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The exponential growth or decay with time of the out-of-time-order commutator(OTOC) is one widely used diagnostic of many-body chaos in spatially-extendedsystems. In studies of many-body classical chaos, it has been noted that onecan define a velocity-dependent Lyapunov exponent, $\lambda({\bf v})$, which isthe growth or decay rate along "rays" at that velocity. We examine the behaviorof $\lambda({\bf v})$ for a variety of many-body systems, both chaotic andintegrable. The so-called light cone for the spreading of operators is definedby $\lambda({\bf \hat n}v_B({\bf \hat n}))=0$, with a generallydirection-dependent "butterfly speed" $v_B({\bf \hat n})$. In spatially localsystems, $\lambda(v)$ is negative outside the light cone where it takes theform $\lambda(v) \sim -(v-v_B)^{\alpha}$ near $v_b$, with the exponent $\alpha$taking on various values over the range of systems we examine. The regimeinside the light cone with positive Lyapunov exponents may only exist forclassical, semi-classical or large-$N$ systems, but not for "fully quantum"chaotic systems with strong short-range interactions and local Hilbert spacedimensions of order one.

Velocity-dependent Lyapunov exponents in many-body quantum, semi-classical, and classical chaos