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Loschmidt Echo and Berry phase of the quantum system coupled to the XY spin chain: Proximity to quantum phase transition

We study the Loschmidt echo (LE) of a coupled system consisting of a central spin and its surrounding environment described by a general XY spin-chain model. The quantum dynamics of the LE is shown to be remarkably influenced by the quantum criticality of the spin chain. In particular, the decaying behavior of the LE is found to be controlled by the anisotropy parameter of the spin chain. Furthermore, we show that due to the coupling to the spin chain, the ground-state Berry phase for the central spin becomes nonanalytical and its derivative with respect to the magnetic parameter $λ$ in spin chain diverges along the critical line $λ=1$, which suggests an alternative measurement of the quantum criticality of the spin chain.