Exact zeros of Loschmidt echo and quantum speed limit time for dynamical quantum phase transition in finite size systems

We study exact zeros of Loschmidt echo and quantum speed limit time for dynamical quantum phase transition in finite size systems. Our results illustrate that exact zeros of Loschmidt echo exist even in finite size quantum systems when the postquench parameter takes some discrete values in regions with the corresponding equilibrium phase different from the initial phase. As the system size increases and tends to infinity, the discrete parameters distribute continuously in the parameter regions. We further analyse the first time for the appearance of the exact zero of Loschmidt echo which is known as the quantum speed limit time {\tau}QSL. We demonstrate that the maximal value of $\tau_QSL$ is proportional to L and approaches infinity in the thermodynamical limit, when we quench the initial non-critical state to the critical phase. We also calculate the minimal value of $\tau_QSL$ and find that its behaviour is dependent on the phase of initial state.