"Operator hydrodynamics, OTOCs, and entanglement growth in systems without conservation laws"
"The scrambling of quantum information and so-called 'many-body quantum chaos' are the subject of much recent study from the perspective of lattice Hamiltonians (i.e. spin chains), quantum field theory and holography. A quantity of interest, proposed to measure such quantum chaotic effects, is the out-of-time-order correlator (OTOC), which exhibits an analogue of the classical Lyapunov exponent in certain systems, such as large N field theories and the Sachdev-Ye-Kitaev model. Much less is known about the spreading of quantum information in lattice systems with locally bounded Hilbert spaces and local interactions. Here we investigate this question in one-dimensional spin chains evolving under random local unitary circuits and prove a number of exact results on the behavior of OTOCs and the growth of entanglement in this setting. These results follow from the observation that the spreading of operators in random circuits is described by a "hydrodynamical'' equation of motion, despite the fact that random unitary circuits do not have locally conserved quantities (e.g., no conserved energy). In this hydrodynamic picture quantum information travels in a front with a `butterfly velocity' that is smaller than the light cone velocity of the system, while the front itself broadens diffusively in time. The OTOC increases sharply after the arrival of the light cone, but we do not observe a prolonged exponential regime for any fixed Lyapunov exponent. We find that the diffusive broadening of the front also has important consequences for entanglement growth, leading to an entanglement velocity that can be significantly smaller than the butterfly velocity. We conjecture that the hydrodynamical description captures certain universal properties of more generic ergodic systems and support this by verifying numerically that the diffusive broadening of the operator wavefront also holds in a more traditional non-random Floquet spin-chain. We also compare our results to Clifford circuits, which have less rich hydrodynamics and consequently trivial OTOC behavior, but which can nevertheless exhibit linear entanglement growth and thermalization."