Entanglement barriers in dual-unitary circuits

Abstract

After quantum quenches in many-body systems, finite subsystems evolve nontrivially in time, eventually approaching a stationary state. In typical situations, the reduced density matrix of a given subsystem begins and ends this endeavor as a low-entangled vector in the space of operators. This means that if its entanglement in the operator space initially grows (which is generically the case), it must eventually decrease, describing a barrier-shaped curve. Understanding the shape of this ``entanglement barrier'' is interesting for three main reasons: (i) it quantifies the dynamics of entanglement in the (open) subsystem; (ii) it gives information on the approximability of the reduced density matrix by means of matrix product operators; and (iii) it shows qualitative differences depending on the type of dynamics undergone by the system, signaling quantum chaos. Here we compute exactly the shape of the entanglement barriers described by different R\'enyi entropies after quantum quenches in dual-unitary circuits initialized in a class of solvable matrix product states (MPS)s. We show that, for free (SWAP-like) circuits, the entanglement entropy behaves as in rational conformal field theories (CFT)s. On the other hand, for completely chaotic dual-unitary circuits, it behaves as in holographic CFTs, exhibiting a longer entanglement barrier that drops rapidly when the subsystem thermalizes. Interestingly, the entanglement spectrum is nontrivial in the completely chaotic case. Higher R\'enyi entropies behave in an increasingly similar way to rational CFTs, such that the free and completely chaotic barriers are identical in the limit of infinite replicas (i.e., for the so called min-entropy). We also show that, upon increasing the bond dimension of the MPSs, the barrier maintains the same shape. It simply shifts to the left to accommodate for the larger initial entanglement.