Mutual information after a local quench in conformal field theory

TL;DR

The paper computes the time evolution of entanglement entropy and mutual information for two disjoint intervals after a local quench in 1+1D CFTs, using the replica trick and BCFT techniques to obtain universal results in specific regimes. It also develops and compares a holographic model based on a falling massive particle in AdS$_3$, computing holographic entanglement entropy and mutual information via geodesics, and contrasts the BCFT and holographic predictions. A key finding is a long-range mutual information in the CFT setup consistent with a quasiparticle picture, whereas the holographic model exhibits phase transitions and vanishing mutual information at large separations, highlighting qualitative differences between the BCFT local quench and its holographic dual. The work also clarifies the roles of Euclidean versus Lorentzian conformal transformations in constructing bulk duals and introduces an efficient alternative method for computing holographic entanglement entropy via Lorentzian maps. These results illuminate how localized energy injections produce intricate dynamical entanglement structures and offer avenues for extending to higher dimensions and bulk quantum corrections.

Abstract

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 formulae 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.

Figures (24)

• 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 lightcone 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 lightcone, the entanglement leads to a mutual information that is larger than the ground state value.
• Figure 2: Illustration of the mapping \ref{['eq:UHPmap']}, 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 \ref{['fig:w-points']}, is mapped to the points in the $z$ plane seen in \ref{['fig:z-points']}. 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: $(i)$ symmetric intervals, $(ii)$ asymmetric intervals and $(iii)$ 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 $A = [-11,-10]$ and $B =[10.5,11.5]$, below $A=[-31,-1]$, $B=[3,33]$. Here $\epsilon =0.01$ and $a, \tilde{c}_1'$ have been eliminated in favor of $\epsilon$ using Eq. \ref{['relationaepsilon']}.
• 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 $A = [10,11]$ and $B =[20,21]$, below $A=[-0.5,1]$, $B=[15,16.5]$. Here $\epsilon =0.01$ and $a, \tilde{c}_1'$ have been eliminated in favor of $\epsilon$ using Eq. \ref{['relationaepsilon']}.