Quantum phase transition in the XXZ central spin model

We investigate the quantum phase transition (QPT) in the XXZ central spin model, which can be described as a spin-21 particle coupled to N bath spins. In general, the QPT is supposed to occur only in the thermodynamical limit. In contrast, we present that the central spin model exhibits a normal-to-superradiant phase transition in the limit where the ratio of the transition frequency of the central spin to that of the bath spins and the number of bath spins tend to infinity. We give the low-energy effective Hamiltonian analytically in the normal phase and the superradiant phase, and we find that the longitudinal interaction A can significantly influence the excitation number and the coherence of the ground state. These two quantities are remarkably enhanced for the negative longitudinal interaction while suppressed for the positive longitudinal interaction. In addition, the finite-size effect on the central spin model is also illustrated through the mean-field analysis. We further exploit the quantum Fisher information to characterize the QPT and propose a measurement scheme that can be applied in practice. This work builds a connection between the qubit-spin systems and the qubit-field systems, which provides a possibility for the realization of criticality-enhanced quantum sensing in central spin systems.