Universality of the negativity in the Lipkin-Meshkov-Glick model

The entanglement between noncomplementary blocks of a many-body system, where a part of the system forms an ignored environment, is a largely untouched problem without analytic results. We rectify this gap by studying the logarithmic negativity between two macroscopic sets of spins in an arbitrary tripartition of a collection of mutually interacting spins described by the Lipkin-Meshkov-Glick Hamiltonian. This entanglement measure is found to be finite and universal at the critical point for any tripartition whereas it diverges for a bipartition. In this limiting case, we show that it behaves as the entanglement entropy, suggesting a deep relation between the scaling exponents of these two independently defined quantities which may be valid for other systems.

Figure 1

Logarithmic negativity $\mathcal{L}$ as a function of magnetic field $h$ for an equal tripartition ${\tau }_1={\tau }_2={\tau }_3=1/3$ and for two different values of the anisotropy parameter $\gamma$. In the thermodynamic limit, corresponding to the red (gray) lines, at the critical point, $\mathcal{L}$ is universal (independent of $\gamma$). Black lines correspond to numerical data for $N=90$, 150, 210, which match the analytical prediction $N=\infty$ increasingly well. Note also that $\mathcal{L}$ vanishes for $h=\sqrt{\gamma }$ where the ground state is separable [12].

Figure 2

Logarithmic negativity $\mathcal{L}$ as a function of the magnetic field $h$ for a bipartition ${\tau }_1=1/3,{\tau }_3=2/3,$ and $\gamma =1/2$, shown as red (gray) line. In the broken phase $h<1$ we plot $\mathcal{L}+\ln 2$ for a reason detailed in the text. Black lines corresponds to exact diagonalization data for $N=180$, 270, 360. Inset: Scaling of $\mathcal{L}$ with system size $N=120,150,\dots ,1080$ from exact diagonalization (black circles) approaching a linear dependence on $\ln N$ with slope $1/6$ (dashed line) for large $N$.