Abstract
In this thesis, we using the tight-binding anisotropic honeycomb lattice to investigate the zero-energy edge states and the bipartite entanglement for the total system at half filling and $T=0$ by the entanglement entropy spectrum. The zero-energy edge states for the system having two infinite and parallel bearded or zigzag edges are found to survive only when the the system with PBC has two Dirac points. On the contrary, there is no zero-energy edge state for the system having two infinite and parallel armchair edges. On the other hand, we numerically calculate the entanglement entropy spectrum of the system $A$ that has finite bearded edges on up and down and finite armchair edges on left and right. The maximal entangled states of the entanglement entropy spectrum are found to have one-to-one correspondence to the zero-energy edge states of the system $A$ with OBC which means we numerically proves the one-to-one coorespondence between the zero-energy edge states and the maximal entangled states even when the edge is finite. In addition, we also find the zero-energy edge state of the system with longer bearded edge can be created by combining the zero-energy edge states of the system with shorter bearded edge.