Evolution of entanglement entropy in the D1-D5 brane system

TL;DR

The paper presents an exact analytic calculation of the time evolution of entanglement entropy for arbitrary spatial intervals in the D1D5 CFT after a localized marginal deformation (local quench). It employs the replica trick and covering-space techniques to compute four-point functions involving twist operators, extracting both Rényi and von Neumann entropies and separating vacuum from quench-induced contributions. The results reveal how entangled pairs generated at the quench propagate at light speed, yielding a time- and interval-dependent entanglement growth consistent with partial thermalization, and they connect this dynamical process to holographic pictures of stringy black hole formation. The study provides a concrete, analytic bridge between microscopic CFT dynamics and emergent thermal behavior, with implications for holography, quantum gravity, and black-hole information concepts.

Abstract

We calculate the evolution of the geometric entanglement entropy following a local quench in the D1D5 conformal field theory, a two-dimensional theory that describes a particular bound state of D1 and D5 branes. The quench corresponds to a localized insertion of the exactly marginal operator that deforms the field theory off of the orbifold (free) point in its moduli space. This deformation ultimately leads to thermalization of the system. We find an exact analytic expression for the entanglement entropy of any spatial interval as a function of time after the quench and analyze its properties. This process is holographically dual to one stage in the formation of a stringy black hole.

Figures (7)

• Figure 1: The contour along the real axis of the complex $\tau$-plane for the Euclidean path integral. The analytic continuation back to Lorentzian time is shown in light gray, which shows how the $\epsilon$ regularization arises.
• Figure 2: The reduced density matrix as a path integral. Note that the flat piece on top is there for illustrative purposes only. The boundary conditions on the bottom two edges are both $\ket{\psi_0}$, whereas the boundaries in region $A$ are "inputs" that determine the matrix element of the reduced density matrix.
• Figure 3: Plot of $\varphi$ (blue, dashed) and $\bar{\varphi}$ (red, dot-dashed) versus time for $\epsilon = 10^{-2}$, where $\eta = \exp 2i\varphi$ and $\bar{\eta} = \exp 2i\bar{\varphi}$. Note that $\varphi$ and $\bar{\varphi}$ obey the strict inequality $-\pi<\varphi<0<\bar{\varphi}<\pi$ for all time. In the limit as $\epsilon$ goes to zero, the function converges pointwise to a piecewise function saturating the inequalities. Indeed, for $\epsilon=10^{-4}$ the plot is (visually) indistinguishable from the corresponding piecewise function.
• Figure 4: $\Delta S_{\text{vN}}$, \ref{['fig:LRent2']}, and $\Delta S_2$, \ref{['fig:S2plot']}, for the interval $[\pi/2, 3\pi/4]$. For $\Delta S_{\text{vN}}$ the right-moving contribution is shown in red, the left-moving in black. For $\Delta S_2$ we cannot separate the left and right-moving contributions.
• Figure 5: Here we show the two circles being joined at $\theta = 0$ and $\theta = 2\pi$. Time is increasing up on the figure. The quench occurs at $t=0$. Light cones are emanating from the two quench sites. The right-moving excitations travel along the green curves, and the left-moving excitations along the blue curves. The grey vertical strip represents the time evolution of the interval $[\theta_1, \theta_2]$, and the red parts of the strip show when we expect nonvanishing entanglement from the particle interpretation.