Equivalence of Critical Scaling Laws for Many-Body Entanglement in the Lipkin-Meshkov-Glick Model

We establish a relation between several entanglement properties in the Lipkin-Meshkov-Glick model, which is a system of mutually interacting spins embedded in a magnetic field. We provide analytical proofs that the single-copy entanglement and the global geometric entanglement of the ground state close to and at criticality behave as the entanglement entropy. These results are in deep contrast to what is found in one- dimensional spin systems where these three entanglement measures behave differently.

Figure 1

Behavior of the geometric entanglement (left panel ) and single-copy entanglement (right panel) as a function of the magnetic field for different system sizes $N=16$, 32, 64, 128, and 256 (from bottom to top). Upper lines are the thermodynamical limit given in Eqs. (6, 12). Here, $\mathcal{G}$ has been obtained by a numerical minimization of the fidelity over all coherent states.

Figure 2

Behavior of $\mathcal{G}$ and $\mathcal{S}$ at the critical point as a function of $N$. The dotted line has a slope $1/6$, as predicted by the scaling hypothesis. The inset shows the behavior of the coefficient $\chi$ (see text) as a function of $N$. The dotted line in the inset also has a slope $1/6$.