Membrane theory of entanglement dynamics from holography

TL;DR

This work demonstrates that holographic entanglement entropy dynamics in the scaling limit can be equivalently described by a minimal timelike membrane in Minkowski space with a tension ${\cal E}(v)$ determined by the final equilibrium black brane geometry. The authors derive ${\cal E}(v)$ from the holographic extremal-surface problem, prove its key properties, and show how the resulting membrane dynamics reproduce known entropy growth and bounds, including universal linear growth in thermofield double setups and tsunami-type bounds. The membrane reformulation provides a unifying framework linking holography, chaotic dynamics, and tensor-network pictures, and offers a tractable route to derive new bounds and explore extensions such as inhomogeneous quenches and operator entanglement. Overall, the results support the membrane theory as a universal description of entanglement spreading in chaotic quantum systems and illuminate connections between bulk geometry and boundary dynamics.

Abstract

Recently, a minimal membrane description of the entanglement dynamics of large regions in generic chaotic systems was conjectured in arXiv:1803.00089. Analytic results in random circuits and spin chain numerics support this theory. We show that the results found by the author in arXiv:1612.00082 about the dynamics of entanglement entropy in theories with a holographic dual can be reformulated in terms of the same minimal membrane, providing strong evidence that the membrane theory describes all chaotic systems. We discuss the implications of our results for tensor network approaches to holography and the holographic renormalization group.

Figures (4)

• Figure 1: Left: A minimal membrane (blue) stretching between the two faces of a slab of Minkowski spacetime of width $t$. On the upper boundary it is an ellipse. Right: At the red point we drew a tangent plane (green), its angle with the vertical plane is $\arctan (v)$. For this membrane $v=v_B$ at every point, and we will refer to such membranes as "light sheets" of slope $v_B$.
• Figure 2: ${\cal E}\left(v\right)$ for different values of the chemical potential in $d=3$. The black brane metrics are given by \ref{['MetricBH']}, with $a(z)=1-(1+3q)z^3+3q z^{4}$ and $b(z)=1$, where the dimensionless boundary theory chemical potential is ${\beta\,\mu}\propto {q / (1-q)}$. We chose $q=0,1/4,1/2,3/4,15/16$ for this plot, larger $q$ corresponds to smaller $v_E={\cal E}\left(0\right)$. The black dashed lines are at $45^\circ$, at $v=v_B$, ${\cal E}\left(v\right)$ by \ref{['Eprops']} is equal to and tangent to this line. For $v\leq v_B$ we use solid, for $v>v_B$ dotted lines, as the latter regime is not important for the dynamics.
• Figure 3: Entropy growth in the thermofield double state in $d=4$ for two concentric spheres with $R_R/R_L=2$ as a function of $T$. The entropy is normalized by the thermal entropy of the left sphere. Representative snapshots of membranes (blue) and the boundaries of the slab (black) are included at different times marked with gray gridlines. There are four regimes drawn by different colors matching the color coding used in Mezei:2016zxg. In the black regime the membrane consists of a horizontal and a "light sheet" section that has slope $v_B$. In the blue section the membrane has increasing cross section as it interpolates between $L$ and $R$, while in the orange section it reaches a minimum in between. Finally, it transitions through a "light sheet" of slope $v_B$ to a disconnected membrane in the red region.
• Figure 4: Entropy growth in a charge neutral quench in $d=4$. Representative membranes shapes are included at different times marked with gray gridlines. The membranes end perpendicular to the lower green end of the time slab, which represents the initial state. Note that at saturation, $T=R/v_B$ the membrane forms a cone.