Entanglement negativity after a local quantum quench in conformal field theories

We study the time evolution of the entanglement negativity after a local quantum quench in (1 + 1)-dimensional conformal field theories (CFTs), which we introduce by suddenly joining two initially decoupled CFTs at their end points. We calculate the negativity evolution for both adjacent intervals and disjoint intervals explicitly. For two adjacent intervals, the entanglement negativity grows logarithmically in time right after the quench. After developing a plateau-like feature, the entanglement negativity drops to the ground-state value. For the case of two spatially separated intervals, a light-cone behavior is observed in the negativity evolution; in addition, a long-range entanglement, which is independent of the distance between two intervals, can be created. Our results agree with the heuristic picture that quasiparticles, which carry entanglement, are emitted from the joining point and propagate freely through the system. Our analytical results are confirmed by numerical calculations based on a critical harmonic chain.

Figure 1

Setup for a local quantum quench. Two separate CFTs defined on two semi-infinite lines are joined together at their end points. Then quasiparticles, which may be viewed as entangled pairs, are generated at the joining point, and propagate freely through the system. The entanglement negativity between two intervals which are far from each other may be built with the help of these propagating entangled pairs.

Figure 2

Path integral representation of (a) $\text{Tr}{\left({\rho }_A\right)}^n$ and (b) $\text{Tr}{\left({\rho }_A^{T_2}\right)}^n$ for two disjoint intervals.

Figure 3

Illustration of the conformal mapping in Eq. (11), based on which the $z$ plane is mapped to a right half plane (RHP) with $\text{Re}w>0$. For later use, we also label ${\tilde{w}}_i=-{\overline{w}}_i$, which is the image of $w_i$.

Figure 4

Configurations of two intervals considered in this work: two adjacent intervals (up) and two disjoint intervals (bottom).

Figure 5

Entanglement negativity $\mathcal{E}$ for two symmetric adjacent intervals as a function of time. Here we choose the central charge $c=1,\epsilon =0.1,l=25$, 50, 75, and 100, respectively. Shown in (a) is the CFT result, and (b) is the numerical calculation based on a critical harmonic chain.

Figure 6

Entanglement negativity $\mathcal{E}$ for two asymmetric adjacent intervals as a function of time. Here we choose central charge $c=1,\epsilon =0.1,(l_1,l_2)=(25,75),(50,75)$, and $(75,75)$, respectively. Shown in (a) is the CFT result, and (b) is the numerical calculation based on a critical harmonic chain.

Figure 7

Entanglement negativity $\mathcal{E}$ for two symmetric disjoint intervals as a function of time. Here we choose the central charge $c=1,\epsilon =1,(d,l)=(140,10),(160,10)$, and $(180,10)$, respectively. Shown in (a) is the CFT result, and (b) is the numerical calculation based on a critical harmonic chain.

Figure 8

Entanglement negativity $\mathcal{E}$ for two asymmetric disjoint intervals as a function of time. Here we choose the central charge $c=1,\epsilon =1,l=15,(d_1,d_2)=(150,150),(150,155),(150,160)$, and $(150,165)$, respectively. Shown in (a) is the CFT result, and (b) is the numerical calculation based on a critical harmonic chain.

Figure 9

Effects of propagating entangled pairs on the entanglement negativity $\mathcal{E}$ of two adjacent intervals and two disjoint intervals in a lattice model. For two adjacent intervals, all four kinds of entangled pairs may contribute to the entanglement negativity. For two disjoint intervals, however, only the fast-fast entangled pair may contribute to the entanglement negativity near $t\sim d$, as discussed in the main text.

Figure 10

Distribution of entangled pairs as a function of $(v_L,v_R)$ for a harmonic chain. The peak at $(-v_L/v_c,v_R/v_c)\simeq (1,1)$ is contributed by the fast-fast pairs. The $\mathrm{region}(-v_L/v_c,v_R/v_c\le 1$) with finite $\mathcal{E}$ is contributed by the other three kinds of entangled pairs, i.e., fast-slow pairs, slow-fast pairs, and slow-slow pairs, which result from the high-energy contribution. The parameters we choose are $l_1=l_2=l=5$ and $t=100$.