Entanglement and correlation functions following a local quench: a conformal field theory approach

TL;DR

This work analyzes the non-equilibrium dynamics after a local quench in a one-dimensional gapless system using conformal field theory. By mapping the quench setup to slit geometries and employing the replica trick, it derives explicit time-dependent entanglement entropies for various bipartitions, uncovering a horizon-like growth and plateaus governed by the central charge $c$ and boundary entropy, with quasiparticles emitted at the joining point driving the dynamics. The study also develops the time evolution of one- and two-point correlation functions, including exact results in the Gaussian model and general CFT predictions via the four-point ratio $\eta$ and a boundary-dependent function $F(\eta)$, highlighting the role of boundary operator content. A finite-interval decoupled case is treated approximately, giving a time-dependent form for $S_A(t)$ with a sinusoidal factor, and the authors discuss the regime of validity and connections to numerics. Overall, the paper provides a coherent, analytic framework linking local quenches, entanglement growth, and correlation functions through boundary CFT and a semiclassical quasiparticle picture, with predictions testable in lattice models and cold-atom experiments.

Abstract

We show that the dynamics resulting from preparing a one-dimensional quantum system in the ground state of two decoupled parts, then joined together and left to evolve unitarily with a translational invariant Hamiltonian (a local quench), can be described by means of quantum field theory. In the case when the corresponding theory is conformal, we study the evolution of the entanglement entropy for different bi-partitions of the line. We also consider the behavior of one- and two-point correlation functions. All our findings may be explained in terms of a picture, that we believe to be valid more generally, whereby quasiparticles emitted from the joining point at the initial time propagate semiclassically through the system.

Figures (3)

• Figure 1: The four different bipartitions of the line we consider here.
• Figure 2: $S_A(t)$ for a general slit $B=[\ell_1,\ell_2]$. The plots are done for finite $\epsilon=10^{-3}$, and they are indistinguishable from the asymptotic result. In all the plots $\ell_1-\ell_2=20$. Left: $\ell_2<0$, for several $\delta= \ell_1+\ell_2$. Right: $\ell_2>0$, for several $\ell_1$.
• Figure 3: Left: Space-time region for the density matrix of a semi-infinite system. Center: Mapping of the $z$-plane under the transformation Eq. (\ref{['mapp1']}). Right: Final map to the half-plane of the $\zeta$ plane given by Eq. (\ref{['mapp2']}) when $\ell\gg\epsilon$.