Spatial dependence of entanglement renormalization in $XY$ model

TL;DR

This study investigates how entanglement behaves across different spatial dimensions in the $XY$ model near quantum phase transitions using Quantum Renormalization Group, revealing dimensional effects on entanglement and critical scaling.

Contribution

It provides a comparative analysis of entanglement renormalization in 1D, 2D, and 3D $XY$ models, highlighting dimensional influences on entanglement and phase transition behavior.

Findings

Maximum concurrence decreases with higher dimensions.

Number of iterations to reach QPT reduces in higher dimensions.

Entanglement scaling behavior varies across dimensions.

Abstract

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.