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

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-extended systems. In studies of many-body classical chaos, it has been noted that one can define a velocity-dependent Lyapunov exponent, $\lambda({\bf v})$, which is the growth or decay rate along "rays" at that velocity. We examine the behavior of $\lambda({\bf v})$ for a variety of many-body systems, both chaotic and integrable. The so-called light cone for the spreading of operators is defined by $\lambda({\bf \hat n}v_B({\bf \hat n}))=0$, with a generally direction-dependent "butterfly speed" $v_B({\bf \hat n})$. In spatially local systems, $\lambda(v)$ is negative outside the light cone where it takes the form $\lambda(v) \sim -(v-v_B)^{\alpha}$ near $v_b$, with the exponent $\alpha$ taking on various values over the range of systems we examine. The regime inside the light cone with positive Lyapunov exponents may only exist for classical, semi-classical or large-$N$ systems, but not for "fully quantum" chaotic systems with strong short-range interactions and local Hilbert space dimensions of order one.