Recent experimental measurements have discovered the fracture toughness of monolayer h-BN is much large than its intrinsic surface energy and based on DFT calculations toughening mechanisms of crack tip bifurcation and edge swapping due to the lattice asymmetry have been proposed. In this study, combining theoretical analysis and phase field (PF) modeling, we study the effect of the polarized edge stress and anisotropic fracture energy on the fracture of h-BN. It is found that polarized edge stress can contribute a mode II stress intensity factor (SIF) travelling with the crack tip and the competition between local lattice asymmetry and gloabl extrenal loading can result in a zigzag crack path with swaping branches.