Anomalous behaviors in dynamical quantum phase transitions of long-range topological chains

The dynamics of quantum phase transitions of topological systems has attracted interest. Here we consider the effect of long-range hopping terms on the dynamic behavior. The long-range hoppings preserve chiral symmetry and decay as a function of distance, with an exponent<i>β</i>. We also consider a situation where the hoppings are modulated by oscillations. We compare the results with those obtained for short-range models such as the Su-Schrieffer-Heeger model or its extension to second neighboring cells. While often the lowest critical times coincide with the first peak of the Loschmidt rate, several anomalous cases are identified. Also, the number of critical times often exceeds the absolute value of the change of the winding number in a quench. Various correlation functions are considered, and their time-dependence may often be used to identify the early-time dynamical phase transitions.