Entanglement membrane in the Brownian SYK chain

TL;DR

This paper derives an entanglement membrane theory in the Brownian SYK chain, revealing how entanglement spreads and splits into wave fronts at the butterfly velocity, linking quantum information dynamics with scrambling.

Contribution

It provides the first derivation of the entanglement membrane in a solvable chaotic large-N model, the Brownian SYK chain, including explicit calculations of membrane velocity and tension.

Findings

The entanglement membrane has finite width for velocities below the butterfly velocity.

For velocities above the butterfly velocity, the membrane splits into two wave fronts.

The membrane's properties are characterized by a velocity and a tension, connecting entanglement dynamics with scrambling.

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 R\'enyi 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.