Mutual information after a local quench in conformal field theory

We compute the entanglement entropy and mutual information for two disjoint intervals in two-dimensional conformal field theories as a function of time after a local quench, using the replica trick and boundary conformal field theory. We obtain explicit formulas for the universal contributions, which are leading in the regimes of, for example, close or well-separated intervals of fixed length. The results are largely consistent with the quasiparticle picture, in which entanglement above that present in the ground state is carried by pairs of entangled freely propagating excitations. We also calculate the mutual information for two disjoint intervals in a proposed holographic local quench, whose holographic energy-momentum tensor matches the conformal field theory one. We find that the holographic mutual information shows qualitative differences from the conformal field theory results and we discuss possible interpretations of this.

Figure 1

Illustration of a local quench: two CFTs on half lines are joined instantaneously at their boundaries. This generates left and right moving excitations that propagate along the light cone of the quench event and are entangled with each other. During the time period, in shaded red, that the two intervals $A$ and $B$ are intersected by this light cone, the entanglement leads to a mutual information that is larger than the ground state value.

Figure 2

Illustration of the mapping (11), with $\epsilon =0.1$. The sample of points in the region $[-0.1,0.1]\times [-0.12,0.12]$ in the $w$ plane, shown in (a), is mapped to the points in the $z$ plane seen in (b). The twist operators are inserted at the points $w_1$, $w_2$, $w_3$ and $w_4$ and the branch cuts they induce are shown with thin curvy lines. The thick curvy lines indicate the boundaries of $W$, which are mapped to the imaginary axis in the $z$ plane. We also show the image points in the left half of the $z$ plane and the conjugate points in the lower half plane.

Figure 3

We illustrate various configurations of intervals and associated parameters: (a) symmetric intervals, (b) asymmetric intervals and (c) intervals on the same side of the quench.

Figure 4

Entanglement entropies (rescaled by $c/6$) for one and two asymmetric intervals, in the universal regimes. In the top plots (a) and (b): $A=[-11,-10]$ and $B=[10.5,11.5]$; below in plots (c) and (d): $A=[-31,-1]$, $B=[3,33]$. Here $ϵ=0.01$ and $a,{{\tilde{c}}_1}^{\prime }$ have been eliminated in favor of $\epsilon$ using Eq. (32).

Figure 5

Entanglement entropies (rescaled by $c/6$) for one and two intervals on the same side of the quench, or overlapping it, in universal regimes. In the top plots (a) and (b): $A=[10,11]$ and $B=[20,21]$; below in plots (c) and (d): $A=[-0.5,1]$, $B=[15,16.5]$. Here $\epsilon =0.01$ and $a,{{\tilde{c}}_1}^{\prime }$ have been eliminated in favor of $\epsilon$ using Eq. (32).

Figure 6

Time evolution of the mutual information (rescaled by $c/6$) for symmetric intervals of length $\ell =1$ at distances $d=20$, 24, 28, 32, 36 from left to right. Here we have set $\epsilon =0.01$.

Figure 7

Mutual information $I_{[-11,-10],[10+j/2,11+j/2]}$ (rescaled by $c/6$) as a function of time for $j=0$, 1, 2, 3, 4 (from left to right) with $\epsilon =0.01$.

Figure 8

Evolution of the mutual information (rescaled by $c/6$) in the universal regime for $d=20$ and (a) $\ell =1$, $x=1$, 2, 3, (b) $\ell =2.5$, $x=-1$, $-0.5$, 0 from left to right, and $\epsilon =0.01$.

Figure 9

The expectation value of the energy density $⟨T_{tt}(x,t)⟩$ given in Eq. (68) with $c=1$ and $\epsilon =0.2$.

Figure 10

Renormalized entanglement entropy $\mathrm{\Delta }S$ (rescaled by $4G_{\mathrm{N}}$) as a function of time $t_{\infty }$, interval length $\ell$ or interval midpoint position $\xi$ for: a symmetric interval with (a) $-{\ell }_1={\ell }_2\equiv \ell =0.5,1.5,3$ from left to right, (b) times $t_{\infty }=0.5$, 1.5, 3 from left to right; an interval with an excited endpoint ${\ell }_1=0$ with (c) ${\ell }_2=0.9$, 3, 5 from the bottom up, (d) $t_{\infty }=0.9$, 3, 5 from left to right; a generic interval with (e) ${\ell }_1=-1.5$, 05, 2 ${\ell }_2={\ell }_1+4$ from left to right, (f) ${\ell }_1=\xi -\frac{3}{2}$, ${\ell }_2=\xi +\frac{3}{2}$ for $t=0$, 5, 10 from left to right. In the plots, we have set the parameters $\alpha =1$ and $M=3/4$.

Figure 11

The time evolution of the entanglement entropy with the early time value subtracted (and rescaled by $4G_{\mathrm{N}}$) for intervals of length 30 and ${\ell }_1=-1$, 4, 9. The universal part of the CFT results are shown with dashed lines for comparison. Here we have set $M=3/4$ and $\epsilon =\alpha =0.001$.

Figure 12

The entanglement entropy with the early time value subtracted (and rescaled by $4G_{\mathrm{N}}$) for intervals (a) of length 5 and ${\ell }_1=0.5$, 25, 50 from left to right; (b) of length 10 and ${\ell }_1=-0.5$, $-1.5$, $-3$ from the outmost to the inmost. In dashed lines, the universal part of the CFT results, which in some cases overlap the holographic curves without appreciable difference. The parameters have been set to $M=3/4$, $\epsilon =\alpha =0.001$.

Figure 13

The entanglement entropy with the early time value subtracted (and rescaled by $4G_{\mathrm{N}}$) for intervals centered at 10 and of length ${\ell }_2-{\ell }_1=1$, 1, 2 from the interior to the exterior. In dashed lines, the universal contributions to the CFT result, which are essentially indistinguishable from the holographic ones. We have set $M=3/4$, $\epsilon =\alpha =0.001$.

Figure 14

A schematic drawing of the three sets of geodesics competing to the holographic entanglement entropy of the union of two intervals $S_{A\cup B}$.

Figure 15

Time evolution of the mutual information with the early time value subtracted (rescaled by $4G_{\mathrm{N}}$) for (a) $\ell =10$, $d=2$, 10, 40, 70 from left to right and (b) $d=2$, $\ell =20$, 35, 50 from left to right. The CFT results are shown with dashed lines and here we have set $M=3/4$ and $\alpha =\epsilon =0.001$.

Figure 16

The two competing configurations of geodesics for the mutual information at $t=5$ for $\ell =6$ and $d=4$ (blue), $d=10$ (red) ($\alpha =1$, $M=3/4$). The configuration of minimal length is in solid lines. As we increase the separation between the two intervals, we observe a transition from the connected towards the disconnected configuration.

Figure 17

Mutual information (rescaled by $4G_{\mathrm{N}}$) for (a) $x=1$, $\ell =10$, $d=0.008$, 0.5, 1 from the top to the bottom, (b) $x=1$, $d=1$, $\ell =5$, 7, 9 from the bottom to the top, as a function of time. The universal parts of the CFT results are shown in dashed lines. These have been individually shifted as to match the holographic results at $t=0$, to better visualize how the time dependence of the two results compares. Here $M=3/4$ and $\alpha =\epsilon =0.001$.

Figure 18

The two competing configurations of geodesics for the mutual information at $t=10\in [{\overline{\ell }}_2,{\overline{\ell }}_3]$ for $x=1$, $\ell =10$ and $d=3$ ($\alpha =1$, $M=3/4$). In this time range, the outmost geodesic develops a long curl and the disconnected configuration (in solid lines) becomes the minimal one.

Figure 19

Mutual information (rescaled by $4G_{\mathrm{N}}$) for asymmetric intervals about the quench with (a) $x=10$, $\ell =2$, $d=18.5$, 19, 19.5 and $x=3$, $\ell =15$, $d=4$, 5 (b) $x=11$, $d=21.5$, $\ell =2$, 3 and $x=2$, $d=3$, $\ell =15$, 20. We have set $M=3/4$ and $\alpha =\epsilon =0.001$ and plotted the CFT results in dashed lines.

Figure 20

Mutual information (rescaled by $4G_{\mathrm{N}}$) for nonzero overlap with the defect and (a) $\ell =10$, $d=1.5$, 2, 2.5, $y=9.5$, 9, 8.5 from the top to the bottom, (b) $y=7.5$, $d=3.5$, $\ell =11.5$, 12, 12.5 from the bottom to the top, as a function of time. The solid lines are the holographic results and the dashed lines the universal part of the CFT ones (the latter have been shifted so as to match the holographic result at $t=0$, to facilitate the comparison between the two). Here $M=3/4$ and $\alpha =\epsilon =0.001$.

Figure 21

Schematic drawing of the geodesics contributing to the entropy formula in Eq. (141). The semicircles ${\tilde{\gamma }}_1$ and ${\tilde{\gamma }}_2$ are mapped to the geodesics ${\gamma }_1$ and ${\gamma }_2$, respectively, under the map $g$ defined on the conformal boundary by Eq. (139).

Figure 22

Comparison of the renormalized entanglement entropy (rescaled by $4G_{\mathrm{N}}$) of a symmetric interval of length 6 computed with the two methods we have described. The dashed lines show the contributions coming from the two competing geodesics ${\gamma }_1$ and ${\gamma }_2$ and the solid red line shows the result computed in Sec. 6b.