Quantum Mpemba effect of localization in the dissipative mosaic model

The quantum Mpemba effect in open quantum systems has been extensively studied, but a comprehensive understanding of this phenomenon remains elusive. In this paper, we conduct an analytical investigation of the dissipative dynamics of single excitations in the The mosaic model. Surprisingly, we discover that the presence of an asymptotic mobility edge, denoted as $E_c^{\infty}$, can lead to unique dissipation behavior, serving as a hallmark of the quantum Mpemba effect. Especially, it is found that the energy level $E_c^{\infty}$ exhibits a global periodicity in the real configuration, which acts to inhibit dissipation in the system. Conversely, when the system deviates from $E_c^{\infty}$, the quasidisorder sets in, leading to increased dissipative effects due to the broken of periodicity. Furthermore, we find that the rate of dissipation is closely linked to the localization of the initial state. As a result, the quantum Mpemba effect can be observed clearly by a measure of localization.