Boundary chaos: Exact entanglement dynamics

Abstract

We compute the dynamics of entanglement in the minimal setup producing ergodic and mixing quantum many-body dynamics, which we previously dubbed boundary chaos. This consists of a free, non-interacting brickwork quantum circuit, in which chaos and ergodicity is induced by an impurity interaction, i.e., an entangling two-qudit gate, placed at the system’s boundary. We compute both the conventional bipartite entanglement entropy with respect to a connected subsystem including the impurity interaction for initial product states as well as the so-called operator entanglement entropy of initial local operators. Thereby we provide exact results in a particular scaling limit of both time and system size going to infinity for either very small or very large subsystems. We show that different classes of impurity interactions lead to very distinct entanglement dynamics. For impurity gates preserving a local product state forming the bulk of the initial state, entanglement entropies of states show persistent spikes with period set by the system size and suppressed entanglement in between, contrary to the expected linear growth in ergodic systems. We observe similar dynamics of operator entanglement for generic impurities. In contrast, for T-dual impurities, which remain unitary under partial transposition, we find entanglement entropies of both states and operators to grow linearly in time with the maximum possible speed allowed by the geometry of the system. The intensive nature of interactions in all cases causes entanglement to grow on extensive time scales proportional to system size.

Hilbert Space Delocalization under Random Unitary Circuits

“…In this work, we investigate Hilbert space delocalization under the dynamics of quantum circuits comprising 2-qudit Haar random gates arranged in a brick wall pattern [ 23 ]. The locality and unitarity of this setup constitute the minimal requirements for chaotic evolution in many-body systems [ 49 , 50 , 51 , 52 , 53 , 54 , 55 , 56 , 57 , 58 , 59 , 60 , 61 , 62 , 63 , 64 , 65 , 66 , 67 , 68 , 69 , 70 ], hence allowing us to gain a phenomenological understanding of the Hilbert space delocalization under generic quantum dynamics. Our analysis combines rigorous analytical methods, based on the mapping between the average of the IPR over the random circuits and a statistical mechanics model, with exact numerical simulations, including tensor network techniques [ 71 , 72 , 73 , 74 ] (here, our implementation is based on the open-source library ITensor [ 75 , 76 ] and available at [ 77 ]).…”