Exploring the membrane theory of entanglement dynamics

Abstract

Recently an effective membrane theory valid in a "hydrodynamic limit" was proposed to describe entanglement dynamics of chaotic systems based on results in random quantum circuits and holographic gauge theories. In this paper, we show that this theory is robust under a large set of generalizations. In generic quench protocols we find that the membrane couples geometrically to hydrodynamics, joining quenches are captured by branes in the effective theory, and the entanglement of time evolved local operators can be computed by probing a time fold geometry with the membrane. We also demonstrate that the structure of the effective theory does not change under finite coupling corrections holographically dual to higher derivative gravity and that subleading orders in the hydrodynamic expansion can be incorporated by including higher derivative terms in the effective theory.

Entanglement Membrane in Chaotic Many-Body Systems

“…The spins take values in S 4 , because in total we need four U layers and four U Ã layers to write Ω 2 C . We see that there is a distinction between spin values in the S 2 × S 2 subgroup generated by (12) and (34) and spin values outside this subgroup. At leading order, the former are forbidden inside the cluster.…”

“…5. In a þ domain, the S 4 spins are fixed to I, and in a − domain, they are fixed to (12) (34). Inside the ⊥ cluster, the spins must be summed over.…”

“…There are in total six classes. Within each class, the configurations are related by the combination of reflection symmetry about the center axis, conjugation by (12), (34), or both.…”

“…In particular, they reveal a generic structure associated with an emergent "membrane" in spacetime [6,18]. The effective statistical mechanics of this membrane determine the production of entanglement after a quantum quench, the spreading of quantum operators, and other aspects of the "scrambling" [7,8,11,[19][20][21][22][23][24][25][26][27][28][29][30][31][32][33][34] of quantum information.…”

“…There are analytic computations in random Floquet circuits in the large Hilbert space dimension (large q) limit [11], showing that the domain wall structure makes sense in that limit. Finally, there is an analytical derivation of the form of the membrane tension in holographic systems [34,37], at leading order in the number of degrees of freedom; this derivation also begins to make a concrete connection between the entanglement membrane and the Ryu-Takayanagi and Hubeny-Rangamani-Takayanagi geometrical pictures for entanglement in AdS space [38][39][40][41]. Lattice toy models for the AdS-CFT correspondence, using random tensor networks, also exhibit domain walls between permutation degrees of freedom [42,43].…”