Ground-state and dynamical properties of a spin-$S$ Heisenberg star

We generalize the Heisenberg star consisting of a spin-1/2 central spin and a homogeneously coupled spin bath modeled by the XXX ring [Richter J and Voigt A 1994 \emph{J. Phys. A: Math. Gen.} \textbf{27} 1139-1149] to the case of arbitrary central-spin size $S<N/2$, where $N$ is the number of bath spins. We describe how to block-diagonalize the model based on the Bethe ansatz solution of the XXX ring, with the dimension of each block Hamiltonian $\leq 2S+1$. We obtain all the eigenenergies and explicit expressions of the sub-ground states in each $l$-subspace with $l$ being the total angular momentum of the bath. Both the eigenenergies and the sub-ground states have distinct structures depending whether $S\leq l$ or $l<S$. The absolute ground-state energy and the corresponding $l$ as functions of the intrabath coupling are numerically calculated for $N=16$ and $S=1,2,\cdots,7$ and their behaviors are quantitatively explained in the weak and strong intrabath coupling limits. We then study the dynamics of the antiferromagnetic order within an XXX bath prepared in the N\'eel state. Effects of the initial state of the central spin, the value of $S$, and the system-bath coupling strength on the staggered magnetization dynamics are investigated. By including a Zeeman term for the central spin and the anisotropy in the intrabath coupling, we also study the polarization dynamics of the central spin for a bath prepared in the spin coherent state. Under the resonant condition and at the isotropic point of the bath, the polarization dynamics for $S>1/2$ exhibits collapse-revival behaviors with fine structures. However, the collapse-revival phenomena is found to be fragile with respect to anisotropy of the intrabath coupling.