Barrier from chaos: operator entanglement dynamics of the reduced density matrix

It is believed that thermalization drives the reduced density matrix of a subsystem to approach a short-range entangled operator. If the initial state is also short-range entangled, it is possible that the reduced density matrix remains low-entangled throughout thermalization; or there could exist a barrier with high operator entanglement between the initial and thermalized reduced density matrix. In this paper, we study such dynamics in three classes of models: the rational CFTs, the random unitary circuit, and the holographic CFTs, representing systems of increasing quantum chaoticity. We show that in all three classes of models, the operator entanglement (or variant of) exhibits three phases, a linear growth phase, a plateau phase, and a decay phase. The plateau phase characterized by volume-law operator entanglement corresponds to the barrier in operator entanglement. While it is present in all three models, its persistence and exit show interesting distinctions among them. The rational CFTs have the shortest plateau phase, followed by the slowest decay phase; the holographic CFTs mark the opposite end, i.e. having the longest plateau phase followed by a discontinuous drop; and the random unitary circuit shows the intermediate behavior. We discuss the mechanisms underlying these behaviors in operator entanglement barriers, whose persistence might serve as another measure for quantum chaoticity.