Bubble quenches in the AdS/BCFT model

In this paper we construct time-dependent solutions of three-dimensional gravity in anti–de Sitter space dual to systems with boundaries (BCFTs), following the AdS/BCFT prescription. Such solutions can be discussed in the context of the dynamics of first-order phase transitions, or more generally, in the description of quantum quenches. As an example, we apply the holographic model to calculate the dynamics of the entanglement entropy of a local quench corresponding to a nucleation of a Euclidean bubble. As in the known $1+1$ conformal field theory examples of local cut and glue quenches, the holographic entanglement entropy grows logarithmically with time with the correct universal coefficient. However, in the bubble quench, the behavior is different at late times. The AdS/BCFT model exhibits the light-cone spreading of correlations and saturation at late times. We also find an analytical formula for the entropy at finite temperature. In the latter case, the initial logarithmic growth is followed by the linear law at intermediate times.

Figure 1

In the AdS/BCFT construction of Ref. [47] the gravity theory in the $d+1$-dimensional space $N$ with boundary $Q$ is expected to be dual to a CFT defined in the $d$-dimensional space $M$, with boundary $P=\partial M=\partial Q$.

Figure 2

Nucleation of a Euclidean bubble of anti–de Sitter space $t<0$, and the creation ($t=0$) and evolution of the real bubble ($t>0$).

Figure 3

Nucleation of a Euclidean bubble of the BTZ spacetime $t<0$, and the creation ($t=0$) and evolution of the real bubble ($t>0$), for $\theta =\pi /2$.

Figure 4

Profiles of $Q$ in the BTZ geometry for different values of $R$ (height of the bubble), at $t=0$ and $\theta =\pi /2$. As $R$ approaches the horizon, the width of the bubble becomes infinite, $l=2\rho \to \infty$.

Figure 5

Left: trajectories of the $T=0$ [Eq. (12)] (blue curve) and $T\ne 0$ [Eq. (16)] (red curve) bubble walls. Right: velocities of the two types of walls as a function of $x$.

Figure 6

Left: geodesics (blue and green lines) and end-of-the-world surface $Q$ (black) in the empty AdS geometry. The initial bubble is shown as the gray dashed arc. Its spatial radius $R$ is shown as a thick gray line. Right: same in the BTZ geometry, $\theta =\pi /2$.

Figure 7

Plot of the time dependence of the entanglement entropy of the two halves of the expanding bubble for different initial sizes of the bubble. The left plot corresponding to empty AdS geometry shows asymptotic logarithmic growth. The right plot corresponding to the finite-temperature geometry shows saturation of the entropy at a value independent of the initial parameters of the bubble.

Figure 8

Calculation of the entanglement entropy of the finite interval bounded by the dashed arc with the rest of the system after the quench. The two competing contributions to the entropy are the disconnected blue piece and the connected dashed piece.

Figure 9

Left: evolution of the entanglement entropy of a finite interval in a bubble quench from the AdS/BCFT calculation (configuration of Fig. 8). The blue curve shows the logarithmic growth from Eq. (29). The growth saturates at $S_0$ (red line) [Eq. (30)]. The gray line is the behavior predicted in a local cut and glue quench [Eq. (36)]. Right: similar plot for $T\ne 0$. The blue line has a linear segment described by Eq. (34). The gray line illustrates the $T=0$ curve.