Entanglement negativity after a global quantum quench

TL;DR

This work investigates how entanglement negativity between two intervals evolves after a global quench in a 1+1D conformal field theory, combining analytic twist-field methods with exact harmonic-chain numerics. The authors show that negativity follows the quasi-particle spreading picture, yielding space-time scaling forms that parallel entanglement entropy and mutual information, in both adjacent and disjoint geometries. They uncover two distinctive lattice effects: a late birth of entanglement, where negativity rises slightly after the continuum prediction, and a sudden death of entanglement, where negativity vanishes at finite times before revivals. These results extend the quasi-particle framework to mixed-state entanglement measures and clarify finite-size and lattice corrections that are relevant for simulations and potential experiments.

Abstract

We study the time evolution of the logarithmic negativity after a global quantum quench. In a 1+1 dimensional conformal invariant field theory, we consider the negativity between two intervals which can be either adjacent or disjoint. We show that the negativity follows the quasi-particle interpretation for the spreading of entanglement. We check and generalise our findings with a systematic analysis of the negativity after a quantum quench in the harmonic chain, highlighting two peculiar lattice effects: the late birth and the sudden death of entanglement.

Figures (10)

• Figure 1: Configurations of two intervals in the infinite line that we consider: adjacent intervals (top) and disjoint intervals (bottom).
• Figure 2: Graphical representation for the quasi-particle spreading of entanglement (for the case with all quasi-particles having the same velocity $v=1$ as in a CFT). The quasi-particles emitted from every point at $t=0$ and reaching one $A_1$ (red) and the other $A_2$ (blue) are responsible of the entanglement between them. The entanglement at a given time $t$ is proportional to the section of the green shaded area, which is the intersection of the light cones starting from all the points of $A_1$ and $A_2$ (in the figure these lengths are the braces). The time-dependence of the entanglement obtained in this way are depicted as purple curves on the right for a single interval in the infinite line (top) and two disjoint intervals (bottom): they are proportional to the CFT calculations in Eqs. (\ref{['SA one interval']}) and (\ref{['logneg N2disj cft t-dep']}) respectively. The regions from where the corresponding quasi-particles have been emitted at $t=0$ are obtained by projecting the intersections at time $t=0$ (vertical dashed lines).
• Figure 3: Left: Temporal evolution of the entanglement entropy for one interval of $\ell$ sites in a periodic harmonic chain with $L$ sites. At $t=0$ the mass is quenched from $\omega_0=1$ to $\omega=0$. The dashed curve is the CFT prediction (\ref{['SA one interval']}) with $c=1$ and the best fitted value for $\tau_0$. Right: Temporal evolution of the Rényi entropies and of the logarithmic negativity $\mathcal{E}_A$, which coincides with $S_A^{(1/2)}$ in this case, for a periodic chain with $L=5000$ and $\ell=400$. In the inset, we report the best fitted values of $\tau_0$ for the values of $n$ displayed in the main plot.
• Figure 4: Adjacent intervals with several equal lengths $\ell_1=\ell_2$ and for various total size $L$ of the periodic harmonic chain. All panels show the data for $\omega_0=1$ and a critical evolution, $\omega=0$. Panels (a) and (c) display the mutual information $I$ while (b) and (d) the logarithmic negativity $\mathcal{E}$. Top and middle panels show different time scales and the revivals due to the finiteness of the system are evident in the middle panels, where a larger range of $t$ is considered. The dashed CFT curves in (a) and (b) are given by (\ref{['MI n N2 cft t-dep adj']}) and (\ref{['neg 2adj t-dep cft']}) respectively. In the last two panels we show the time evolution of the Rényi mutual information $I^{(n)}$ (e) and of the replicated negativity $\mathcal{E}^{(n)}$ (f) for various values of $n$.
• Figure 5: Adjacent intervals with different lengths $\ell_1=2\ell_2$ for different intervals lengths and total size $L$ of the periodic harmonic chain. Critical evolution of the mutual information $I$ (panel (a), (c)) and of the logarithmic negativity $\mathcal{E}$ (panels (b), (d)). The revivals are reported in panels (c) and (d). Notice that outside the light cone $\mathcal{E}$ always decays while $I$ reaches a plateau (apart from some finite size effects responsible of a very slow increase) (panels (c) and (d)). Compared to Fig. \ref{['fig N2 L200L200d0']} for the case of intervals with equal length, we observe a plateau starting at $t \simeq \textrm{min}(\ell_1,\ell_2)/2$ with temporal width $|\ell_2-\ell_1|/2$. The panel (e) and (f) report the ratios $R_n$ for several values of $n$.