Evolution of Entanglement Entropy in One-Dimensional Systems

TL;DR

This work analyzes how entanglement entropy between an interval and its complement evolves after a quantum quench in one-dimensional systems, using both conformal-field-theory path-integral methods and exact results for the transverse-field Ising chain. It reveals a universal linear growth with time up to t ~ ℓ/2, followed by saturation proportional to ℓ, with coefficients set by the initial state and regularization scale ε; a quasiparticle picture based on causality explains this dynamics. In the Ising chain, the asymptotic entropy density S_A(∞)/ℓ is nonzero and depends on h and h_0, with a maximum at 2 log 2 − 1 for certain quenches, while finite-time behavior shows robust linear growth and a rich dependence on quench protocol. The results bridge nonequilibrium quantum dynamics and quantum information measures, highlighting the role of high-energy regularization, and point to interesting extensions to higher dimensions and more complex geometries.

Abstract

We study the unitary time evolution of the entropy of entanglement of a one-dimensional system between the degrees of freedom in an interval of length l and its complement, starting from a pure state which is not an eigenstate of the hamiltonian. We use path integral methods of quantum field theory as well as explicit computations for the transverse Ising spin chain. In both cases, there is a maximum speed v of propagation of signals. In general the entanglement entropy increases linearly with time $t$ up to t=l/2v, after which it saturates at a value proportional to l, the coefficient depending on the initial state. This behavior may be understood as a consequence of causality.

Figures (7)

• Figure 1: Space-imaginary time regions for the density matrix in (\ref{['dm0']})
• Figure 2: Leading asymptotic entanglement entropy for unit length of the interval, as function of $h$ Eq. (\ref{['Stinfinito']}). (a) The case $h_0=\infty$ (full line) is compared with the numerical results (crosses) obtained by diagonalizing $\Gamma^A_\ell$ for $\ell=200$ and using the exact coefficients Eq. (\ref{['glh0infinity']}). (b) As in (a), but for several finite values of $h_0$.
• Figure 3: Entanglement entropy for the quench from $h_0=\infty$ to $h=1$, for various $\ell$. The dashed lines are the leading asymptotic results for large $\ell$, cf. Eq. (\ref{['Stinfinito']}). The inset shows the derivative with respect to the time of $S_{100}(t)$.
• Figure 4: $S_{60}(t)$ for the quench from $h_0=\infty,\,5,\,2,\,1.5,\,1.1$ to $h=1$. The dashed lines are the leading asymptotic results for large $\ell$ Eq. (\ref{['Stinfinito']}).
• Figure 5: $S_{100}(t)$ for the quench from $h_0=\infty$ to $h=5,2,1.5,1.01,1$. The dashed lines are the leading asymptotic results for large $\ell$ Eq. (\ref{['Stinfinito']}). The inset shows the rescaling of the curves, according to the asymptotic value $S_{100}(\infty)$.