There is a typing mistake in the expression of γ 4 given in the Appendix. The actual expression is γ (5+α 1 +γ 2) 2 √ 2α 2 , which is used in the calculations throughout the paper. In the caption of Fig. 6, "(see Fig. 6)" should read "(see Fig. 4)".

We investigate entanglement and quantum phase transition (QPT) in a two-dimensional Heisenberg anisotropic spin-1/2 XY model, using 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 non zero value at the critical point more rapidly as compared to 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 first 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 behaviour of the system is investigated by computing the first derivative of the concurrence in terms of the system size. * Electronic address: kk@qau.edu.pk arXiv:1708.03365v1 [quant-ph]

In this article a comparative study of the renormalization of entanglement in one, two and three dimensional space and its relation with quantum phase transition (QPT) near the critical point is presented by implementing the Quantum Renormalization Group (QRG) technique. Adopting the Kadanoff's block approach, numerical results for the concurrence are obtained for the spin -1/2 XY model in all the spatial dimensions. The results show similar qualitative behavior as we move from the lower to the higher dimensions in space but the number of iterations reduces for achieving the QPT in the thermodynamic limit. We find that in the two dimensional and three dimensional spin -1/2 XY model, maximum values of the concurrence reduce by the factor of 1/n (n = 2, 3) with reference to the maximum value of one dimensional case. Moreover, we study the scaling behavior and the entanglement exponent. We compare the results for one, two and three dimensional cases and illustrate how the system evolves near the critical point.1 arXiv:1610.00761v1 [quant-ph]

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