Abelian Moore-Read State using Density-Matrix Renormailzation Group