We derive an exact renormalization group recursion relation for the Loschmidt amplitude of the quantum $Q$-state clock model and the quantum $Q$-state Potts model in one dimension. The renormalization group flow is discussed in detail. The fixed points of the renormalization group flow are found to be complex in general. These fixed points control the dynamical phases of the two models, giving rise to nonanalyticities in its Loschmidt rate function, for both the pure and the disordered system. For the quench protocols studied, dynamical quantum phase transitions are found to occur in the clock model for all $Q\mathrm{s}$ considered, while in the Potts model, they only occur when $Q<4$.

• Received 19 June 2019
• Revised 11 January 2020

DOI:https://doi.org/10.1103/PhysRevB.101.014305