Generation of concurrence in a generalized central spin model with a three-spin interacting environment

We consider the three-spin Ising model to study the effect of three-spin interacting term on bi-partitie entanglement between adjacent spins. The three-dominated disordered region has tri-partite entanglement causing a vanishingly small concurrence, while it acquires maximum value around the critical points. Considering the above model as an environment, we construct a generalized central spin model where two central spins, initially in an unentangled pure state, are coupled locally to two distinct sites of the environmental spin chain. We study the generation of mixed state entanglement between the central spins when the transverse field of the environment is kept fixed, and suddenly quenched, referring to equilibrium and non-equilibrium dynamics of the central spins, respectively. For the critical environment in the equilibrium, the concurrence shows a dip-revival structure governed by quasi-particle movement. In the non-equilibrium study, we find an initial growth of concurrence followed by a two-stage fall for the inter-phase quench which is governed by dynamic decoherence channels. The central spins are maximally entangled for a quench in the vicinity of a multicritical point, which arises due to three-spin interaction only. The concurrence becomes long-lived for an intra-phase quench, and this sustainability depends on the strength of the three-spin interaction. Therefore, the three-spin interaction indeed helps in generating bi-partite entanglement in the central spins.