Julia Set in Quantum Evolution: The case of Dynamical Quantum Phase Transitions

Abstract: Dynamical quantum phase transitions (DQPTs) are a class of non-equilibrium phase transitions that occur in many-body quantum systems during real-time evolution, rather than through parameter tuning as in conventional phase transitions. This paper presents an exact analytical approach to studying DQPTs by combining complex dynamics with the real-space renormalization group (RG). RG transformations are interpreted as iterated maps on the complex plane, establishing a connection between DQPTs and the Julia set, the fractal boundary separating the basins of attraction of the stable fixed points. This framework is applied to a quantum quench in the one-dimensional transverse field Ising model, and the sensitivity of DQPTs to changes in boundary conditions is examined. In particular, it is demonstrated how the topology of the spin chain influences the occurrence of DQPTs. Additionally, aqualitative argument based on quantum speed limits is provided to explain the suppression of DQPTs under certain boundary modifications.