Orthogonality catastrophe and the speed of quantum evolution in a qubit-spin-bath system

The orthogonality catastrophe(OC) of quantum many-body systems is an important phenomenon in condensed matter physics. Recently, an interesting relationship between the OC and the quantum speed limit(QSL) was shown(Fogarty 2020 Phys. Rev. Lett. 124 110601). Inspired by the remarkable feature, we provide a quantitative version of the quantum average speed as another different method to investigate the measure of how it is close to the OC dynamics. We analyze the properties of an impurity qubit embedded into an isotropic Lipkin-Meshkov-Glick spin model,and show that the OC dynamics can also be characterized by the average speed of the evolution state. Furthermore, a similar behavior of the actual speed of quantum evolution and the theoretical maximal rate is shown which can provide an alternative speed-up protocol allowing us to understand some universal properties characterized by the QSL.