The entanglement membrane in deterministic systems

In certain analytically-tractable quantum chaotic systems, the calculation of out-of-time-order correlation functions, entanglement entropies after a quench, and related dynamical observables, reduces to an effective statistical mechanics of an ‘entanglement membrane’ in spacetime. These tractable systems involve an average over random local unitaries defining the dynamical evolution. We show here how to make sense of this membrane in more realistic models, which do not involve an average over random unitaries. Our approach relies on introducing effective ‘pairing’ degrees of freedom in spacetime, inspired by the structure emerging in random unitary circuits. We also provide an efficient algorithm for determining the ‘line tension’ of the entanglement membrane in 1+1D models.