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The relaxation of many-body system is still a challenging problem that hasnot well been understood. In this work we exactly calculate the dynamics of thequantum $XY$ model with boundary dissipation, in which the density matrix interms of Majorana operators can be decoupled into independent subspacesrepresented by different number of Majorana fermions. The relaxation ischaracterized by multiply time scales, and in the long-time limit it isdetermined by the single particle relaxation process in a typical time scale$T^*$. For the bulk bands, we find $T^* \propto N^3/\gamma n^2 $ in the weakdissipation limit; and $T^* \propto \gamma N^3/ n^2$ in the strong dissipationlimit, where $N$ is the chain length, $\gamma$ is the dissipation rate and $n$is the band index. For the edge modes $T^* \propto 1/\gamma$, indicating ofmost vulnerable to dissipation in the long chain limit. These results arecounter-intuitive because it means any weak dissipation can induce relaxation,while strong dissipation can induce weak relaxation. We find that these twolimits correspond to two different physics, which are explained based on thefirst and second-order perturbation theory in an equivalent non-Hermitianmodel.Furthermore, we show that even in the long chain limit the relaxation mayexhibit strong odd-even effect. These results shade new insight into thedynamics of topological qubits in environment.

Anomalous relaxation and multiply time scales in the quantum $XY$ model with boundary dissipation