Bi-partite and global entanglement in a many-particle system with collective spin coupling

Bipartite and global entanglement are analyzed for the ground state of a system of $N$ spin 1/2 particles interacting via a collective spin-spin coupling described by the Lipkin-Meshkov-Glick (LMG) Hamiltonian. Under certain conditions which includes the special case of a super-symmetry, the ground state can be constructed analytically. In the case of an anti-ferromagnetic coupling and for an even number of particles this state undergoes a smooth crossover as a function of the continuous anisotropy parameter $\gamma $ from a separable ($\gamma =\infty $) to a maximally entangled many-particle state ($\gamma =0$). From the analytic expression for the ground state, bipartite and global entanglement are calculated. In the thermodynamic limit a discontinuous change of the scaling behavior of the bipartite entanglement is found at the isotropy point $\gamma =0$. For $% \gamma =0$ the entanglement grows logarithmically with the system size with no upper bound, for $\gamma \neq 0$ it saturates at a level only depending on $\gamma $. For finite systems with total spin $J=N/2$ the scaling behavior changes at $\gamma =\gamma _{\mathrm{crit}}=1/J$.