Entanglement membrane in the Brownian SYK chain

Abstract

There is mounting evidence that entanglement dynamics in chaotic many-body quantum systems in the limit of large subsystems and long times is described by an entanglement membrane effective theory. In this paper, we derive the membrane description in a solvable chaotic large-$N$ model, the Brownian SYK chain. This model has a collective field description in terms of fermion bilinears connecting different folds of the multifold Schwinger-Keldysh path integral used to compute Rényi entropies. The entanglement membrane is a traveling wave solution of the saddle point equations governing these collective fields. The entanglement membrane is characterised by a velocity $v$ and a membrane tension ${\cal E}(v)$ that we calculate. We find that the membrane has finite width for $v<v_B$ (the butterfly velocity), however for $v > v_B$, the membrane splits into two wave fronts, each moving with the butterfly velocity. Our results provide a new viewpoint on the entanglement membrane and uncover new connections between quantum information dynamics and scrambling.

Figures (20)

• Figure 1: The $n$-th Rényi entropy of sites to the left of site $x$ at $t = T$ is related to a membrane of velocity $v$ that intersects the chain at $t = 0$ at site $x- vT$. The velocity of the membrane is determined by the minimization in equation \ref{['eq:entanglement_growth_conjecture']}.
• Figure 2: A maximally entangled state in $\mathcal{H}_R \otimes \mathcal{H}_L$. The dashed lines illustrate maximal entanglement between sites in $R$ and $L$ at $T = 0$. $S^{(n)}(x,y,T)$ is the entanglement entropy of the region $A \cup B \cup \bar{A}$. (Note that the bar stands for the subsystem in the $L$ chain and not for complement.) We illustrate the entanglement membrane by a black line of slope $v$ connecting $x$ and $y$.
• Figure 3: Left: Path integral contour for preparation of the state $\rho(T)_{A \cup B \cup \bar{A}}$. The solid arcs at the bottom of the contour illustrate maximally entangled states between the subregions of the chains $R$ and $L$. The grey boxes illustrate the joint unitary evolution of fermions in chain $R$. The open legs on the left and right denote the ket and bra states of the density matrix $\rho(T)_{A \cup B \cup \bar{A}}$ respectively. Right: We consider two copies of $\rho(T)_{A \cup B \cup \bar{A}}$ and cyclically contract all the open legs in $A \cup B \cup \bar{A}$. This defines the Schwinger-Keldysh contour for $\text{Tr}\,\rho^2(T)_{A \cup B \cup \bar{A}}$.
• Figure 4: Left: Contour for the second Rényi entropy ignoring $A$. Right: The OTOC contour. The second Rényi contour resembles the OTOC contour when $A$ is ignored.
• Figure 5: Saddle point solution for $v >v_B$. The solution consists of two fronts: (i) originating from the domain wall $A|B$ at $t = 0$, (ii) originating from $B|C$ at $t = T$. Both fronts propagate with the butterfly velocity towards region $B$, one forward and one backward in time as indicated by the arrows.