Emergent statistical mechanics of entanglement in random unitary circuits

Random unitary circuits are powerful minimal models for chaotic nonequilibrium evolution. Here, the authors establish an exact mapping between dynamical observables in random circuits and a statistical mechanical model of interacting spins, and use it to study systematically the entanglement generated by a random circuit consisting of local gates. A domain wall line tension in this spin model sets the entanglement growth rate. By a careful analysis of the interaction between domain walls, the authors derive explicitly the Kardar-Parisi-Zhang equation that governs the evolution of second Renyi entropy and identify a phase transition of the growth-rate function.