Multipartite Entanglement Signature of Quantum Phase Transitions

We derive a general relation between the nonanalyticities of the ground state energy and those of a subclass of the multipartite generalized global entanglement (GGE) measure defined by de Oliveira et al. [Phys. Rev. A 73, 010305(R) (2006)] for many-particle systems. We show that GGE signals both a critical point location and the order of a quantum phase transition (QPT). We also show that GGE allows us to study the relation between multipartite entanglement and QPTs, suggesting that multipartite but not bipartite entanglement is favored at the critical point. Finally, using GGE we were able, at a second-order QPT, to define a diverging entanglement length (EL) in terms of the usual correlation length. We exemplify this with the $XY$ spin-$1/2$ chain and show that the EL is half the correlation length.

Figure 1

Upper and lower bounds of $\mathcal{G}(2,1)$ for the $XY$ chain for three values of the anisotropy: $\gamma =1$ (red solid line), 0.6 (blue dashed line), and 0.2 (black dotted line).

Figure 2

Derivative of the lower bound of $\mathcal{G}(2,1)$ for three values of anisotropy: $\gamma =1$ (red solid line), 0.6 (blue dashed line), and 0.2 (black dotted line). The second phase transition is also imprinted for the $\gamma =0.2$ as the curve crosses the abscissa at $\lambda =1/\sqrt{1-{\gamma }^2}$.

Figure 3

Lower bound of $\mathcal{G}(2,n)$ for $\gamma =0.6$ and for three values of $n$: $n=1$ (solid line), 2 (dashed line), and 7 (long-dashed line).