Global entanglement and quantum phase transitions in the transverse XY Heisenberg chain

TL;DR

This paper investigates quantum phase transitions in the XY Heisenberg chain using the Meyer-Wallach measure of global entanglement, providing analytic expressions and demonstrating its effectiveness in identifying critical points and exponents.

Contribution

It introduces analytic formulas for global entanglement in finite and infinite systems and shows its utility in characterizing various quantum phase transitions in the XY Heisenberg chain.

Findings

Global entanglement accurately identifies critical points.

Analytic expressions for entanglement in finite and infinite systems.

Distinct behavior of entanglement at phase transition points.

Abstract

We provide a study of various quantum phase transitions occurring in the XY Heisenberg chain in a transverse magnetic field using the Meyer-Wallach (MW) measure of (global) entanglement. Such a measure, while being readily evaluated, is a multipartite measure of entanglement as opposed to more commonly used bipartite measures. Consequently, we obtain analytic expression of the measure for finite-size systems and show that it can be used to obtain critical exponents via finite-size scaling with great accuracy for the Ising universality class. We also calculate an analytic expression for the isotropic (XX) model and show that global entanglement can precisely identify the level-crossing points. The critical exponent for the isotropic transition is obtained exactly from an analytic expression for global entanglement in the thermodynamic limit. Next, the general behavior of the measure is calculated in the thermodynamic limit considering the important role of symmetries for this limit. The so-called oscillatory transition in the ferromagnetic regime can only be characterized by the thermodynamic limit where global entanglement is shown to be zero on the transition curve. Finally, the anisotropic transition is explored where it is shown that global entanglement exhibits an interesting behavior in the finite-size limit. In the thermodynamic limit, we show that global entanglement shows a cusp singularity across the Ising and anisotropic transition, while showing non-analytic behavior at the XX multicritical point. It is concluded that global entanglement, despite its relative simplicity, can be used to identify all the rich structure of the ground-state Heisenberg chain.