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Operator spreading under unitary time evolution has attracted a lot ofattention recently, as a way to probe many-body quantum chaos. While quantitiessuch as out-of-time-ordered correlators (OTOC) do distinguish interacting fromnon-interacting systems, it has remained unclear to what extent they can trulydiagnose chaotic {\it vs} integrable dynamics in many-body quantum systems.Here, we analyze operator spreading in generic 1D many-body quantum systemsusing a combination of matrix product operator (MPO) and analytical techniques,focusing on the operator {\em right-weight}. First, we show that while smallbond dimension MPOs allow one to capture the exponentially-decaying tail of theoperator front, in agreement with earlier results, they lead to significantquantitative and qualitative errors for the actual front -- defined by themaximum of the right-weight. We find that while the operator front broadensdiffusively in both integrable and chaotic interacting spin chains, the preciseshape and scaling of the height of the front in integrable systems is anomalousfor all accessible times. We interpret these results using a quasiparticlepicture. This provides a sharp, though rather subtle signature of many-bodyquantum chaos in the operator front.

Operator front broadening in chaotic and integrable quantum chains