Quantum renormalization group of the XY model in two dimensions

We investigate entanglement and quantum phase transition in a two-dimensional Heisenberg anisotropic spin-1 /2 XY model, using the quantum renormalization-group method (QRG) on a square lattice of N ×N sites. The entanglement through geometric average of concurrences is calculated after each step of the QRG. We show that the concurrence achieves a nonzero value at the critical point more rapidly as compared to the one-dimensional case. The relationship between the entanglement and the quantum phase transition is studied. The evolution of entanglement develops two saturated values corresponding to two different phases. We compute the derivative of the concurrence, which is found to be discontinuous at the critical point γ =0 , and indicates a second-order phase transition in the spin system. Further, the scaling behavior of the system is investigated by computing the derivative of the concurrence in terms of the system size.