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

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 σ 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, where l is the size of the subsystem.

Universal behavior of the Shannon mutual information in nonintegrable self-dual quantum chains

“…It is interesting to note that this leading behavior is the same as the Rényi entanglement entropy with Rényi index n = 2 [12]. The above conjecture was tested analytically and numerically for a large number of exactly integrable quantum chains [11,13,14], namely, a set of coupled harmonic oscillators (Klein Gordon theory), the XXZ quantum chain, the Ashkin-Teller, the spin-1 Fateev-Zamolodchikov, the Q-state Potts models (Q = 2, 3, 4) and the Z(Q) parafermionic models (Q = 5 − 8). Up to now, except for the chain of coupled harmonic oscillators, this conjecture was only tested numerically.…”

“…The connection between the quantum correlations and the entanglement properties of quantum many body systems provided us, in recent years, a powerful tool to detect [1] and classify quantum phase transitions (see [2] and references therein). Several measures of the entanglement were proposed along the years, like the von Neumann and Rényi entanglement entropies [2][3][4], the concurrence [5], the fidelity [6], etc. Among these measures the von Neumann and Rényi entanglement entropies are the most popular since in one dimension, where most of the critical chains are conformally invariant, they provide a way to calculate the central charge of the underlying conformal field theory (CFT), identifying the universality class of critical behavior.…”

“…Among these measures the von Neumann and Rényi entanglement entropies are the most popular since in one dimension, where most of the critical chains are conformally invariant, they provide a way to calculate the central charge of the underlying conformal field theory (CFT), identifying the universality class of critical behavior. Although interesting proposals were presented [7][8][9][10] it is quite difficult to measure these quantities in the laboratory, and the central charge of a critical chain has never been measured experimentally.…”

“…It is important to notice that differently from the von Neumann entanglement entropy and the von Neumann mutual information, which are basis independent, the Shannon entropy Sh and the Shannon mutual information I(A, B) are basis dependent quantities. In [11] it was conjectured that, for periodic critical quantum chains in their ground state, the Shannon mutual information shows universal features provided the ground state is expressed in some special bases, called conformal basis. A given basis of the Hilbert space of the quantum chain is related to a certain boundary condition in the time direction of the underlying (1+1)-Euclidean CFT.…”

“…The conformal basis are related to the boundary conditions that do not destroy the conformal invariance, as happens in the case of Dirichlet and Neumann boundary conditions. It was conjectured [11] that whenever the ground state wavefunction is expressed in the conformal basis, the leading finite-size scaling behavior of the Shannon mutual infor-mation for large systems and subsystem sizes is given by…”