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

In this paper, we study various quantum phase transitions in the XY Heisenberg chain in the presence of a transverse magnetic field using a multi-partite entanglement (Meyer-Wallach) measure. Providing an analytical expression of the measure, we are able to extract critical exponents of Ising quantum phase transition via finite-size scaling for finite-size systems. Then, the general behavior of the measure is considered in the thermodynamic limit indicating that 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. We also calculate an analytic expression for the isotropic (XX) model and show that global entanglement can precisely identify the level-crossing points. Finally, it is shown that global entanglement exhibits an interesting behavior around the anisotropic transition line in the finite size limit. In the thermodynamic limit, we show that global entanglement exhibits a cusp-singularity across the Ising and anisotropic transition. 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.