Entanglement in XYZ model on a spin-star system: Anisotropy vs. field-induced dynamics

We consider a star-network of n=n₀+n_p spin-frac{1}{2} particles, where interaction between n₀ central spins and n_p peripheral spins are of the XYZ-type. In the limit n₀/n_pll 1, we show that for odd n, the ground state is doubly degenerate, while for even n, the energy gap becomes negligible when n is large, inducing an effective double degeneracy. In the same limit, we show that for vanishing xy-anisotropy γ, bipartite entanglement on the peripheral spins computed using either a partial trace-based, or a measurement-based approach exhibits a logarithmic growth with n_p, where the sizes of the partitions are typically sim n_p/2. This feature disappears for γneq 0, which we refer to as the anisotropy effect. Interestingly, when the system is taken out of equilibrium by the introduction of a magnetic field of constant strength on all spins, the time-averaged bipartite entanglement on the periphery at the long-time limit exhibits a logarithmic growth with n_p irrespective of the value of γ. We further study the n₀/n_pgg 1 and n₀/n_p→ 1 limits of the model, and show that the behaviour of bipartite peripheral entanglement is qualitatively different from that of the n₀/n_pll 1 limit.