Abstract

Identifying observables that distinguish integrable from chaotic dynamics remains a central challenge in non-equilibrium quantum physics. In this context, the Operator Space Entanglement Entropy (OSEE) has played a primary role over the past two decades, quantifying the computational resources required to classically simulate the Heisenberg time evolution of quantum operators. It has long been conjectured that chaotic systems typically exhibit a linear growth of the OSEE for local operators, while in integrable models, this growth is bounded logarithmically in time. However, this conjecture has recently shown some cracks: for instance, it was shown that the OSEE of specific local operators can exhibit logarithmic growth even in strictly chaotic systems. In this talk, we challenge this paradigm from the opposite direction, demonstrating that a local operator can exhibit an OSEE growing faster than logarithmically in an integrable setting.
We investigate a discrete-time, U(1) number-conserving free-fermionic quantum circuit built with a brickwall architecture of random two-site Haar unitaries. This model proves to be particularly rich: depending on the specific application of spatio-temporal disorder, the system displays distinct dynamical regimes, ranging from ballistic to localized and diffusive behaviors. By using numerical tools alongside recent analytical techniques for random circuit dynamics, we show that in the presence of full spatio-temporal randomness, the operator entanglement of a Jordan-Wigner string grows diffusively (~√t), despite the underlying free-fermionic nature of the circuit.