Formation probabilities and Shannon information and their time evolution after quantum quench in the transverse-field XY chain

We first provide a formula to calculate the probability of occurrence of different configurations (formation probabilities) in a generic free fermion system. We then study the scaling of these probabilities with respect to the size in the case of the critical transverse-field XY chain in the sigma(z) bases. In the case of the transverse field Ising model, we show that all the crystal configurations follow the formulas expected from conformal field theory (CFT). In the case of the critical XX chain, we show that the only configurations that follow the formulas of the CFT are the ones which respect the filling factor of the system. By repeating all the calculations in the presence of open and periodic boundary conditions we find further support for our classification of different configurations. Using the developed technique, we also study Shannon information of a subregion in our system. In this respect we distinguish particular configurations that are more important in the study of the scaling limit of the Shannon information of the subsystem. Finally, we study the evolution of formation probabilities, Shannon information, and Shannon mutual information after a quantum quench in free fermion system. In particular, for the initial state considered in this paper, we demonstrate that the Shannon information after quantum quench first increases with the time and then saturates at time t* = l/2, where l is the size of the subsystem.