Logarithmic growth of operator entanglement in a clean non-integrable circuit

TL;DR

This paper investigates a non-integrable quantum circuit where operator entanglement grows logarithmically over time, revealing intermediate dynamical behavior between chaos and integrability, with implications for understanding quantum thermalization.

Contribution

It demonstrates that operator entanglement in a non-integrable, disorder-free circuit grows logarithmically, challenging prior expectations of rapid entanglement growth in such systems.

Findings

Operator entanglement grows at most logarithmically in time.

Auto-correlation functions can be expressed using $SO(3)$ matrices.

Operator size distribution becomes bimodal at certain times.

Abstract

We study a so-called semi-ergodic brickwork dual-unitary circuits where, in the infinite volume limit, the two-point correlation functions of single-site operators exhibit ergodic behavior along one light ray and non-ergodic behavior along the other light ray. Here, however, we study intermediate and long-time dynamics of a system in a finite, large volume. Under such dynamics, the Heisenberg evolution of a single traceless single-site operator lies within a restricted subspace, and this time evolution can be mapped to a simpler problem of a single qutrit scattering with a bunch of qubits sequentially. Despite the model being non-integrable and free from any quenched disorder, the operator entanglement grows at most logarithmic in time, contrary to prior expectations. The auto-correlation function can be written in terms of a sum of products of $SO(3)$ matrices, allowing for a random matrix prediction for the auto-correlation function at late times. The operator size distribution also becomes bimodal at certain times, displaying intermediate behavior between chaotic and free systems.