Velocity-dependent Lyapunov exponents in many-body quantum, semiclassical, 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, $\lambda \left(\mathbf{v}\right)$, which is the growth or decay rate along rays at that velocity. We examine the behavior of $\lambda \left(\mathbf{v}\right)$ for a variety of many-body systems, both chaotic and integrable. The so-called light cone for the spreading of operators is defined by $\lambda \left(\widehat{\mathbf{n}}{v}_B\left(\widehat{\mathbf{n}}\right)\right)=0$, with a generally direction-dependent butterfly speed $v_B\left(\widehat{\mathbf{n}}\right)$. In spatially local systems, $\lambda \left(v\right)$ is negative outside the light cone where it takes the form $\lambda \left(v\right)\sim -{(v-v_B)}^{\alpha }$ 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, semiclassical, or large-$N$ systems, but not for “fully quantum” chaotic systems with strong short-range interactions and local Hilbert space dimensions of order one.

Figure 1

Schematic illustration of velocity-dependent Lyapunov exponents $\lambda \left(v\right)$ along rays $\left|x\right|=vt$ in a 1D system. The color scale encodes $\lambda \left(v\right)$, which is positive inside the light cone for $v<v_B$ in classical and semiclassical/large-$N$ quantum systems, but is ill-defined inside the cone in “fully” quantum thermalizing spin systems that do not see an extended period of exponential growth for $v<v_B$. All local systems have negative $\lambda \left(v\right)<0$ for $v>v_B$ following Lieb-Robinson, and $\lambda \left(v\right)$ smoothly interpolates from positive to negative in semiclassical/large $N$ systems, passing through zero at $v_B$. The OTOC $C(x_0,t)$ for a fixed position $x_0$ and increasing time (green arrow) cuts through rays with different $\lambda \left(v\right)$'s and thus will not show a simple-exponential-in-time growth unless $\lambda \left(v\right)$ scales linearly with $v$ (5).

Figure 2

Schematic illustration of $\lambda \left(v\right)$ for the different models considered here. (a) In classical and semiclassical weak-scattering systems, $\lambda \left(v\right)$ smoothly interpolates from positive to negative, passing through zero at $v_B$. (b) In large $N$ holographic models and chains of coupled SYK dots, the OTOC has a simple exponential form $C\sim \epsilon e^{2\pi T(1-v/v_B)t}$ at low $T$ and leading order in $1/N$. For cases (a) and (b), $\lambda \left(v\right)$ scales linearly with $v$ near $v_B$: $\lambda \left(v\right)\sim (v-v_B)$, corresponding to $\alpha =1$ and a simple exponential in time growth for $C(x_0,t)$ near $v_B$. (c) $\lambda \left(v\right)$ in fully quantum thermalizing systems with $d<4$ and in integrable models. In these cases, a negative $\lambda \left(v\right)$ for $v>v_B$ smoothly approaches zero as $v\to v_B+$ with exponent $\alpha >1$ due to broadening of the operator front. As a result, these models do not show a simple exponential in time growth in $C\left(\right|{\mathbf{x}}_{\mathbf{0}}|,t)$ outside the front. (d) In higher dimensions $(d>4)$, the KPZ model capturing the dynamics of operator spreading in chaotic quantum systems has a flat phase which may allow for $\alpha =1$ and a simple exponential growth in $C\left(\right|{\mathbf{x}}_{\mathbf{0}}|,t)$ outside the front. Cases (c) and (d) do not show an extended period of exponential growth inside the light cone due to the absence of additional small parameters $\epsilon$, and hence these do not have a well-defined $\lambda \left(v\right)$ for $v<v_B$.