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

The relaxation of many-body system is still a challenging problem that has not well been understood. In this work we exactly calculate the dynamics of the quantum $XY$ model with boundary dissipation, in which the density matrix in terms of Majorana operators can be decoupled into independent subspaces represented by different number of Majorana fermions. The relaxation is characterized by multiply time scales, and in the long-time limit it is determined 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 weak dissipation limit; and $T^* \propto \gamma N^3/ n^2$ in the strong dissipation limit, 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 of most vulnerable to dissipation in the long chain limit. These results are counter-intuitive because it means any weak dissipation can induce relaxation, while strong dissipation can induce weak relaxation. We find that these two limits correspond to two different physics, which are explained based on the first and second-order perturbation theory in an equivalent non-Hermitian model.Furthermore, we show that even in the long chain limit the relaxation may exhibit strong odd-even effect. These results shade new insight into the dynamics of topological qubits in environment.