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

TL;DR

This paper develops formulas for formation probabilities in free fermion systems, analyzes their scaling and conformal field theory correspondence in the transverse-field XY chain, and studies the time evolution of Shannon information after a quantum quench.

Contribution

It introduces a general formula for formation probabilities in free fermion systems and explores their scaling, boundary effects, and relation to conformal field theory, along with the dynamics of Shannon information post-quench.

Findings

Formation probabilities follow CFT predictions in the Ising model.

Only configurations respecting the filling factor follow CFT in the XX chain.

Shannon information saturates at half the subsystem size after a quantum quench.

Abstract

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 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 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 to 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 intial state considered in this paper, we demonstrate that the Shannon information after quantum quench first increases with the time and then saturates at time $t^*=\frac{l}{2}$, where $l$ is the size of the subsystem.